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The STEP Method Library

STEP does not reward knowing more mathematics. It rewards recognising which shape of argument a question wants, and then writing that argument so someone else can follow it. These are the shapes.

How this STEP material was checked

What this is: a taxonomy of the argument shapes a long-form mathematics paper asks for, with what triggers each one, the move itself, the trap it punishes, and what the marks are actually for.
Who wrote it: the argument shapes are standard, the same ones any STEP guide would name; the wording and the choice of what to emphasise are ours.
How to use it: when a question defeats you, do not look up the answer. Work out which family it belongs to, read the entry, and go back to the question. The recognition is the skill being tested.
Practice to go with it: our STEP mock papers are being rewritten, so no entry links to a question yet. Work the entries alongside the official STEP papers from past years, which remain the final word on the exam itself, and our ESAT library for the pure content the two share.
How we check our material in general is on how we work.
Found an error? Tell us at hello@mintnib.com and it will be fixed or withdrawn.

Before the families: the habit that earns the most marks

Most candidates lose more marks to abandoning questions than to being unable to do them, and more to unjustified steps than to wrong answers. Three habits are worth more than any single technique below.

Read every part before you start. The last part usually tells you what the first part was for. A question that looks impossible at part three is normally reachable through what part one handed you.
Say what you are doing and why. An examiner reads the reasoning. Naming the move, for example "we prove the contrapositive" or "by the inductive hypothesis applied to the first n terms", costs one line and secures the method marks.
Finish fewer questions, properly. A complete, well-argued answer beats fragments. A partial attempt that states clearly what has been proved and what remains scores better than most candidates expect.

Structure

Reading the ladder

The parts of a STEP question are rungs. An early part almost always exists to make a later one reachable, and the last part usually tells you what the first part was for.

When it fires. Any multi-part question, and especially one where a later part looks impossible.
The move. Read every part before starting. Then, for each later part, ask which earlier result it is meant to consume: the phrase 'hence' is an instruction, and 'deduce' means the work is already done somewhere above.
The trap. Treating the parts as separate questions and re-deriving from scratch what part (i) handed you. It costs time and usually fails, because the general case is genuinely harder without the concrete one.
What the marks are for. Examiners expect the earlier result to be cited by name. A later part that silently re-proves an earlier one is not wrong, but it is slower and it loses the method marks attached to the intended route.

Symmetry and without loss of generality

Using a symmetry of the problem to halve the work, and saying why the reduction is legitimate.

When it fires. An expression unchanged by swapping two variables, a geometric configuration with a reflection, or a sum over an index set that maps to itself.
The move. Name the symmetry, state the assumption it licenses, for example $x \geq y$ , and confirm that the other case follows by relabelling. Pairing a sum with its own reversal is the same idea in miniature.
The trap. A 'without loss of generality' that is not: assuming an ordering that the expression is not actually symmetric under is a false step that a marker will find.
What the marks are for. The justification is the mark. 'WLOG $a \geq b$ since the expression is unchanged by swapping $a$ and $b$ ' scores; a bare 'WLOG' does not.

Invariants and monovariants

A quantity that never changes under the allowed moves, or one that only ever moves one way.

When it fires. A process, a game, a sequence of operations, or a question asking whether one configuration can be reached from another.
The move. Compute the candidate quantity before and after a single move. If it is unchanged and differs between the two configurations, the answer is no, and that is the whole proof.
The trap. Finding an invariant that fails to distinguish the two configurations, and concluding nothing while believing you have concluded something.
What the marks are for. Marks are for showing the quantity is invariant under every allowed move, not just the typical one, and then for the separate step of evaluating it at both ends.

Proof frames

Induction

Proving a statement for every integer beyond a base by showing it propagates. Includes strong induction, two-step induction, and induction on something other than the obvious variable.

When it fires. A statement indexed by $n$ , a recurrence, a product or sum with $n$ terms, or a construction that adds one object at a time.
The move. State the proposition $S (n)$ precisely, prove the base case, assume $S (k)$ and derive $S (k + 1)$ , and say which step used the hypothesis. When the step needs more than the previous case, use strong induction and say so.
The trap. Assuming what is to be proved by writing the inductive step as a chain of equalities that starts from $S (k + 1)$ and ends at something true, without saying the steps are reversible.
What the marks are for. Marks are for the frame as much as the algebra: a stated proposition, a checked base case, an inductive step that visibly uses the hypothesis, and a closing sentence. Dropping the base case is a common and expensive omission.

Strong induction and descent

Assuming every case below $k$ , not just $k - 1$ ; and its mirror image, infinite descent, where a counterexample is used to build a smaller counterexample.

