- What Makes a Strong Maths AA IA Idea?
- Maths AA IA Requirements at a Glance
- How to Choose Your Topic Area
- Calculus and Optimisation IA Ideas
- Functions and Modelling IA Ideas
- Sequences, Series and Finance IA Ideas
- Geometry and Trigonometry IA Ideas
- Probability and Statistics IA Ideas
- Pure Mathematics and Proof IA Ideas
- Example: Turning an Idea Into an Exploration
- Criterion E: Depth Is Not the Same as Complexity
- How to Score on Criteria A to D
- Maths AA Key Concepts and Your IA
- New Maths AA Guide: What Changes for Exams From 2029
- Common Maths AA IA Mistakes
Quick Answer
Strong Maths AA IA ideas start from a question you genuinely want answered that needs course-level mathematics to answer it: calculus optimisation, modelling with functions, series, trigonometry, probability or proof. The exploration is worth 20% at SL and HL, runs to 12-20 double-spaced pages and is marked out of 20 on criteria A to E. Below are 50 ideas with the method and data each needs.
What You'll Learn
- The Maths AA exploration is worth 20% at SL and HL, is about 12-20 double-spaced pages and is marked out of 20 on criteria A to E
- Start with a question, then find the maths: the best ideas need mathematics at the level of your course to answer them
- Criterion E (use of mathematics) carries 6 of the 20 marks, and at HL the top band needs sophistication and rigour
- Personal engagement comes from your own data, conjectures and decisions, not from saying the topic interests you
- New maths guides apply from first exams in 2029; if you sit exams in 2026, 2027 or 2028, the current criteria apply
What Makes a Strong Maths AA IA Idea?
A strong idea passes four tests. First, it asks a question with a clear aim, for example 'what is the cheapest shape for a 330 ml can?' rather than 'cans and maths'. Second, answering it needs mathematics commensurate with Maths AA, which the guide describes as part of the syllabus or at a similar level (HL can go slightly beyond). Third, you can make it your own by collecting data, choosing between models or generalising a result. Fourth, it fits in 12-20 pages, which means one line of reasoning explored properly, not a tour of five topics.
Pro Tip
Test an idea by writing its aim in one sentence and listing the three main mathematical steps. If you cannot name the steps, or they all use maths from before the Diploma (basic algebra, percentages, simple averages), refine the idea before you start.
Maths AA IA Requirements at a Glance
The Maths AA internal assessment is called the mathematical exploration. It is a written report on a topic you choose, marked by your teacher and moderated by the IB. These requirements come from the current guide (first assessment 2021), which applies if you sit your exams in 2026, 2027 or 2028.
- Weighting: 20% of your final grade at both SL and HL
- Length: approximately 12-20 pages with double line spacing, including diagrams and graphs but excluding the bibliography
- Class time: the guide recommends about 10-15 hours of teaching time for the exploration
- Criterion A, Presentation: 4 marks for an exploration that is coherent, well organised and concise
- Criterion B, Mathematical communication: 4 marks for correct notation, defined variables and appropriate representations
- Criterion C, Personal engagement: 3 marks for thinking independently or creatively and making the mathematics your own
- Criterion D, Reflection: 3 marks for critical reflection on results, methods and limitations throughout
- Criterion E, Use of mathematics: 6 marks, with different descriptors at SL and HL, for a total of 20 marks
Watch Out
The guide states that you will not receive a grade for your maths course if you do not submit an exploration. Treat the deadline as seriously as an exam.
How to Choose Your Topic Area
Start from a part of the syllabus you enjoy and a context you know, then look for the question where they meet. The 50 ideas below are grouped by syllabus area. Each says why it works, what data or method it needs, and whether it suits SL and HL or HL only. HL only means the core mathematics sits in the additional higher level content, such as complex numbers, vectors, proof by induction, differential equations or Maclaurin series.
