IB Math AA HL Internal Assessment — the complete guide
The AA HL exploration is the same 20-mark, 12–20 page task as every other IB Mathematics course — but Criterion E is marked against the HL syllabus, and the top two marks require sophistication and rigour. This page explains exactly what that means, gives you six topics that sit at genuine HL level, and walks through two complete explorations.
Worth 20% of your final grade. Marked by your teacher, moderated by the IB.
What Criterion E expects at AA HL
Criterion E (6 marks) is where AA HL students win or lose. "Commensurate with the level of the course" means the HL syllabus: calculus on the HL options list, proof, complex numbers, vectors in 3D, or statistics beyond the SL core. An exploration built entirely on SL mathematics is capped at 3/6 for E, however well it is written.
| Clearly at HL level | Improper integrals and convergence; volumes and surface areas of revolution; systems of differential equations (Euler, phase portraits); Maclaurin series and error bounds; proof by induction or contradiction with a real result; complex numbers and roots of unity; eigenvalues; integration by parts, substitution and partial fractions in a real model. |
|---|---|
| Borderline — needs an HL extension | Single first-order separable ODE (SL); optimisation with one variable (SL); regression and correlation (SL); binomial or normal probability (SL). Any of these becomes HL if you add a second-order model, derive the formula, prove a general result or analyse convergence. |
| Not enough on its own | Quadratics, trigonometric graphs, arithmetic/geometric series, descriptive statistics, financial formulas from the booklet. Use them as tools inside a bigger exploration, never as the exploration. |
For the full level descriptors of every criterion, see the IB Math IA criteria and rubric; for the 6-mark criterion in depth, see Criterion E — Use of Mathematics.
Six IA topic ideas at AA HL level
Each one has a research question you could use tomorrow, the mathematics that makes it course-level, and where the data comes from. None of them appears in the general topic list, so they are not the ones every examiner has seen.
Why a hanging chain is not a parabola — the catenary
What curve does a chain hanging between two points really make, and how far is it from the parabola it is often assumed to be?
Hyperbolic functions, calculus of variations lite (derive y = a cosh(x/a) from a force balance), arc length by integration, numerical solving for a.
Photograph a real chain or cable, overlay the fitted catenary and the best parabola, compare residuals.
The Basel problem and how fast series converge
How many terms of the sum of 1/n² are needed for a given accuracy, and can Euler's method of proof be reconstructed?
Series convergence, Maclaurin expansion of sin x, comparison with an integral to bound the error, proof.
Pure mathematics IA — strong when the derivation is your own and the error bounds are proved, not just observed.
Damped oscillation of a real spring or pendulum
Which damping model — linear, quadratic or dry friction — best describes the decay of a pendulum I measured?
Second-order differential equations, auxiliary equation with complex roots, Euler and improved Euler for the non-linear case, residual analysis.
Phone slow-motion video gives position–time data free of charge.
Complex numbers and the geometry of a rotating fan or drone
How can complex numbers describe the combined motion of a rotor that spins while the drone turns?
Complex numbers in polar form, de Moivre's theorem, roots of unity, composition of rotations, parametric curves and their arc length.
An unusual application of a core HL topic; pair with a GeoGebra animation.
Volumes of revolution and the design of a wine glass or vase
What profile curve gives a glass that holds exactly 250 mL and is as tall as possible, and how sensitive is the volume to the shape?
Volumes of revolution, integration by parts or substitution, optimisation with a constraint, sensitivity analysis.
Measure a real glass; the "how sensitive" question is what lifts Criterion D.
Proof by induction and the Tower of Hanoi family
What is the minimum number of moves for the Tower of Hanoi with four pegs, and can a formula be proved rather than conjectured?
Recurrence relations, proof by induction, strong induction, the Frame–Stewart conjecture, comparison with computer search.
Pure maths with a genuine open question at the end — ideal for a student who prefers proof to data.
Two complete AA HL explorations, summarised
These are two of the 16 worked IA examples on this site, chosen because their core technique sits squarely at AA HL level. Here you get the idea, the mathematics and the reflection; the full page shows every calculation step.
Worked example 5: Gabriel's Horn — Finite Volume, Infinite Surface Area
"How can a solid of revolution have finite volume but infinite surface area, and what does this reveal about the nature of infinity?"
Mathematical concepts used
- Volume of revolution: V = π∫[a→b] [f(x)]² dx
- Surface area of revolution: SA = 2π∫[a→b] f(x)√(1 + [f′(x)]²) dx
- Improper integrals (limits at ∞) and convergence vs divergence
- Comparison theorem for improper integrals
What the answer means — and how to reflect on it
Resolve the paradox rather than just stating it. Volume converges because the discs shrink like 1/x², whose integral converges; surface area diverges because the bands shrink only like 1/x, whose integral diverges. The deep point is that convergence depends on how fast a quantity shrinks, and ‘1/x’ is exactly the boundary case — the harmonic series in integral form. Explain that painting the inside would need a coat of paint thinner than any positive thickness, which is why ‘fill it but can't paint it’ is not a real contradiction.
