HookThe spreadsheet formula that helped justify austerity
In 2010 two Harvard economists, Carmen Reinhart and Kenneth Rogoff, published a paper arguing that once a government's debt passes about 90% of its national income, economic growth tends to turn negative. It became one of the most-cited economics papers of its decade, quoted in parliaments and finance ministries to argue for cuts. Then in 2013 a 28-year-old PhD student, Thomas Herndon, asked for their spreadsheet — and found that the formula meant to average the growth figures had been pointed at the wrong block of cells. It stopped five rows short, silently leaving five countries out of the average. Correcting that error, along with other issues, softened the headline result the world had built policy on.
A formula is a machine that turns inputs into an output, and it is only ever as trustworthy as the terms you feed it and the way you write it down. That is what this whole section trains. You will learn what algebraic notation really says (ab, 3y, a², a/b), how to substitute numbers into a formula without tripping over the order of operations, and the precise vocabulary — term, expression, equation, formula, identity, factor — that examiners hand marks for using correctly. Then you will manipulate expressions: collecting like terms, expanding brackets, factorising, and using the index laws. Finally you will rearrange a formula to change its subject, tell an equation from an identity, and read a simple expression as a function with an input and an output. Algebra is not a harder kind of arithmetic — it is arithmetic with the numbers left as letters until the last possible moment.
ModelThe notation — what 3y, a² and a/b actually say
Algebra hides its multiplication signs to save ink, and that is where beginners stumble. ab means a × b; 3y means 3 × y; and a coefficient like the 3 in 3y is simply how many of that letter you have. A fractional coefficient works the same way: x/2 is one-half of x, the same thing as ½x. Division is written as a fraction, so a/b means a ÷ b, and brackets group things that must be dealt with together.
The notation that catches the most students is the power. a² means a × a, not a × 2, and a³ means a × a × a. The little raised number counts how many copies of the base are multiplied, exactly as it did with ordinary numbers in section N1.1 — the letters change nothing about what an index means. So 5² is 25, but 5x² is 5 × x × x, with only the x squared, because the power sits on the x alone unless a bracket says otherwise.
Getting fluent at reading notation is the quiet foundation of everything that follows. If you know at a glance that 4(2x − 3) means 'four lots of the whole bracket' and that 2x² is not the same as (2x)², the expanding, factorising and substituting later in this section stop being guesswork and become bookkeeping.
Read each piece of notation as a full instruction. 3y = 3 × y — three lots of y. a² = a × a. The expression 5x² means 5 × x × x, so if you ever put x = 2 into it you get 5 × 4 = 20, whereas (5x)² would mean 10² = 100 — different machines entirely. And x/2 + 3 means 'halve x, then add 3', so brackets are not needed but the order is: the division binds to the x only.
MechanismSubstitution — feeding numbers into the machine
To substitute is to replace each letter with a given number and then just do the arithmetic — but in the right order. Put the number in with brackets around it, especially if it is negative, and then follow BIDMAS: brackets and indices before multiplication, multiplication before addition. This is the single most reliable way to avoid the classic slip. In 5x² with x = 3, the index acts first: x² = 3² = 9, and only then do you multiply by 5 to get 45. Squaring the whole 5x by mistake would give 225 — a different, wrong answer, and the reason the exam rewards writing the substituted line out in full.
Substitution is the bridge from algebra back to the real world, because scientific formulae are algebra with a job. The physics of motion gives v = u + at (final speed = starting speed + acceleration × time); the area of a rectangle is A = lw; the cost of a taxi might be C = 2.5 + 1.2m. Each is a machine waiting for numbers, and substituting correctly is how you run it. Negative values are where care pays off: a squared negative becomes positive, so (−3)² = 9, while −3² conventionally means −(3²) = −9.
Use v = u + at with u = 3, a = 9.8 and t = 2. Substitute with the values in place: v = 3 + 9.8 × 2. Multiplication before addition: 9.8 × 2 = 19.6, then 3 + 19.6 = 22.6. Now the notation trap head-on: evaluate 5x² when x = 3. The power acts first — 3² = 9 — then multiply: 5 × 9 = 45. It is not (5 × 3)² = 225; only the x is squared. Writing the line '5 × 3² = 5 × 9 = 45' shows the examiner exactly which operation you did first.