- Calculus: optimisation, rates of change, areas and volumes, clearly at course level for both SL and HL
- Functions and modelling: fitting and comparing models to real data, then judging where they fail
- Sequences, series and finance: convergence, sums and real loan or savings calculations
- Geometry and trigonometry: angles, distances and three-dimensional problems, with vectors at HL
- Probability and statistics: expected value, distributions and conditional probability
- Pure mathematics and proof: patterns, proofs and numerical methods, where the reasoning is the exploration
Calculus and Optimisation IA Ideas
Calculus explorations do well on criterion E because differentiation and integration are clearly at course level. The risk is that the exploration reads like a textbook exercise. Your own measurements, a comparison with reality or a generalisation fixes that.
- The cheapest drinks can (SL and HL): find the radius that minimises surface area for a fixed volume, then measure 5-10 real cans and explain why manufacturers do not use the optimum. Needs differentiation and your own measurements.
- The lifeguard problem (SL and HL): find the point where a lifeguard should enter the water to reach a swimmer fastest when running and swimming speeds differ, then show the answer matches Snell's law. Needs optimisation of a time function.
- Volume of a real vase, bottle or glass (HL): photograph the profile, fit functions to it, find the volume of revolution and compare with the water it actually holds. Needs AHL volumes of revolution; at SL, find a cross-sectional area with definite integrals instead.
- Area of a lake, leaf or island (SL and HL): fit functions to the outline on a scaled image, integrate, and compare with a grid count or the official figure. Needs definite integrals and piecewise functions.
- Best launch angle with and without air resistance (SL and HL): model a shot put or basketball shot, maximise its range, then compare with video of real throws. SL uses kinematics and optimisation; HL can add a differential equation for drag.
- How fast does the water level rise? (HL): use related rates to model filling a cone-shaped or real container, then film the container filling to test the model. Needs implicit differentiation and related rates.
- Cooling of a hot drink (HL): set up Newton's law of cooling as a differential equation, solve it by separating variables, compare with Euler's method and your own temperature readings. Needs first-order differential equations.
- The most economical driving speed (SL and HL): fit a function to fuel consumption at different speeds, from published tests or a car's trip computer, then minimise the cost per kilometre including the value of time. Needs function fitting and optimisation.
- The best seat in the cinema (SL and HL): find the distance at which a screen subtends the largest angle at your eye, a version of Regiomontanus' problem, then apply it to measured seat rows in a real cinema. Needs trigonometry and differentiation.
- The best gutter shape (SL and HL): fold a sheet of fixed width into a channel and maximise the cross-sectional area, first as a rectangle, then as a trapezium with a variable angle. Needs trigonometry and calculus, with a clear generalisation.
- Length of a cable or roller-coaster track (HL): derive the arc length formula from first principles and apply it to a fitted curve. This is slightly beyond the syllabus, which the HL criterion allows, so justify every step.
Functions and Modelling IA Ideas
Modelling ideas work when you compare at least two models and judge them on more than a fit statistic. Explain what each parameter means in context and where the model breaks down, because that is where criterion D marks come from.
- Daylight hours across the year (SL and HL): fit sinusoidal models to sunrise and sunset times for your city and one at a different latitude, interpret amplitude and phase shift, and test predictions. Needs trigonometric functions and published daylight tables.
- Tides in a harbour (SL and HL): model published tide-table heights with a sine function, then test whether a second sinusoidal term improves the fit. Needs trigonometric modelling and analysis of residuals.
- Parabola or catenary? (SL and HL): photograph a hanging chain or bridge cable, fit a quadratic and a hyperbolic cosine curve, and decide which fits better and why. The catenary goes beyond the syllabus, so explain it carefully.
- Medicine levels with repeated doses (SL and HL): model the amount of a drug in the body with exponential decay and a geometric series, find the long-run level and how the dosing interval changes it. Needs published half-life data and infinite geometric series.
- Population growth, exponential or logistic? (SL and HL): fit models to a country's population data from the World Bank or the UN and compare their long-term predictions. HL students can derive the logistic model from a differential equation.