When it fires. A recurrence that reaches back more than one step, a factorisation argument, or a minimal-counterexample setup.
The move. For strong induction, assume $S (m)$ for all $m < k$ and derive $S (k)$ . For descent, take a counterexample minimal in some positive integer measure and construct a strictly smaller one, which is the contradiction.
The trap. Descent arguments that produce a smaller object without checking it is still a counterexample, or that fail to say what quantity is decreasing.
What the marks are for. Naming the measure that decreases, and confirming it is a positive integer so it cannot decrease forever, is a creditable step in its own right.

Contradiction

Assume the negation, derive something false, conclude the original.

When it fires. A statement of the form 'no such object exists', an irrationality claim, or a uniqueness claim.
The move. Negate the statement carefully, including its quantifiers, then work forwards to a contradiction with a stated fact. Say at the end exactly which assumption fails.
The trap. Negating only part of the statement. The negation of 'for all $x$ there is a $y$ with $P$ ' is 'there is an $x$ such that for all $y$ , not $P$ ', and getting that wrong makes the whole argument prove nothing.
What the marks are for. The negation itself carries marks. So does identifying the contradiction precisely rather than waving at 'a contradiction'.

Contrapositive and converse

Proving 'if $P$ then $Q$ ' by proving 'if not $Q$ then not $P$ ', and keeping the converse separate from the statement.

When it fires. An implication whose hypothesis is awkward to use directly, or an 'if and only if' where the two directions want different arguments.
The move. Write both directions explicitly and prove them separately. When one direction is hard, try its contrapositive: the negation of the conclusion is often a concrete object you can work with.
The trap. Proving the converse and thinking the job is done. An 'if and only if' is two theorems, and a proof of one of them scores half.
What the marks are for. STEP mark schemes almost always split an 'if and only if' into two blocks. A candidate who proves one direction well gets those marks in full and none of the others, which is worth knowing when time is short.

Inequalities

Bounding and estimation

Replacing a quantity you cannot compute by one you can, in a direction that still settles the question.

When it fires. An inequality to prove, a sum or integral with no closed form, or a question asking for the least or greatest value.
The move. Bound term by term, and check the direction of every replacement. For a sum, compare with an integral or with a geometric series; for an integral, bound the integrand on the range.
The trap. A bound that is true but too weak to conclude, or bounding in the wrong direction halfway through. Both produce a page of correct work that proves nothing.
What the marks are for. The bound has to be justified where it is applied, not asserted. Marks are for the inequality chain being valid at every link, and for the final step actually reaching the target.

Classical inequalities and equality cases

AM-GM, Cauchy-Schwarz in Engel form, rearrangement, and the trick of reducing an inequality to a square being non-negative.

When it fires. A symmetric expression to be bounded, a sum of fractions, or an inequality that becomes an identity at a single point.
The move. Clear denominators by multiplying by a quantity you have said is positive, and aim to land on a sum of squares. Then read the equality case straight off the squares.
The trap. Multiplying by a quantity whose sign is unknown, which reverses the inequality silently. And forgetting that the equality case is usually separately marked.
What the marks are for. Establishing when equality holds, and proving it holds only there, is typically worth as much as the inequality itself.

Machinery

Change of variable

Choosing a substitution that makes a symmetry visible or removes a denominator.

When it fires. An integral with a symmetric range, an expression in $x + 1 / x$ , a homogeneous equation, or a recurrence in disguise.
The move. Pick the substitution that the structure suggests, for example $u = a + b - x$ on the range $[a, b]$ , or $t = \tan (x / 2)$ , and carry the limits and the differential with it.
The trap. Substituting into the integrand and forgetting the limits, or using a substitution that is not one-to-one on the range without splitting the range first.
What the marks are for. The transformed integral, with correct limits, is usually a marked intermediate result. Say why the substitution is valid when the map is not obviously invertible.

Telescoping and partial fractions

Writing a general term as a difference so that a sum collapses to its ends.

When it fires. A sum whose terms are rational in the index, or a product of consecutive integers in a denominator.
The move. Split the term into $f (k) - f (k + 1)$ , then write out the first two and last two terms to show the cancellation rather than asserting it.
The trap. Assuming the cancellation without checking the ends, which is where an off-by-one error lives.
What the marks are for. The identity for the general term is one mark block, the cancellation another, and the surviving terms a third. Writing the sum out is the cheapest way to secure the middle one.

Recurrences and reduction formulae

Relating the $n$ th case to earlier cases, then either iterating or solving.

When it fires. An integral $I_{n}$ depending on an integer parameter, a counting problem built one step at a time, or a sequence defined implicitly.
The move. For integrals, integrate by parts choosing the factor that reduces the index. Establish the recurrence, state the initial value, and only then iterate. Keep the parity cases separate when the recurrence steps by two.
The trap. Iterating a recurrence without a base case, or merging the odd and even chains when they do not meet.
What the marks are for. The recurrence, the base case and the closed form are three separate results and are marked separately. A correct closed form with no stated base case loses marks it did not need to.

