IA ideas library
239 Maths IA ideas, with data you can actually get
Every idea below has a precise research question, the mathematics it needs (labelled AA/AI, SL/HL), data sources we have checked exist, an outline, the pitfalls that cost marks, and ways to show personal engagement. Use them as a starting point — the best IAs change the question until it is their own.
Levels: Accessible ideas suit a first IA with course-level maths; Solid ideas need more independent work; Ambitious ideas suit confident students (often HL).
By course: AA SL IA ideas · AA HL IA ideas · AI SL IA ideas · AI HL IA ideas
By interest: Computer science · Physics and engineering · Psychology and social sciences · Geography and travel · Chemistry · Biology · Economics · Business management · Sports, exercise and health science · Environmental systems and societies · Music · Visual arts · Design technology · Sport
Modelling with functions
- Is a hanging chain a parabola? Catenary versus quadratic modelsAA SLAA HLAI SLAI HLSolid
- Which function best describes the arch of a famous bridge?AA SLAI SLAccessible
- How long does a bottle take to drain? Testing Torricelli's lawAA SLAA HLAI SLAI HLSolid
- Does my phone battery really drain in a straight line?AI SLAI HLAA SLAccessible
- How far does a bouncing ball really travel?AA SLAI SLAA HLAI HLAccessible
- Why do stopping distances grow so fast? Modelling the Highway Code tableAA SLAI SLAccessible
- Exponential or logistic? Modelling my country's populationAA SLAA HLAI SLAI HLSolid
- Is Moore's law still true?AA SLAI SLAA HLAI HLAccessible
- Does a pendulum's period really depend on the square root of its length?AA SLAI SLAA HLAI HLAccessible
- Does light really follow an inverse-square law?AA SLAI SLAccessible
- How many times can you fold a sheet of paper?AA SLAA HLSolid
- Can you predict a tree's height from its trunk?AI SLAI HLAA SLSolid
- Which mug keeps tea hot longest? Newton's law of cooling, comparedAA SLAA HLAI SLAI HLAccessible
Calculus & optimisation
- Why aren't drinks cans the optimal shape?AA SLAI SLAA HLAI HLAccessible
- What is the best shape for a rain gutter?AA SLAI SLAA HLAI HLAccessible
- What is the volume of an egg?AA HLAI HLAA SLSolid
- Where should the 125 ml line go on a glass?AA HLAI HLSolid
- Designing a smooth roller-coaster drop with piecewise functionsAA SLAA HLAI SLAI HLAmbitious
- How big is my local lake? Estimating area with the trapezoidal ruleAI SLAI HLAA SLAA HLAccessible
- The lifeguard problem: what is the fastest route to a swimmer?AA SLAA HLAI SLAI HLSolid
- Will the sofa fit round the corner? The longest object through a corridorAA SLAA HLAI SLAI HLSolid
- Where is the best seat in the cinema?AA SLAA HLAI SLAI HLSolid
- How fast is a sprinter at each moment? Differentiating 100 m splitsAA SLAA HLAI SLAI HLSolid
Differential equations & dynamics
- How long does a skydiver take to reach terminal velocity?AA HLAI HLAmbitious
- Linear or quadratic drag? Testing air resistance with falling coffee filtersAA HLAI HLSolid
- How much of cycling effort goes on air resistance? A coast-down testAA HLAI HLAmbitious
- How fast would flu spread through my school? An SIR modelAI HLAA HLAmbitious
- How long would a lake take to flush out a pollutant?AA HLAI HLSolid
- Can the Lotka–Volterra model explain the lynx and hare cycles?AI HLAA HLAmbitious
- Can mathematics predict a battle? Lanchester's laws and Iwo JimaAI HLAA HLAmbitious
- How does bread dough rise? A logistic model in the kitchenAA SLAA HLAI SLAI HLAccessible
- How fast does news spread through a year group?AI SLAI HLAA SLAA HLSolid
- Does caffeine build up if you drink coffee every few hours?AA SLAA HLAI SLAI HLSolid
Statistics & hypothesis testing
- Do students who sleep less react more slowly?AI SLAI HLAA SLAA HLAccessible
