IB Maths AA HL Paper 3: how to approach the problem-solving paper

Where Paper 3 sits in AA HL assessment

From the current Analysis and Approaches guide (first assessment 2021), which applies to exams up to and including November 2028.

What Paper 3 is really testing

Papers 1 and 2 mostly test whether you can do mathematics you have practised. Paper 3 tests whether you can use that mathematics in a situation you have not seen before. A typical question opens with a definition or a setting, asks you to compute a few small cases, then asks you to notice what is happening, state a conjecture, and justify it, often with proof by induction, algebra, calculus or a probability argument. Later parts frequently extend the idea to a more general case.

Because each part builds on the one before, the paper rewards students who keep going. Earlier parts are usually more accessible, and later parts often give you the result you were meant to find, so you can use it even if you could not derive it. With 55 marks in 60 minutes, you have about one minute per mark, and the two questions are each long enough that running out of time on the first is the biggest risk.

Worked investigation (in the Paper 3 style): S(n) = 1 × 2 + 2 × 3 + ... + n(n + 1)

This question was written for this page to show the typical structure. It is not an IB question.

1. Small cases: S(1) = 2, S(2) = 2 + 6 = 8, S(3) = 8 + 12 = 20, S(4) = 20 + 20 = 40. Lay them out in a table so the pattern is easy to see.
2. Look for structure: the values 2, 8, 20, 40 are 6/3, 24/3, 60/3 and 120/3, and 6, 24, 60, 120 are 1 × 2 × 3, 2 × 3 × 4, 3 × 4 × 5 and 4 × 5 × 6.
3. Conjecture: S(n) = n(n + 1)(n + 2)/3. Check n = 4: 4 × 5 × 6/3 = 40. Correct.
4. Prove by induction. Base case n = 1: 1 × 2 × 3/3 = 2 = S(1). Assume S(k) = k(k + 1)(k + 2)/3 for some positive integer k.
5. Inductive step: S(k + 1) = S(k) + (k + 1)(k + 2) = k(k + 1)(k + 2)/3 + 3(k + 1)(k + 2)/3 = (k + 1)(k + 2)(k + 3)/3, which is the formula with n = k + 1. Since it is true for n = 1, and true for n = k + 1 whenever it is true for n = k, it is true for all positive integers n.
6. Use the result: since n(n + 1) = n^2 + n, the sum of k^2 from 1 to n equals S(n) minus the sum of k, which is n(n + 1)(n + 2)/3 - n(n + 1)/2 = n(n + 1)(2n + 1)/6. Check n = 3: 1 + 4 + 9 = 14 and 3 × 4 × 7/6 = 14.
7. Generalise: the same pattern suggests 1 × 2 × 3 + 2 × 3 × 4 + ... + n(n + 1)(n + 2) = n(n + 1)(n + 2)(n + 3)/4. Check n = 2: 6 + 24 = 30 and 2 × 3 × 4 × 5/4 = 30. A real Paper 3 would then ask you to prove this or state the general form.

How to approach any Paper 3 question

1. Read the whole question once, quickly, before writing. Knowing where it is heading tells you why the early parts matter.
2. Do the small cases carefully and present them in a table. Errors here ruin every later part.
3. When asked for a conjecture, write it as a clear statement in terms of n (or whatever the variable is), not as a description.
4. Use the GDC freely to generate cases, plot graphs and test conjectures, but write down what you entered and what you found.
5. If you cannot do a part, move on and use the given result. Many Paper 3 parts say 'show that', which tells you the answer.
6. Keep an eye on time: about 30 minutes per question. Leave a half-finished part and return if time allows.

Common mistakes that cost marks

• Stopping after the first difficult part and never seeing the easier parts that follow.
• Guessing a pattern from two cases. Test a conjecture on at least one more value.
• An induction proof without a concluding sentence, or with an assumption that is never used.
• Using the GDC to 'prove' a result. Numerical checks support a conjecture; they do not prove it.
• Spending 40 minutes on the first question and rushing the second.
• Not linking parts: the result from part (c) is often the key to part (e).

How to prepare, and how a tutor helps

Paper 3 rewards regular practice from DP1 rather than a burst before the exams. Use the specimen paper and past Paper 3 questions from May 2021 onwards, and time yourself strictly. Mixed problem-solving from other sources helps too, because the skill is reasoning across topics: sequences and proof, calculus, complex numbers, vectors and probability all appear.

Make sure the HL tools Paper 3 leans on are secure, especially proof by induction, sums of series and the main calculus techniques. A student who has to think hard about how to set up an induction proof has no time left to think about the problem itself.

In one-to-one lessons a tutor works through Paper 3 style questions alongside the student on the shared whiteboard, asking the questions a strong problem-solver asks: what happens for small values, what stays the same, what would make this easier. Over time the student learns to ask these questions without prompting. The tutor also tracks timing, so the student practises moving on rather than getting stuck.