Found a topic you like?
Drop it straight into the free Maths AA Exploration frame. Planning your topic and question is free — unlock the full Exploration frame (the mathematics, communication and reflection) to build the whole IA across all five criteria.
Start this Exploration in the Maths AA frame →CALCULUS IN THE REAL WORLD
Optimisation, volumes of revolution and modelling rates of change give you genuine, sophisticated AA calculus to carry out yourself.
1 · Optimising the dimensions of a drinks can to minimise material for a fixed volume.
A clean optimisation: form the cost/area function, differentiate, find and justify the minimum, then compare your "ideal" can to real cans — the gap (lids, seams, branding) drives genuine reflection.
2 · Modelling the volume of a wine bottle (or vase) by rotating a curve and integrating.
Photograph a real object, fit a function to its profile, then integrate to predict its volume and check against the true capacity — a personal, measurable test of your model.
3 · How does a cup of coffee cool? Modelling temperature with Newton's law of cooling.
Collect your own cooling data, derive and solve the differential equation, fit the exponential constant and reflect on where the model breaks down (ambient changes, stirring) — calculus you genuinely own.
ALGEBRA, SEQUENCES & PROOF
Series, recurrence and genuine proof are exactly the rigour the Use of mathematics criterion rewards at the top band.
4 · Proving and exploring the convergence of an infinite series behind a familiar constant.
Investigate how fast a chosen series converges, prove a result about its sum, and compare competing series — real analysis-flavoured rigour that shows command well above quoting a formula.
5 · How quickly does a continued fraction converge to the golden ratio?
Derive the recurrence, prove the limit, and link it to Fibonacci ratios — a tightly focused question with proof, sequences and a satisfying convergence story to reflect on.
6 · A proof you find beautiful: exploring and extending the proof of Pythagoras (or irrationality of √2).
Take a classic proof, present it rigorously in your own words, then push it — generalise, find a second proof, or test where the argument fails — turning a known result into a genuine investigation.
7 · Modelling compound growth: how does the frequency of compounding approach continuous interest?
Build the sequence (1 + r/n)ⁿ from a real savings question and show it converges to eʳ — a personal finance hook that lands on a genuine limit and the number e.
Ready to investigate it properly?
The Maths AA Exploration frame walks you through every criterion — and the paid unlock helps you build the mathematics, communication and critical reflection into one export-ready document.
Open the Maths AA frame →GEOMETRY, TRIGONOMETRY & MODELLING CURVES
Fitting functions and trig models to real curves and cycles gives precise, testable mathematics you can graph and evaluate.
8 · Modelling a suspension cable: parabola versus catenary y = a·cosh(x/a).
Hang a chain, measure its shape, and decide whether a parabola or the catenary fits better and why — two competing models, real data, and a clean reflection on which assumptions matter.
9 · Using trigonometric functions to model the hours of daylight across a year.
Fit a sine model to daylight data for your own city, test it against a second latitude, and reflect on where the simple sinusoid fails — a focused, data-anchored trig investigation.
10 · The mathematics of a parabolic dish (or whispering gallery): why the focus works.
Prove the focusing property of a parabola or ellipse from its equation and tangents, then connect it to a satellite dish or whispering gallery you find interesting — proof grounded in a real object.
11 · Modelling a roller-coaster (or ski-jump) profile with piecewise smooth functions.
Design a track that is continuous and smooth at every join by matching values and derivatives — genuine calculus constraints, with a strong personal-design angle to reflect on.
PROBABILITY & STATISTICS
Conditional probability, distributions and regression let you build models and test them — rich material for reflection on assumptions.
12 · Why does the Monty Hall problem feel wrong? A conditional-probability exploration.
Work the probabilities rigorously, generalise to n doors, and back the theory with your own simulation — a famous puzzle turned into a focused, self-driven investigation.
13 · Does the birthday problem hold up? Probability theory versus a real class.
Derive the shared-birthday probability, approximate it, then test it against real groups and reflect on the gap — accessible to set up but rich enough to develop carefully.
14 · Modelling rare events with the Poisson distribution (e.g. goals, buses, or texts per hour).
Collect your own count data, fit a Poisson model, and test the fit with χ² — a genuine statistical investigation that reflects honestly on whether the assumptions hold.
15 · How well does a normal distribution model real heights, reaction times or exam marks?
Gather your own sample, standardise it, and test normality rather than assuming it — the critical evaluation of fit is exactly what the Reflection criterion rewards.
Found your question?