- Is home advantage in football disappearing?AI SLAI HLAA SLAA HLAccessible
- Do left-handed tennis players have an advantage?AI SLAI HLAA HLSolid
- Is crime seasonal? Testing monthly crime countsAI SLAI HLAccessible
- How much is being near a station worth?AI SLAI HLAA SLSolid
- Does money buy a longer life? GDP and life expectancyAI SLAI HLAA SLAA HLSolid
- Does Benford's law hold for the populations of the world's towns?AI SLAI HLAA HLSolid
- Are elite footballers more likely to be born early in the selection year?AI SLAI HLAA SLAA HLAccessible
- Can statistics tell two authors apart?AI SLAI HLAA SLSolid
- Are our summers getting wetter or drier?AI SLAI HLAA SLAccessible
- How often do big earthquakes happen? The Gutenberg–Richter lawAI SLAI HLAA SLAA HLSolid
- Are NBA players getting taller?AI SLAI HLAA SLAA HLAccessible
- Do TV series really get worse over time?AI SLAI HLAccessible
Probability & chance
- How much does a small edge on each point matter in tennis?AA SLAA HLAI SLAI HLSolid
- Is a penalty shoot-out fair to the team that shoots second?AA SLAA HLAI SLAI HLSolid
- Radioactive decay with dice: does chance produce an exponential curve?AA SLAI SLAA HLAI HLAccessible
- The birthday problem, extended: what about birthdays within a day?AA SLAA HLAI SLAI HLSolid
- How many packs does it take to finish a sticker album?AA SLAA HLAI SLAI HLSolid
- Are arrivals at the school canteen random? A Poisson model of a queueAI HLAI SLSolid
- When should you stop looking? The 37% rule for choosing the bestAA HLAA SLAI HLAmbitious
- How many extra tickets should an airline sell?AA SLAA HLAI SLAI HLSolid
- Is the “hot hand” real? Streaks in my free throwsAI SLAI HLAA SLAA HLSolid
- Estimating π by dropping needles: how fast does the estimate improve?AA HLAA SLAI SLAI HLSolid
- The Monty Hall problem when the host is forgetfulAA SLAA HLAI SLAI HLSolid
- Is a lottery ticket ever worth buying? Expected value and rolloversAA SLAA HLAI SLAI HLAccessible
- What is the chance of hitting every green light on my route?AI SLAI HLAA SLAA HLAccessible
Geometry, trigonometry & Voronoi diagrams
- Where should my town put its next defibrillator? A Voronoi analysisAI SLAI HLSolid
- Do school catchment areas match the Voronoi diagram of the schools?AI SLAI HLSolid
- Who lives furthest from a supermarket? Food access with Voronoi diagramsAI SLAI HLSolid
- Voronoi diagrams in a grid city: Euclidean or taxicab distance?AI SLAI HLAA HLAmbitious
- Which airport is really closest? Voronoi diagrams on a curved EarthAI SLAI HLAmbitious
- Would a Voronoi diagram have pointed John Snow to the Broad Street pump?AI SLAI HLSolid
- Should bigger hospitals get bigger regions? Weighted Voronoi diagramsAI HLAI SLAmbitious
- Which method measures a building's height most accurately?AA SLAI SLAccessible
- Designing a working sundial for my latitudeAA SLAA HLAI SLAI HLSolid
- Why do flight paths curve on the map?AA SLAA HLAI SLAI HLAmbitious
- Modelling the sun's height from a stick's shadowAA SLAI SLAA HLAI HLAccessible
- Is a satellite dish really a parabola, and where is its focus?AA SLAA HLSolid
Finance & economics
- Should I lease or buy my first car?AI SLAI HLSolid
- Will I ever repay my student loan?AI SLAI HLAA SLSolid
- Is a longer mortgage ever the better deal?AI SLAI HLAA SLAA HLAccessible
- How much does it cost to start saving ten years later?AI SLAI HLAA SLAA HLAccessible
- Has my pocket money kept up with inflation?AI SLAA SLAI HLAccessible
- Measuring inequality: fitting Lorenz curves and computing the Gini coefficientAA SLAA HLAI SLAI HLSolid
- Is the euro–dollar exchange rate a random walk?AI SLAI HLAA HLAmbitious
- Contract or SIM-only? The real cost of a new phoneAI SLAA SLAccessible
- What price maximises profit at our school bake sale?AA SLAI SLAA HLAI HLSolid