- A damped pendulum or spring (HL): use video analysis to model the motion as an exponential multiplied by a sine function, then use derivatives to find velocity and turning points. Needs the product rule and trigonometric functions.
- Inverse-square law with a phone light sensor (SL and HL): measure light intensity at different distances, linearise with logarithms and estimate the exponent. Needs logarithms and regression on transformed data.
- Kepler's third law from planetary data (SL and HL): use NASA planetary data, take logarithms of orbital period and distance from the Sun, and find the power law that links them. Needs logarithmic transformation and regression.
- Modelling a logo or a car's profile (SL and HL): fit piecewise functions to a shape, but only if you use the model for something, such as finding an area or checking with derivatives that the pieces join smoothly.
Sequences, Series and Finance IA Ideas
Sequence ideas invite generalisation. Once you have solved one case, find the formula for every case and justify it, which lifts both criterion E and criterion C.
- Student loan or mortgage plans (SL and HL): model repayments with geometric series and recursive formulas, then compare the total cost under different interest rates and salary paths. Needs published loan terms and financial applications of geometric series.
- Saving for university (SL and HL): find how long regular deposits take to reach a target with compound interest and inflation, solving for the number of years with logarithms. Needs geometric series and logarithms.
- The Koch snowflake and other fractals (SL and HL): show that the perimeter grows without limit while the area converges, then generalise to a fractal you design. Needs geometric series and limits; fractal dimension is a possible extension.
- The total distance of a bouncing ball (SL and HL): measure rebound heights from video, estimate the common ratio and sum the infinite series, then compare with the measured total time. Needs geometric series and your own data.
- Approximating pi (SL and HL): compare Archimedes' polygon method, which uses trigonometry, with infinite series such as the arctan expansion (Maclaurin series at HL), and measure how fast each converges.
- Where does e come from? (SL and HL): explore (1 + 1/n)^n through compound interest paid more and more often, measure how quickly it approaches e, and at HL prove the limit using L'Hôpital's rule.
- Fibonacci and the golden ratio (HL): heavily overdone, so only choose it if you go further, for example proving Binet's formula by induction and testing claims about the golden ratio in nature with your own measurements.
- Equal temperament in music (SL and HL): show that piano note frequencies form a geometric sequence with ratio the twelfth root of 2, compare with whole-number ratios from just intonation, and measure the differences with logarithms.
Geometry and Trigonometry IA Ideas
Geometry ideas can look simple, so the depth has to come from generalising, from working in three dimensions, or from comparing two methods.
- Great-circle distance between two cities (SL and HL): convert latitude and longitude to 3D coordinates and find the shortest route over the Earth's surface, then compare with the distance on a flat map. SL uses 3D distance and trigonometry; HL can use the scalar product.
- Closest approach of two aircraft or ships (HL): write each path as a vector equation of a line with time as the parameter, then minimise the distance between them. Needs vectors and calculus.
- Packing cans or oranges (SL and HL): compare square and hexagonal packing densities, then check real boxes or displays. Needs trigonometry and area; HL can extend the reasoning to spheres.
- Shadows and the Sun's position (HL): model the direction of sunlight as a vector at different times and predict a shadow's length, then check with measurements. Needs vectors, trigonometry and published solar elevation data.
- Measuring a tall building (SL): only works if you go beyond basic trigonometry, for example by analysing how errors in angle and distance affect the height, and using calculus to choose the viewing distance that minimises the error.
Probability and Statistics IA Ideas
Probability suits Maths AA well. A single correlation or regression often sits closer to Maths AI, so for AA add depth with distributions, conditional probability or a model you derive yourself.
- Is going first an advantage in a penalty shoot-out? (SL and HL): build a tree-diagram model, compare it with historical shoot-out records and use conditional probability. HL can add Bayes' theorem.
