Technique Masterclasses

Pure Mathematics 2

Vocabulary Precision List

Provenance: written 2026-09-29, against research/veridian/WMA12-verified-facts.md (read in full, then re-searched by keyword for every entry below — every citation traces to a line in that file). Cross-checked for duplication against the existing technique-hub documents for this paper — forge-battle-card.md, forge-exam-technique-guide.md, signal-frequency-and-boundaries.md, forge-forty-eight-hour-checklist.md, codex-formula-and-method-bank.md, and the ten codex-deep-dive-*.md files (all read or grep-checked in full) — plus the relevant lesson files under content/veridian/curriculum/wma12/ and content/veridian/question-bank/wma12.ts. Where a distinction below already appears elsewhere in this hub as a worked example or checklist item (the "hence factorise" trap, the "list vs product" answer form, the discriminant "prove no turning points" question), this document does not re-derive the worked mechanics — it isolates the single word or phrase at stake and states the definitional distinction on its own, which no existing document in this hub does as a standalone reference. Two candidate entries were dropped rather than padded in: a "verify" vs "show that" contrast (no real WMA12 mark scheme or examiner report in the facts bank uses "verify" as a distinct command word) and a "hence, or otherwise" contrast (that exact phrase does not appear anywhere in the facts bank for this paper) — both would have been invented pairs, which the brief for this document rules out.

WMA12 — Vocabulary Precision List

Pure Mathematics 2 (Pearson Edexcel IAL) — command words and terms the mark schemes are precise about

What this is, and isn't

This is a list of specific words — command words in question stems, and ordinary words the mark scheme uses more narrowly than everyday English — where the exact meaning is what a real mark scheme pays for, and a plausible-sounding blur of that meaning is what a real examiner report records candidates losing marks over. It is not a glossary of mark-scheme abbreviations in general (that already exists, compressed to one line, in forge-battle-card.md's "Tariff & mark-code reminders" section) — the handful of abbreviations covered here (cao, cso, ag, isw, oe) are covered because a specific pair of them is easy to conflate in a way the battle card's one-line list doesn't have room to unpack. Every entry below is grounded in a real, named series and a real quoted line from a mark scheme or examiner report. Where the facts bank does not evidence a pairing that would otherwise fit this document's shape, that pairing is left out rather than invented.

A. Command words that decide whether a method counts at all

"Prove" over an infinite domain vs. "prove by exhaustion" vs. "show that... is false"

These are three different techniques triggered by three different phrasings, and treating any two as interchangeable is a real, documented failure mode. A bounded claim ("for these three positive integers...") is proved by exhaustion — list every case. An unbounded or continuous claim ("for all real values of x...") that is true is proved by general algebraic manipulation or contradiction — exhaustion is not available because the domain cannot be listed. A claim that is false, on any domain, is disproved by one counter-example with a stated reason. The costliest real confusion runs the other way: candidates reach for exhaustion-style numerical checking on an unbounded true claim. October 2021's Q9(a) — "prove... for all positive values of x and y" — drew this examiner-report line: "several numerical examples of the inequality holding constituted a proof." That question scored 0/4 for 47% of candidates, the worst-performing question on that paper. Separately, attempting an algebraic proof where exhaustion was actually wanted costs time rather than the mark itself — confirmed independently in two series' reports on exhaustion questions (October 2022, October 2023) as "unnecessarily long-winded" and "more demanding than... a numerical one."

"Show that" / "Hence [find/factorise/solve]" as an answer-given instruction

When the target result is already printed on the paper, the mark is for the route, not the number — this is what the mark-scheme code cso/ag means in practice (see section D). A real report records marks lost specifically for "not showing the = 0 or intermediate lines" on a "show that" part (October 2023, factor theorem). Separately, on a circle question where the equation still contained an unresolved constant, the entire final accuracy mark was for the algebra between substitution and the printed line, not for confirming the point satisfied the equation (January 2021, Q9(b)(i)) — a subtly different use of "show that" from confirming a fact already established, which is worth noticing precisely because the surface instruction reads identically in both cases.