ModelThe vocabulary examiners actually mark
AQA gives marks for using these words precisely, and confusing them is a genuine, avoidable loss. A term is a single number-and-letter chunk separated by + or − signs, like 3x or −5 or 2ab. An expression is terms strung together with no equals sign, like 3x + 5 — it can be evaluated but not solved. An equation has an equals sign and can be solved for an unknown, like 3x + 5 = 20 (true only for x = 5). A formula is a rule connecting two or more quantities, like A = lw, that you can substitute into. An inequality uses <, >, ≤ or ≥ instead of =. A coefficient is the number multiplying a letter (the 3 in 3x), and a factor is something that divides exactly into an expression.
The subtle one is the identity, marked with the ≡ sign: a statement true for every value of the letter, not just one special value. 3x + 5 = 20 is an equation because it only works when x = 5; but 2(x + 3) ≡ 2x + 6 is an identity, because expanding the bracket makes the two sides genuinely the same expression, true whatever x is. Telling an equation from an identity — the heart of leaf A6 — comes down to that question: true for one value, or true for all of them? (Formal algebraic proofs of identities are Higher-tier; at Foundation you argue by expanding or simplifying to show the two sides match.)
Classify each statement. 4x − 7 is an expression — no equals sign, nothing to solve. 4x − 7 = 5 is an equation, true only for x = 3. P = 2(l + w) is a formula — a rule for the perimeter of a rectangle you substitute into. And 4(x − 2) ≡ 4x − 8 is an identity: expand the left side and it becomes the right side, so it holds for every value of x. In the expression 4x − 7, the 4 is the coefficient and −7 is a constant term.
MechanismSimplifying, expanding and factorising
Four manipulation skills sit at the core of this section, and each undoes another. Collecting like terms tidies an expression by adding terms of the same type: 3a + 5b − a + 2b = 2a + 7b, because you can only combine 'a's with 'a's and 'b's with 'b's — 3a and 2b stay apart, exactly as three apples and two oranges never merge into five of anything. Expanding a single bracket multiplies every term inside by the term outside: 4(2x − 3) = 8x − 12. The mirror-image skill, factorising, pulls the highest common factor back out: 8x − 12 = 4(2x − 3), and 4x² + 6x = 2x(2x + 3).
The index laws carry straight over from numbers to letters: multiplying powers of the same base adds the indices (y³ × y⁴ = y⁷), dividing subtracts them (p⁶ ÷ p² = p⁴), and a power of a power multiplies them ((m²)³ = m⁶). They work because the powers are just shorthand for repeated multiplication — y³ × y⁴ is seven y's multiplied together.
The headline Foundation skills are the two bracket operations that reverse each other. Expanding two binomials (double brackets) multiplies each term in the first bracket by each in the second: (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15. Factorising x² + bx + c runs it backwards — find two numbers that multiply to c and add to b. The special case worth memorising is the difference of two squares: x² − 9 = (x − 3)(x + 3), because the two middle terms cancel. (Factorising ax² + bx + c where a is not 1, and algebraic fractions, are Higher-tier only.)
Expand and simplify (x + 3)(x + 5). Multiply each pair: x × x = x², x × 5 = 5x, 3 × x = 3x, 3 × 5 = 15. Collect the middle: 5x + 3x = 8x, giving x² + 8x + 15. Now factorise it back — find two numbers multiplying to 15 and adding to 8: that is 3 and 5, so x² + 8x + 15 = (x + 3)(x + 5), exactly where we started. And the difference of two squares: x² − 9 = (x − 3)(x + 3), which you can check by expanding — the −3x and +3x cancel, leaving x² − 9.
MechanismChanging the subject of a formula
Standard formulae — for area, perimeter, speed, the cost of a phone plan — are the reason algebra earns its keep, and rearranging one to change its subject is leaf A5. The subject is the letter sitting alone on one side; to make a different letter the subject you undo whatever is happening to it, using inverse operations, doing the same thing to both sides so the balance is never broken. It is the same move you will use to solve equations, done with letters instead of numbers.
Work outwards, undoing the operations in reverse order. To make x the subject of y = 3x + 4, first subtract 4 from both sides (y − 4 = 3x), then divide both sides by 3 (x = (y − 4)/3). The order matters: you peel off the +4 before the ×3, because when you built the expression you multiplied first and added second, so unwinding runs backwards. Keep each step on its own line — the mark scheme follows your inverse operations one at a time, and a clear layout banks method marks even if the final line has a slip.
Make r the subject of the circle formula C = 2πr. The r is multiplied by 2π, so divide both sides by 2π: r = C/(2π). One step, because only one operation was applied to r. A two-step example: make x the subject of y = 3x + 4. Subtract 4 from both sides: y − 4 = 3x. Divide both sides by 3: x = (y − 4)/3. Undo addition before multiplication — the reverse of the order that built the right-hand side.
