Lesson 19: Paper 1, 2, 3 Exam Strategy

Examination technique converts knowledge into marks, and it is trainable (AA assessment overview). Know the terrain precisely. SL: Paper 1 — 80 marks, 1 hour 30 minutes, *no calculator*; Paper 2 — 80 marks, 1 hour 30 minutes, GDC required. HL: Paper 1 — 110 marks, 2 hours, no calculator; Paper 2 — 110 marks, 2 hours, GDC; Paper 3 — 55 marks, 1 hour, GDC, two compulsory extended-response problems built around a theme, with guided exploration steps. Each paper has Section A (short-response) and Section B (extended-response) — Paper 3 excepted. The IA contributes the remaining 20% of the grade. Grade boundaries vary by session, but the strategy below is boundary-proof: maximise method marks, never leave blanks, spend time in proportion to marks.

Time is a budget

The working rule is about one minute per mark (SL: 90 minutes for 80 marks; HL Paper 3 is the tightest at 60 for 55). Consequences: a 4-mark question owns roughly 4–5 minutes; when it has consumed 10, write your best attempt, mark it with a star, and move — the marks at the start of the next question are cheaper than the last mark of this one. Reserve the final 5–8 minutes for checking: units, three significant figures, whether each question's parts are all attempted. Examiners report the same tragedy every session: blank final questions from strong candidates who over-invested early.

Command terms are instructions, not decoration

• Calculate / Find / Determine — obtain the answer, showing working; "determine" implies justification is expected.
• Show that — the answer is *given*; your job is the derivation. Two gifts: the target tells you if your route is wrong, and if you cannot complete it, you may still use the given result in later parts — the examiners design for this.
• Hence — use the previous result; starting from scratch wastes time and usually misses the point. Hence or otherwise releases you to any method.
• Sketch — shape, key points, intercepts and asymptotes labelled; not a plotted masterpiece, but axes and features must be right.
• Solve — find all values in the stated domain (Lesson 5/6 discipline: families, intervals, rejected roots).
• Prove (HL) — a logical chain (Lessons 12/24), not examples. Justify / Explain — mathematics in sentences: the reasoning mark lives here.

Method marks: how scripts actually score

IB marking allocates M (method), A (accuracy) and R (reasoning) marks. A correct final answer with no working earns its A marks; a *wrong* answer with no working earns nothing at all, while the same wrong answer beneath correct method lines keeps its M marks and often follow-through A marks downstream. Practical rules: write the formula or setup line *before* substituting; keep exact values until the final line (Paper 1 answers are exact — fractions, surds, multiples of π); never round mid-question (store full GDC values); if a multi-part question goes wrong early, continue with your value consistently and say so in writing — follow-through marking rewards consistent use of a stated, even wrong, value.

Worked Example 1 — salvaging a "show that". Part (a) asks you to show that a certain integral equals 32/3, and part (b) builds on that value — but the derivation of (a) will not come. Correct play: write down the method you *would* use for (a) (the setup lines — intersections found, integral written), leave the unfinished step visibly unfinished, then in (b) use 32/3 as given and answer fully. Candidates who abandon the question forfeit (b)'s independent marks; candidates who fake a derivation ("...therefore as required") fool nobody and risk academic-integrity flags. Honest scaffolding scores; theatre does not.

GDC fluency (Papers 2 and 3)

The calculator is a method, and the method must be visible: write the equation you solved or the function you graphed ("solved 2^x = 3x + 1 graphically: intersection at x ≈ …"), because the M mark attaches to the setup. Core fluencies to drill until automatic: graphing with sensible windows; numeric solve and intersections; tables of values; nDeriv and ∫f(x)dx; 1-Var and 2-Var statistics from lists; binompdf/cdf, normalcdf, invNorm; matrix basics (HL systems). Pre-paper ritual: check radian mode, clear lists you will use, fresh batteries/charge, and know your keystrokes for the five functions above without looking them up — fluency bought in practice is time banked in the exam.

Paper 3 (HL): exploration under time pressure

Paper 3's two problems (≈27–28 marks each) introduce unfamiliar mathematics in guided steps — a new definition or context, then questions that build: compute small cases, spot the pattern, generalise, prove, apply. Strategy: the early parts are deliberately accessible — bank them; the generalisation is usually foreshadowed verbatim ("hence prove for general n…"); if a late part resists, mine the earlier results, because Paper 3 is cumulative by design: every part is a tool for a later one. Keep answers structured like mini-explorations: define, compute, conjecture, verify.

Common mistakes (exam technique)

• No working on Paper 2 because "the calculator did it" — the calculator earns nothing; the written setup earns everything.
• Premature rounding compounding through multi-part questions (keep ≥ 5 figures in working).
• Answer-only Section B responses collapsing at the first slip, with no follow-through trail.
• Misreading the domain/interval on solve questions; answering in degrees in a radians course (check the mode line; report trig answers in the unit the question uses).
• Time bankruptcy: 25 minutes on one 6-mark problem while two 10-mark problems sit blank.
• Ignoring "show that" gifts — re-deriving (or worse, doubting) a given result instead of building on it.