Extend by generalising: rotate y = 1/xp and find for which values of p the volume converges (p > ½) and the surface area converges (p > 1). Tabulate the three regimes. This is your own investigation, it uses the same techniques with more sophistication, and it shows precisely the kind of rigour the HL Criterion E descriptors ask for.
See every calculation step for this exploration, explained line by line →Worked example 6: Lotka-Volterra Predator-Prey Dynamics
"Can the Lotka-Volterra system of differential equations model the observed population cycles of lynx and snowshoe hare in Canada?"
Mathematical concepts used
- Coupled first-order ODEs (system of two equations)
- Equilibrium analysis: finding fixed points and classifying stability
- Euler's method for numerical approximation of the solution
- Phase plane analysis (x vs y trajectory, not vs time)
What the answer means — and how to reflect on it
The exploration has produced a genuine prediction from first principles and compared it with history. The discrepancy is where the value lies. The linearisation is only valid for small oscillations; the real cycles are large, and for large orbits the Lotka–Volterra period is longer than 2π/√(αγ) — so part of the gap is the approximation itself. Simulate the full system numerically (Euler or a GDC solver) with realistic starting values and measure the period directly; compare with the linear estimate.
Then challenge the assumptions. Hares do not have unlimited food: replace αx with logistic growth αx(1 − x/K) and show that the equilibrium moves and the eigenvalues acquire a negative real part — the cycles now decay towards a stable point, which is closer to how real ecosystems behave. Explaining what each change does to the eigenvalues, and therefore to the behaviour, is sophisticated mathematics used with purpose: exactly what the HL Criterion E descriptors for 5–6 marks describe.
See every calculation step for this exploration, explained line by line →How the AA HL IA is marked — the five criteria
20 marks in total, identical for every course except the level descriptors of Criterion E. Your teacher marks it; the IB moderates a sample.
| Criterion | Marks | What full marks look like |
|---|---|---|
| A — Presentation | 4 | Coherent, well organised and concise. Clear sections, numbered figures, 12–20 pages. |
| B — Mathematical communication | 4 | Correct notation, variables defined with units, graphs and tables labelled, consistent throughout. |
| C — Personal engagement | 3 | Your own question, your own data or decisions, independent thinking that is visible in every section. |
| D — Reflection | 3 | Critical reflection on results, assumptions, limitations and extensions — throughout, not just at the end. |
| E — Use of mathematics | 6 | Relevant mathematics at HL level, correct, precise, thoroughly understood, with sophistication and rigour. |
| Total | 20 | 20% of the final grade |
How to start your AA HL IA this week
AA HL IA — word count, format and other questions
Does my AA HL IA have to use HL-only mathematics?
Not every line — but the core of the exploration must. Examiners ask whether the main technique would be examinable in AA SL; if it would, the ceiling for Criterion E is 3/6. Build the IA around one HL technique (a second-order ODE, an improper integral, a proof) and let SL tools support it.
Is a pure-mathematics IA acceptable at AA HL?
Yes, and AA HL is the course where it works best. Proof-based explorations (series, number theory, geometry) score well when the student derives results and reflects on the mathematics itself. Personal engagement is shown through your own conjectures and the routes you tried, not through a personal story.
Is there a word count for the IB Math IA?
No — the IB sets a page limit, not a word count. The exploration should be 12–20 pages including graphs and tables but excluding the bibliography and appendices. Most strong IAs land at 14–18 pages and roughly 2,500–4,000 words of prose; if you are over 20 pages you lose the "concise" descriptor in Criterion A, so the maximum there drops to 3/4.
What format should the IA be in?
A typed document (Word, Google Docs or LaTeX) exported to PDF, with page numbers, a title that states the research question, headed sections, numbered figures and tables, and an equation editor for every piece of mathematics. Double spacing is not required; 11–12 pt text is normal. Raw data and repetitive calculations go in an appendix.
How much of my grade is the IA worth?
The exploration is worth 20% of the final grade for every IB Mathematics course (AA SL, AA HL, AI SL and AI HL). The remaining 80% comes from the written papers. It is marked by your teacher out of 20 and then moderated by the IB.
Can I use technology — a GDC, Excel, Desmos, Python?
Yes, and for most explorations you should. Technology is expected for data fitting, graphs and repetitive calculation, but Criterion E rewards the mathematics you show: formula → substitution → result, with technology used to check or to extend. "My calculator gave r = 0.81" with no formula earns little; showing the formula, one worked calculation and then the technology output earns full credit.
How many drafts can my teacher look at?
The IB allows the teacher to give written feedback on one draft. They may discuss ideas and answer questions throughout, but only one full draft receives written comments, so make that draft as complete as possible — including the bibliography and appendix.
Want a second pair of eyes on your AA HL IA?
Mr Ahmad has guided IB Mathematics: Analysis and Approaches HL students in Dubai and online from topic choice to final draft — including the honest "is this at course level?" conversation that saves marks in Criterion E.
IA tutoring — how it works →