CaseEquation, identity, and the think-of-a-number trick
Here is a trick you may have met at primary school. Think of a number. Double it. Add ten. Halve the result. Now subtract the number you first thought of. Whatever you started with, you always end on 5. It feels like magic, but it is an identity — a statement true for every possible starting number — and algebra is how you prove it, which is exactly the reasoning leaf A6 asks for.
Let the starting number be n. Double it: 2n. Add ten: 2n + 10. Halve it: (2n + 10)/2 = n + 5. Subtract the original n: (n + 5) − n = 5. The n's cancel completely, leaving the constant 5 no matter what n was — that cancellation is the proof, and it is why the answer is always 5 and never anything else. This is the difference between an equation and an identity made concrete: an equation like 2n + 10 = 16 pins n to a single value (n = 3), while the trick works for all of them.
The same argument shows two expressions are equivalent — the everyday version of A6. To show (x + 2)(x + 3) ≡ x² + 5x + 6, you expand the brackets and watch the left side turn into the right side, which is a Foundation-level argument for equivalence. (Dressing the same idea up as a formal, generalised proof is Higher-tier; here you argue by expanding or simplifying until both sides visibly match.)
Prove the think-of-a-number trick always gives 5. Start with n. Double: 2n. Add ten: 2n + 10. Halve: (2n + 10)/2 = n + 5. Subtract the original: (n + 5) − n = 5. The variable disappears, so the result is 5 for every starting number — that is what makes it an identity, not an equation. Check with n = 7: double to 14, add ten to 24, halve to 12, subtract 7 → 5. It works because the algebra says it must.
ModelExpressions as functions — input and output
A function is a rule that takes an input, does something to it, and returns an output — a machine, drawn in GCSE as a function machine with the operations in boxes. The expression 3x + 2 is such a rule: it says 'multiply the input by 3, then add 2'. Feed in 4 and it multiplies to 12, adds 2, and returns 14. Substitution, from earlier in this section, is simply running a function forwards, which is why the two ideas belong together.
The machine also runs backwards, and this is where Foundation students can shine without any Higher-tier machinery. If the output is 20, retrace the steps in reverse, undoing each operation: subtract the 2 to get 18, then divide by 3 to get 6 — so the input was 6. That is the same inverse-operation logic that changed the subject of a formula, applied to a chain of boxes. (Writing the inverse formally as a new function, and combining two functions into a composite, are both Higher-tier — at Foundation you work the machine forwards and backwards.)
Functions matter because they are how mathematics models cause and effect: put in the number of hours, get out the pay; put in a temperature in Celsius, get out Fahrenheit. Reading an ordinary expression as a function — input here, output there — is the mental habit that later makes graphs, sequences and real-world modelling feel like the same idea wearing different clothes.
Take the function machine 'input → ×3 → +2 → output', the expression 3x + 2. Forwards: an input of 4 gives 3 × 4 + 2 = 14. Backwards: if the output is 20, undo the machine in reverse — subtract 2 to get 18, then divide by 3 to get 6, so the input was 6. Check forwards: 3 × 6 + 2 = 20. The forward run is substitution; the backward run is the inverse-operation reasoning you used to change the subject of a formula.
VocabularyKey terms the mark scheme pays for
TrapsMisconceptions that cost marks
ExamWhat examiners want
Algebra runs across all three papers, much of it on the non-calculator Paper 1, and almost all of it is marked method-first — so show the working the mark scheme is hunting for. When you expand double brackets, write all four products before collecting, so both middle terms are visible even when they cancel (as in the difference of two squares). When you factorise x² + bx + c, the pair of numbers that multiply to c and add to b is itself worth a mark, so note them down rather than leaping to the answer.
For substitution, write the line with the number physically in place of the letter and brackets round any negative, then apply BIDMAS — the index before the multiply. Examiners report that 5x²-type questions are where careless candidates square the coefficient by accident, so the visible ordered line ('5 × 3² = 5 × 9 = 45') is what protects the mark. When changing the subject, undo one operation per line using inverse operations, doing the same to both sides.
Use the vocabulary precisely, because AQA sets questions that hand marks simply for knowing an identity from an equation or a term from an expression, and use ≡ when a relationship holds for all values. Above all, watch the negative sign when a minus or a subtraction sits outside a bracket — −(x − 4) = −x + 4 — because that single sign flip is one of the most reliable discriminators between a Grade 3 and a Grade 5 script on this topic.