IB DP Mathematics: Analysis and Approaches – Paper-by-Paper Exam Preparation

The assessment revision notes set out what each paper weighs, times and tests. These notes turn that into an exam-day plan – building genuine by-hand fluency for Paper 1, showing full working on “show that” questions, and what HL Paper 3 specifically demands – with a worked example and a checklist, alongside the full syllabus guide already on the site.

Paper 1 (40% SL / 30% HL, no calculator): build genuine by-hand fluency

Because Paper 1 tests algebraic manipulation and proof entirely without a calculator or other technology, the single most valuable exam preparation is revising without reaching for a calculator, even for arithmetic that would normally be checked electronically. A student who only ever checks algebra with a calculator during revision often finds Paper 1 significantly harder than expected on exam day – not because the mathematics is unfamiliar, but because manual verification was never practised as a habit. Build this habit deliberately in the weeks before the exam, not on exam day itself.

Command terms and what full working actually means

Calculate, find and solve need a direct computational or algebraic answer. Hence or hence or otherwise require building explicitly on a result from an earlier part of the same question. Show that requires a fully justified derivation, with every algebraic step shown, reaching the given result – stating the correct final answer without the intermediate steps loses marks even when that answer is right, since the mark scheme awards credit for the steps themselves. Prove, tested most often at HL, requires a rigorous, logically complete argument.

Paper 2 (40% SL / 30% HL, technology allowed): don’t skip working here either

Even though technology is permitted, “show that” and multi-step questions on Paper 2 still require shown working, not just a correct final value read off a calculator or graphing tool – a calculator can confirm an answer is right, but it cannot substitute for the algebraic or logical steps a mark scheme is built around. Use technology to check work, not to replace showing it.

HL Paper 3 (20%, 1h 15min): extended, multi-step problem-solving

Paper 3 consists of two extended-response problem-solving questions, using technology, and rewards sustained work through a multi-part problem rather than isolated short answers. This 1 hour 15 minute duration applies to the current course, through its final examination session in November 2028; the successor course, first assessed in 2029, shortens Paper 3 to 1 hour. Exam-preparation priority: practise full past-style extended questions under timed conditions, specifically building the stamina and structure needed to carry a solution through several connected parts, since this is a genuinely different demand from Paper 1 and Paper 2’s more contained questions.

The mathematical exploration: start early, revise once

The exploration is worth 20% at both SL and HL, is completed without exam-day time pressure, and is one of the most controllable components of the final grade if work begins early. A first draft written and checked against the criteria well before the deadline, then revised at least once, consistently outperforms one rushed in the final weeks – treat exploration milestones with the same seriousness as an exam date. A strong exploration in this course typically includes some element of rigour beyond the initial numerical case investigated – a proof, a derivation, or a genuine generalisation.

Worked practice scenario: a full “show that” derivation

Question: “Show that the derivative of f(x) = x³eˣ is f’(x) = eˣ(x³ + 3x²).”

Let u = x^3,   du/dx = 3x^2
Let v = e^x,   dv/dx = e^x

Product rule: f'(x) = u(dv/dx) + v(du/dx)
            = x^3 . e^x + e^x . 3x^2
            = e^x(x^3 + 3x^2)          <- factor out e^x

This matches the given result, so the derivation is complete.

Every step here is shown explicitly: the product rule stated, both derivatives identified separately, substitution shown, and the final factorisation step made explicit rather than skipped – this is exactly the level of working a “show that” question requires, and each shown step is itself worth marks independently of the final line matching the given answer.

Before/during exam checklist

• Before the exam: practise algebra, differentiation and integration by hand without a calculator, specifically for Paper 1; practise full “show that” questions showing every step, not just the final answer; HL students should practise full extended, multi-step problems under timed conditions for Paper 3; start the mathematical exploration early and plan at least one full revision before the deadline.
• During Paper 1: work entirely by hand, and double-check algebraic manipulation manually rather than assuming it’s correct.
• During Paper 2: use technology to verify an answer, but still write out the algebraic or logical steps a “show that” or multi-part question requires.
• During HL Paper 3: work through each part of an extended question in order, since later parts often build on earlier results (“hence” logic even when not stated explicitly).
• On every paper: for a “show that” question, write every intermediate step, even ones that seem obvious – the steps themselves carry marks.

Self-test

1. Why is practising without a calculator specifically important for Paper 1?
2. What does “show that” require that “calculate” does not?
3. In the worked scenario, what two derivatives are needed before applying the product rule, and what are they?
4. Why is starting the mathematical exploration early one of the most controllable ways to protect the final grade?

Answers: 1. Because Paper 1 is sat entirely without a calculator or other technology, so revision that always relies on a calculator to check arithmetic and algebra does not build the manual fluency the exam actually requires. 2. A fully justified derivation with every algebraic step shown, reaching the given result – not just a correct final answer, since the mark scheme awards credit for the shown steps themselves. 3. The derivative of x³ (which is 3x²) and the derivative of eˣ (which is eˣ), needed as u and v’s derivatives before applying the product rule. 4. Because, unlike a timed exam paper, the exploration is completed over an extended period without exam-day time pressure, so starting early leaves time for a full draft to be checked against the criteria and revised at least once, which consistently outperforms work rushed in the final weeks.

Official syllabus

International Baccalaureate Organization, Diploma Programme Subject Brief – Mathematics: Analysis and Approaches, first assessment 2021, published 2019 – the same source cited by the assessment revision notes and full syllabus guide. The worked derivation above is an original example written for this resource, not a reproduction of any official past or sample paper question.