The examiner-written Maths AA Exploration frame takes you through every criterion with the rubric, worked examples and the traps that cost marks. Planning your topic and question is free — unlock the full frame to develop the mathematics, communication and reflection.
Open the Maths AA frame →MATHS IN SPORT, MUSIC, ART & FINANCE
A personal passion is the surest route to genuine engagement — provided you build real AA mathematics on top of it.
16 · Optimising the angle of a basketball free throw (or football free kick).
Derive the trajectory, optimise the angle for a given release height and speed, and compare to your own measured shots — calculus and a sport you care about, with honest reflection on air resistance.
17 · The mathematics of musical tuning: equal temperament and the frequencies of a scale.
Show why equal temperament uses ¹²√2, compare the resulting frequencies to pure ratios, and reflect on the compromise — a focused, original take if you play an instrument.
18 · Tessellations and transformations: the geometry behind an Islamic or Escher-style pattern.
Reconstruct a real pattern from transformations, prove why only certain tilings are possible, and design your own — geometry with a strong artistic, personal hook.
19 · Pricing a fair bet or insurance premium with expected value.
Model a real game, lottery or premium, compute the expected value and the house edge, and reflect on risk versus return — a focused finance question with genuine probability behind it.
20 · How fair is a ranking? Modelling a league or tournament with matrices.
Build a results matrix for a sport you follow, derive a ranking, and reflect on what the model rewards and ignores — a personal, data-rich algebra investigation.
21 · The geometry of a perfect penalty zone: optimising shot placement.
Model the angle a goal subtends from different positions and optimise it — a tightly focused, original sports-geometry question with real trig and calculus to develop.
22 · Fractals and self-similarity: measuring the dimension of a coastline or fern.
Estimate a fractal dimension from your own measurements using a log–log plot, and reflect on what "dimension" really means — visually striking and genuinely investigative.
23 · Modelling the spread of a trend (or rumour) with the logistic function.
Derive the logistic model, fit it to real adoption data you gather, locate the inflection point with calculus, and reflect on the carrying-capacity assumption.
24 · The mathematics of a spiral: modelling a shell, sunflower or galaxy with logarithmic spirals.
Fit a logarithmic spiral to a real photograph in polar form, test the fit, and connect it to the golden angle — an original, beautiful topic with genuine functions and reflection.
From a topic to a top-band Exploration
A topic is the easy part — the marks are in how you investigate it. The Exploration is scored out of 20 across five criteria: A Presentation /4, B Mathematical communication /4, C Personal engagement /3, D Reflection /3 and E Use of mathematics /6. Whichever topic you pick, the same moves win: a focused question you genuinely care about, sophisticated AA-level mathematics (calculus, algebra, functions, trig, proof) that you carry out yourself, correct notation with defined variables and labelled graphs, and critical reflection on assumptions, limitations and extensions — not just restating the answer.
Build your chosen topic into a full Exploration
The examiner-written Maths AA Exploration frame takes you through every section with the rubric, worked examples and the traps that cost marks. Planning your topic and question is free — unlock the full Exploration frame (the mathematics, communication and reflection) to finish the whole IA and export it to Word or PDF.
Open the Maths AA frame →Maths AA Exploration ideas — FAQ
What makes a good IB Maths AA Exploration topic?
Genuine personal interest, a focused question narrow enough to investigate in depth, and room to develop sophisticated AA-level mathematics — calculus, algebra, functions, trigonometry or proof — that you carry out yourself with correct notation. Avoid topics you can only describe or that simply quote a known result; choose one you can genuinely investigate.
How personal and focused should my Exploration be?
Very. Personal engagement and reflection carry a real share of the marks, so the topic should be visibly yours — a hobby, a puzzle, something from sport, music, art or finance — and the question should be narrow enough that 12–20 pages can explore it properly. A focused question you care about beats a broad, generic one every time.
Can I just copy one of these topics?
Use them as a launchpad, but make the Exploration your own: narrow the research question to something you care about, bring your own data or context, and do the mathematics yourself rather than quoting results. That ownership is exactly what Personal engagement, Use of mathematics and Reflection reward.
How much and what level of mathematics do I need?
Mathematics that is relevant, correct and commensurate with the Analysis & Approaches course or beyond — for top marks, sophisticated or rigorous: genuine calculus, algebra and sequences, functions, trigonometric modelling, probability and statistics, or a real proof. Build it section by section in the free Maths AA Exploration frame — focused question, mathematics, communication and critical reflection.
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