IB Exam Experience

Question 1 [2 marks]. A Paper 1 question asks for the exact value of ∫₁³ (2x + 1) dx and a candidate writes 16.0. Explain the two faults. *Notes:* the integral is [x² + x]₁³ = 12 − 2 = 10, so the value is wrong; and "16.0"-style decimal presentation would in any case violate the exact-answer convention — Paper 1 answers are exact (integers, fractions, surds, π terms); decimals appear only when the question demands an approximation, then to three significant figures.

Question 2 [3 marks]. In a "show that" part you are stuck after five minutes. Describe the optimal sequence of actions for parts (a) and the following (b), and why it scores best. *Notes:* write the setup/method for (a) (banking any M marks), leave space, proceed to (b) *using the given result as true*, complete (b) fully; return to (a) only with spare time; never fabricate steps. Full credit for naming follow-through and the given-result rule.

Question 3 [4 marks]. A 12-mark Section B question has parts (a) 2, (b) 4, (c) 6 marks. You have spent 16 minutes and finished (a) and (b) only; 20 minutes remain in the paper and one 8-mark question is untouched. Allocate the time and justify. *Notes:* the untouched 8-marker offers ~8 marks at roughly a minute per mark from a standing start — its early marks are the cheapest in the room; the residual (c) typically has its hardest marks last. Optimal: move to the 8-mark question now (≈10–12 minutes), return to (c) with what's left (≈6–8 minutes) to harvest its setup marks. Any allocation that finishes the paper with no blanks and protects cheap marks earns credit; "finish (c) first" does not.

Extended task (Paper 3 simulation, HL) [10 marks]. A sequence is defined by u₁ = 1 and uₙ₊₁ = 2uₙ + 1. (a) Find u₂, u₃, u₄. [2] (b) Conjecture a formula for uₙ in terms of n. [2] (c) Prove your formula by induction. [4] (d) Hence find the smallest n with uₙ > 10⁶. [2] *Notes (this mirrors Paper 3's compute → conjecture → prove → apply arc):* (a) 3, 7, 15. (b) uₙ = 2ⁿ − 1 — accept the conjecture however spotted (each term is one less than a power of 2). (c) Base n = 1: 2¹ − 1 = 1 ✓; inductive step: assume u_k = 2^k − 1; u_{k+1} = 2(2^k − 1) + 1 = 2^{k+1} − 1 ✓; conclusion stated (M1 base, M1 assumption-to-step algebra, A1 manipulation, A1 conclusion sentence). (d) 2ⁿ > 10⁶ + 1 → n ≥ 20 since 2²⁰ = 1 048 576; n = 20. Paper 3 marking rewards exactly this: patterns noticed, claims proved, results used.

Exploration link: Paper 3 themes (dynamical systems, graph theory, number patterns) overlap IA territory — a Paper 3 topic you enjoyed is a legitimate IA seed, now with personal data or a new context added (Lesson 17).

Going deeper: the last ten minutes and the first ten minutes

First ten minutes of the paper: skim the whole paper once (2 minutes) to locate your strengths and the Section B structures; start with a question you can finish — early completed marks settle nerves and calibrate your pace; write the mode checklist on your rough paper (radians for Papers 2/3, exact-form for Paper 1). Last ten minutes: stop starting new parts; audit instead — every question attempted? units present where the context demands them? answers to three significant figures? "show that" parts used as givens downstream? Check the one calculation per question most likely to carry a sign error by redoing just its first line.

Extra worked example — a pace plan. SL Paper 2, 80 marks, 90 minutes. Plan: reading/skim 3 min; Section A (40 marks) at 1 min/mark ≈ 40 min; Section B (40 marks across typically two long questions) ≈ 42 min; buffer/checking 5 min. Total 90. The plan's point is not the arithmetic but the *decision rules* it encodes: Section A must not bleed into Section B's time; the buffer is sacred; a part stuck beyond twice its mark-value in minutes gets starred and abandoned until the sweep.

Extra exam practice [3 marks]. Mid-paper, you realise your GDC has been in degree mode for two completed trig answers (a period and an amplitude question; the equation-solving parts are unaffected since you worked in exact form). State what is wrong, what is not, and the recovery plan. *Notes:* amplitude unaffected (it is a y-value); period read from a degree-mode graph *is* wrong (the graph's x-axis is in degrees, so the period reads 360 where 2π was meant); recovery: switch to radians, re-graph only the affected part, correct the period; the exact-form equation work stands. Knowing *which* answers a mode error contaminates is the mark-worthy insight.