"Hence" — the specific cost changes by topic, but the rule doesn't

"Hence" names a required input, and reaching the right answer by any other route scores zero on that part regardless of correctness — a mark-scheme-mandated instruction, not a style preference. The clearest statement of this is the factor-theorem case: "there is an expectation that they use the factor... which is given," with a fully correct factorisation reached by calculator-solving the cubic directly scoring "M0 and therefore A0 A0" (October 2023, Q4(c)(ii)). The same word appears in a structurally different guise in trigonometric equations, where "hence solve" after a compound- angle substitution requires the solution range to be re-scaled by the substitution's own coefficient, not left at the original range — confirmed independently across three series (June 2025, October 2022, October 2023). Same word, same underlying principle (use the thing "hence" points at), two different concrete failure shapes depending on what "hence" is attached to.

"Justify... by further calculus" vs. "deduce"

"Justify," attached to the nature of a stationary point, names the method as part of the mark: the real mark scheme for this exact phrasing states plainly that a correct conclusion reached by reasoning about the shape of a cubic "scores no marks" — only a second-derivative or sign-either- side calculus check is credited (January 2021, Q2(b)). "Deduce," by contrast, is used on a lighter, more computational step — "deduce the range of values of x for which y is decreasing" — where the mark scheme explicitly accepts either of two equivalent presentations of the critical values, "with consistent inequality use," as sufficient (June 2025, Q8(a)). The two words sit at different points on the same scale: "justify" demands the reasoning chain be calculus all the way through; "deduce" accepts a correctly-drawn conclusion from an already-established result.

"Write down" / "state" vs. "find"

"Write down" and "state" mark a result that should require no working — and real candidates treat them that way successfully: "the majority... achieved two marks for correctly writing down the centre coordinate" on a circle whose centre-radius form was given directly (January 2021, Q9(a)). The mark loss on that same question was not at this step at all — it came three parts later, at a "find" step that required substituting a specific numeric value back into the earlier symbolic answer, which several candidates skipped or mishandled. The distinction matters for how a candidate should spend time: a "write down" step is not a checkpoint to labour over; a "find" step built on it is.

B. Words that fix the exact shape of the final answer

"Exact" vs. a correctly-rounded decimal

The general marking guidance states the rule directly: "marks will normally be lost if the candidate resorts to using rounded decimals." This is not a one-off penalty; it recurs by name across stationary-point x-values, circle radii left as decimals instead of surd form, and — the specific, verified trap — radian answers requested "in the form kπ." On June 2025's Q8(i), the examiner report records candidates who "did not take note of the required form... kπ" and used decimals instead, with the mark scheme's own accuracy line reading "must be exact and in radians." A decimal that rounds correctly to the exact value is not a partial match for this mark — it is the wrong object.

"Giving your answer in the form x = log_a k, a rational" — combined vs. merely correct

"In the form" narrows "any mathematically correct exact expression" down to one specific shape, and the two are scored as genuinely separate accuracy marks on the real anchor for this: "any correct exact expression for x, such as... log₇3 − 2" earns one A1, while the fully-combined single logarithm "log₇(3/49)" earns a second, distinct "correct answer only" A1 (January 2021, Q3(i)). An unsimplified exact answer is not almost the required form — it is a different, separately-scored answer that happens to sit one step before the one actually asked for.

"Factorise" vs. "fully factorise" / "factorise completely"

A cubic factored into a linear factor and a quadratic that itself factorises further is not "fully factorised" even though every step shown is correct — a real report records marks lost for "leaving the answer... as (2x+1)(2x² + x − 15)," a quadratic that itself reduces to (2x−5)(x+3) (October 2023, Q4). The word "completely" or "fully" is not emphasis; it is a separate condition that isn't satisfied while any factor shown still factorises.

"A product of factors" vs. "a list of factors" — not the same object

A factorised cubic answer must be written "as a product of factors (not as a list)" to earn the mark (January 2020) — three correctly-identified factors written with commas between them, rather than multiplied together as (x−1)(x−2)(x−3)(x-1)(x-2)(x-3), is not treated as an equivalent presentation of the same information. The question asked for the factorised expression itself, not the set of values that make it zero, and a mark scheme that credits isw (ignore subsequent working) once the product is correctly written down will not retroactively credit a list that never became a product.