- The birthday problem in your school (SL and HL): derive the probability of a shared birthday, extend it to birthdays within the same week, and test it against anonymised class data. HL can use counting principles.
- How reliable is a positive test? (HL): use Bayes' theorem with the published sensitivity and specificity of a screening test, then explore how the disease's prevalence changes the chance that a positive result is a true positive.
- The hot hand in basketball (SL and HL): record 200 or more of your own free throws, or use published player data, model successes with a binomial distribution and test whether streaks are longer than chance predicts.
- Do betting systems work? (SL and HL): calculate the expected value and variance of strategies such as the Martingale on roulette, then simulate them to compare. Needs discrete random variables.
- Are heights or leaf lengths normally distributed? (SL and HL): collect 100 or more measurements, fit a normal model, use z-scores and check how well the model fits in the tails.
- Waiting times for a bus (HL): build a probability density function from your own timed waits, find its mean and median, and compare with the timetable. Needs continuous random variables.
- Buffon's needle (SL and HL): drop a needle on lined paper 500 or more times to estimate pi, then derive the probability with integration and analyse how the error shrinks with more throws.
- How long does a game of snakes and ladders last? (SL and HL): estimate the expected number of turns by simulation and by reasoning about expected values, then test the effect of moving one snake.
Pure Mathematics and Proof IA Ideas
Pure explorations suit students who enjoy reasoning for its own sake. Personal engagement comes from your own conjectures, examples and proofs rather than a real-world context.
- Sums of powers (HL): find formulas for the sums of squares, cubes and fourth powers, spot the pattern and prove each by induction. Needs proof by mathematical induction.
- Newton's method for solving equations (SL and HL): use tangent lines to approximate roots, compare its speed with interval halving, and find equations where it fails. Uses SL differentiation; HL can analyse the error formally.
- How many terms do you need? (HL): use Maclaurin series to approximate sin x or e^x and find how many terms give six decimal places of accuracy across a range of x. Needs Maclaurin series and error analysis.
- Patterns in Pascal's triangle (SL and HL): shade the odd entries to reveal a Sierpinski pattern, explain why using binomial coefficients, and generalise to other divisors.
- Regular polygons and roots of unity (HL): use complex numbers and De Moivre's theorem to find the product of the distances from one vertex of a regular polygon to all the others, conjecture the pattern and prove it.
- Continued fractions for square roots (SL and HL): approximate the square root of 2 with continued fractions, compare the speed of convergence with Newton's method, and generalise to other square roots.
- Perfect riffle shuffles (HL): find how many perfect shuffles return a 52-card deck to its original order, test it with a real deck and generalise to n cards. This goes beyond the syllabus, so justify every step.
- Public-key cryptography with small primes (HL, with care): explain why RSA works using modular arithmetic and test it with small numbers. Risky at SL because the mathematics is outside the course; at HL you must prove the key steps.
Example: Turning an Idea Into an Exploration
Here is an illustrative example of how one idea becomes a focused exploration. Say a student who swims in a local lake wants to explore its area. 'The maths of lakes' is too broad. A workable aim is: how accurately can definite integrals of fitted functions estimate the lake's area, compared with the published figure?
- 1Aim: state the question and why it matters to you in two or three sentences, which is where criterion C starts
- 2Data: take a scaled satellite image, place axes and record 20-30 points on the shoreline
- 3Model: fit piecewise polynomial functions to the north and south shores and justify each choice
- 4Mathematics: integrate the difference between the functions, showing working that proves you understand it (criterion E)
- 5Check: compare with a grid estimate and the official area, and calculate the percentage error
- 6Reflect: explain which assumptions caused the error and how more points or different functions would change it (criterion D)
- 7Present: keep each graph next to the text it supports and define every variable (criteria A and B)
Criterion E: Depth Is Not the Same as Complexity
Criterion E is worth 6 of the 20 marks. At SL, the top band needs relevant mathematics commensurate with the course that is correct, with thorough knowledge and understanding. At HL, the top band also needs mathematics that is precise and shows sophistication and rigour. The guide says that overly complicated mathematics, where simple mathematics would do, is not relevant, and that it is better to do a few things well than many things badly.