Generating functions

Encoding a sequence as the coefficients of a series and manipulating the series instead of the sequence.

When it fires. A counting problem with a convolution structure, a recurrence with constant coefficients, or a probability question about a sum of independent variables.
The move. Write the generating function, use the recurrence to get a closed form, and extract the coefficient. For probability, use the probability generating function and read the mean off the derivative at 1.
The trap. Ignoring convergence when the question needs it, and shifting an index without shifting the powers to match.
What the marks are for. Setting up the generating function correctly is a marked step by itself. So is justifying the coefficient extraction rather than pattern-matching it.

Complex numbers and roots of unity

Using de Moivre, roots of unity and the algebra of $e^{i θ}$ to turn a trigonometric identity into a polynomial one.

When it fires. A sum of cosines in arithmetic progression, a power of a trigonometric function, or a polynomial whose roots are evenly spaced.
The move. Write the trigonometric sum as the real part of a geometric series, or write $2 \cos θ = z + z^{- 1}$ and expand. Roots of unity sum to zero, which is often the entire point.
The trap. Taking real parts too early, before the geometric series has been summed, which loses the structure that made the method work.
What the marks are for. The identification of the sum as a real part, and the correct handling of the case where the common ratio is 1, are both separately creditable.

Analysis

Calculus as an argument

Using derivatives to prove something rather than to compute something: monotonicity, an auxiliary function, a stationary point that has to be classified.

When it fires. An inequality between two functions, a claim about the number of roots, or a maximum that has to be shown to be a maximum.
The move. Build the auxiliary function $h = f - g$ , evaluate it at an endpoint, and use the sign of $h^{'}$ to control it on the whole interval. Say what happens at the ends of the range.
The trap. Concluding a global maximum from a single stationary point without checking the ends of the interval or the behaviour as $x \to \infty$ .
What the marks are for. The construction of the auxiliary function is the insight and is marked as such. The classification of stationary points and the endpoint check are separate blocks.

Number theory

Modular arithmetic and divisibility

Working with remainders to prove that something cannot happen, or to pin down the residues that can.

When it fires. An equation in integers, a claim that no solutions exist, or a statement about the last digit or about squares.
The move. Choose the modulus from the squares or powers involved, for example 4 or 8 for squares and 9 for digit sums, then tabulate the possible residues and eliminate.
The trap. Tabulating only some of the residues, so that one case is never ruled out, and treating a congruence as an equation by dividing through by something not coprime to the modulus.
What the marks are for. The choice of modulus and the residue table are both marked. A conclusion that a case is impossible must say which residue rules it out.

Discrete

Counting by bijection and conditioning

Counting a set by pairing it with one you can already count, or by splitting on a first move.

When it fires. A combinatorial identity, a probability with a natural first step, or a count that is hard forwards and easy backwards.
The move. Describe the map explicitly and show it is one-to-one and onto, or condition on the first event and sum the cases with their probabilities.
The trap. A map that is described but never shown to be a bijection, and a conditioning that double counts because the cases overlap.
What the marks are for. Marks are for the map being well defined and invertible, or for the case split being exhaustive and disjoint. The arithmetic afterwards is usually the cheap part.

Mechanics

Setting up a mechanics model

Turning a described physical situation into equations, and choosing between forces, energy and momentum.

When it fires. Any mechanics question. The modelling decision usually matters more than the algebra that follows.
The move. Draw the diagram, choose axes, and decide early: forces and acceleration for an instantaneous question, energy for a start-to-finish question with no friction, momentum when there is an impulse or a collision.
The trap. Using energy conservation through an impact, or resolving along axes that make the constraint awkward when a different pair makes it trivial.
What the marks are for. The equation of motion or the energy equation is the marked result, with separate marks for the correct resolution of forces. A stated modelling assumption, for example that a string is light and inextensible, is creditable where the question turns on it.

Probability

Probability structure

Choosing the right description of a random experiment: a partition to condition on, an indicator to sum, or a distribution to recognise.

When it fires. A probability or expectation that is hard to compute directly, or one indexed by an integer parameter.
The move. Condition on a well-chosen first event, or write the count as a sum of indicators and use linearity of expectation, which does not need independence.
The trap. Assuming independence to multiply probabilities when the events are only conditionally independent, and using linearity of expectation for a variance.
What the marks are for. Naming the partition and showing it is exhaustive is marked. So is the statement that linearity of expectation needs no independence, when the question is built on that fact.