A mathematically equivalent factor in the wrong coefficient form

(x−7/2)(x - 7/2) and (2x−7)(2x - 7) have the same root, but a mark scheme requiring integer coefficients marks only the second as correct: "This is incorrect as integer values were required" (June 2025 Regional, Q8). When a root comes out as a fraction b/a, the factor is written (ax−b)(ax - b), not (x−b/a)(x - b/a) — a genuinely non-obvious case of "mathematically equivalent" not meaning "equally acceptable."

C. Ordinary words the mark scheme uses more narrowly than everyday English

"Product" vs. "sum"

On a proof-by-exhaustion question asking candidates to show abc is always a multiple of a given number, one examiner report states plainly: "the definition of 'product' is not widely understood, with many students considering the sum" (October 2022, Q1). This is not an arithmetic slip inside a correct method — it is evaluating the wrong operation throughout an otherwise well-organised table, because the word itself was read as its more familiar neighbour.

"Coefficient" vs. "term"

On a binomial-expansion question asking for "the coefficient of the term in x⁷," one report notes: "a high proportion of candidates did not understand the word coefficient," writing "96228x⁷" rather than the bare number "96228" the question asked for (January 2024, Q2). This entry carries an honest caveat the source itself states: this specific misreading was flagged by examiners but not used as a scored-mark citation on that series — it is recorded here as a genuinely confirmed misunderstanding of the term, not as proof it costs a mark on every occasion it occurs.

"...is a multiple of k" — needs the quotient shown, not just asserted

Stating that a computed value "can be divided by 4" is not, on its own, sufficient evidence that it is a multiple of 4 — the real mark scheme requires "further clarification such as 'to give whole numbers'" (October 2023, Q1). The word "multiple" names a claim about an integer quotient existing; asserting divisibility without producing or naming that quotient states the conclusion without the one piece of evidence the mark scheme is actually checking for.

D. A stated reason vs. a restated conclusion

The single hardest mark documented anywhere in this course's WMA12 research is a reasoning mark, not a computational one: on the trapezium-rule over/underestimate question, one real report states that saying the answer is an underestimate "because it is less than the true area is not giving a reason" (October 2023). The credited reason has to point at the geometric cause — that the sum of trapezia areas found is itself less than the shaded area they approximate — not restate the comparison already given in the question. A sentence that repeats "it's smaller so it's an underestimate" contains the conclusion twice and the reason never.

E. Three mark-scheme codes that look interchangeable and aren't

cao vs. cso vs. ag. All three gate a mark on correctness, but at different scopes. cao ("correct answer only") gates the final number — stated in the mark scheme's own general guidance as the default for every A mark unless flagged otherwise. cso ("correct solution only") gates the entire part: "there must be no errors in this part of the question" to earn it — a strictly wider requirement than cao, since a solution can arrive at the right final number while still containing an earlier, uncorrected error that cso would catch and cao alone would not. ag ("answer given") marks are, in practice, cso marks on a target that happens to already be printed on the paper — the circle-lesson audit for this paper states this relationship directly: an ag mark "is what 'cso' means in practice." The three terms describe one increasingly strict idea, not three unrelated ones.

isw vs. oe. Both look like leniency codes, but they apply at different moments. oe ("or equivalent") is checked at the answer line — a correct answer can be written more than one way and still earn the mark, verified in a real scheme on a logarithm-law step ("states or uses log₂4x² = 2log₂2x (oe)," January 2020) and an exact-surd answer ("a = ±2 must be exact. Accept ±⁴√4 oe," January 2020). isw ("ignore subsequent working") is checked after the answer line — it protects a mark already correctly earned from being undone by something written afterward, as in a factorised-cubic answer where the mark scheme notes the correct product form "and isw" (January 2020). Confusing the two the wrong way round matters in exactly one direction: oe cannot rescue a wrong final answer no matter what came before it, and isw cannot excuse an answer that was never correctly reached in the first place.

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