- 1Use mathematics from the syllabus or at a similar level; work based only on prior learning cannot reach the top bands
- 2Show your reasoning, not just answers: the guide says a correct answer alone does not demonstrate understanding
- 3At HL, justify or prove each claim you rely on (rigour), and use HL-level ideas or SL ideas in a complex way (sophistication)
- 4Use technology for heavy calculation, but explain what each output means
- 5Show one repeated calculation in full, then summarise the rest in a table
How to Score on Criteria A to D
Criteria A to D are the same at SL and HL and carry 14 of the 20 marks. Many of these marks are won or lost in the writing, not the mathematics.
- A, Presentation (4 marks): include an introduction with a clear aim, a logical body and a conclusion; put large data sets in an appendix but keep relevant graphs in the text
- B, Mathematical communication (4 marks): use proper notation rather than calculator syntax, define every variable, label every graph and use more than one representation where it helps
- C, Personal engagement (3 marks): show your own decisions, such as collecting data, testing a prediction or trying another method; the guide says textbook-style explorations are unlikely to reach the higher levels
- D, Reflection (3 marks): reflect as you go, not only at the end; discuss limitations, compare methods and say what the results mean
Pro Tip
Write for a classmate. The guide says the exploration should be written so that your peers could follow it fairly easily, which is a good test for criteria A and B.
Maths AA Key Concepts and Your IA
The current Maths AA guide lists twelve key concepts that connect the five syllabus topics: approximation, change, equivalence, generalisation, modelling, patterns, quantity, relationships, representation, space, systems and validity. They are not an IA criterion, but they help you find an angle. A modelling idea is really about validity and approximation; a series idea is about patterns and generalisation. Naming the concept behind your exploration can sharpen your aim and give your reflection a focus.
New Maths AA Guide: What Changes for Exams From 2029
The IB has published a subject brief for the new Maths AA course. The full guide launches in February 2027, for first teaching in August 2027 and first assessment in May 2029. It affects students whose final exams are in 2029 or later, most of whom start the course in 2027. If you sit your exams in 2026, 2027 or 2028, the requirements above apply. According to the subject brief, the exploration stays at 20% at SL and HL, the recommended time rises to 30 teaching hours, and the criteria follow four stages of mathematical inquiry, the same at SL and HL (check the exact wording in the full guide once it is out):
- A, Problem specification: 4 marks (problem in context, desired outcomes)
- B, Abstraction: 6 marks (assumptions, selected techniques and tools, mathematical form)
- C, Computation: 4 marks (calculations, mathematical communication)
- D, Interpretation: 6 marks (interpretation of results, evaluation of desired outcomes, refinement)
Pro Tip
The ideas on this page work under both guides. Under the new criteria you state the desired outcome at the start and evaluate whether you reached it, so build that into your aim from day one.
Common Maths AA IA Mistakes
These problems weaken explorations before the writing even starts, or cost marks at the end. If you are unsure whether your idea has enough mathematical depth, an IB Maths AA tutor can help you scope it before you start.
- Choosing a topic with too little mathematics, or with mathematics only from prior learning
- Picking advanced mathematics you cannot explain, which the guide treats as not relevant if simpler maths would do
- Pasting calculator or software syntax instead of proper notation (criterion B)
- Copying a popular idea, such as Fibonacci, Monty Hall or a Ferris wheel, without adding your own data, generalisation or proof
- Leaving all reflection to the final paragraph instead of reflecting as you go (criterion D)
- Running well past 20 pages: conciseness is part of criterion A
- Missing references: they are not assessed, but leaving them out can be treated as an academic integrity issue
Watch Out
Do not reuse an exploration you found online. Your teacher and the IB check authenticity, and the work must reflect your own choices and understanding.