- How fast do cars lose value? Modelling depreciation from listingsAI SLAI HLAA SLAccessible
- Where does e come from? Compounding more and more oftenAA SLAA HLSolid
Sport
- What is the best angle to shoot a free throw?AA SLAA HLAI SLAI HLSolid
- Why don't long jumpers take off at 45°?AA SLAA HLAI SLAI HLSolid
- Is there a limit to how fast humans can run 100 m?AA SLAA HLAI SLAI HLSolid
- When should a Formula 1 driver pit? Modelling tyre degradationAA SLAI SLAA HLAI HLAmbitious
- Where should a winger shoot from? Maximising the angle to goalAA SLAA HLAI SLAI HLSolid
- Are cricket batting scores geometrically distributed?AI SLAI HLAA HLAA SLSolid
- Can a simple rating system predict football results?AI SLAI HLAA HLAmbitious
- Will women's Olympic times ever match men's?AI SLAI HLAA SLAA HLSolid
- Where should I aim on a dartboard?AI SLAI HLAA HLAmbitious
- Where should a pool ball hit the cushion? The geometry of bank shotsAA SLAI SLAccessible
Environment & climate
- Modelling the Keeling curve: trend plus seasonsAA SLAA HLAI SLAI HLSolid
- When might the Arctic be nearly ice-free in September?AA SLAA HLAI SLAI HLSolid
- How strongly is global temperature linked to ln(CO₂)?AI SLAI HLAA SLAA HLAmbitious
- Is solar power growing logistically?AA SLAA HLAI SLAI HLSolid
- Is deforestation in [a country] slowing down?AI SLAI HLAA SLAccessible
- What is the best tilt for a solar panel at my home?AA SLAI SLAA HLAI HLSolid
- Are Atlantic hurricanes becoming more frequent?AI SLAI HLAmbitious
- Can a sum of sine waves predict the tides?AA SLAA HLAI SLAI HLSolid
- How does the shape of the daylight curve change with latitude?AA SLAI SLAA HLAI HLAccessible
- How much does a cold month cost my family?AI SLAI HLAA SLAccessible
- How much does rain reduce bike-share use?AI SLAI HLSolid
Health, biology & medicine
- A positive test result: how likely is it that you really have the condition?AA HLAA SLAI SLAI HLSolid
- How much vaccination is enough to stop measles?AI SLAI HLAA SLSolid
- How fast does my heart rate recover after exercise?AA SLAI SLAA HLAI HLAccessible
- Is BMI's “height squared” right? Evidence from OlympiansAI SLAI HLAA SLAA HLSolid
- Modelling bacterial growth in the biology labAA SLAA HLAI SLAI HLSolid
- Do more doctors mean fewer infant deaths?AI SLAI HLAA SLAccessible
- How fast did COVID-19 spread at the start? Comparing doubling timesAA SLAA HLAI SLAI HLSolid
- Can height predict lung capacity?AI SLAI HLAA SLAccessible
- How do doctors estimate your skin area? Comparing body-surface-area formulasAA SLAA HLAI SLAI HLSolid
Networks, graphs & algorithms
- What is the shortest route to walk every corridor in my school?AI HLSolid
- What is the cheapest way to connect villages with fibre broadband?AI HLAI SLSolid
- Planning the shortest delivery round: bounds for the travelling salesman problemAI HLSolid
- How does Google rank pages? PageRank on a small websiteAI HLAA HLAmbitious
- How connected is my year group? Analysing a friendship networkAI HLAI SLSolid
- Which Underground station is the most central?AI HLAI SLSolid
- Will one phone brand take over? Brand loyalty as a Markov chainAI HLSolid
- How many flights does it take to get anywhere?AI HLAI SLAmbitious
- What is the fewest number of exam sessions my school needs?AI HLAA HLAI SLSolid
Pure maths, number & proof
- Beyond Fibonacci: what ratio do other recurrence sequences approach?AA HLAA SLSolid
- Why does Pascal's triangle make a fractal?AA HLAA SLSolid
- The Collatz conjecture: what can we say about stopping times?AA SLAA HLAI SLAI HLSolid
- Finding formulas for 1ᵏ + 2ᵏ + … + nᵏAA HLAA SLSolid
- The product of a regular polygon's diagonalsAA HLAmbitious
- Why is 355/113 such a good approximation to π?AA HLAA SLSolid
- How many primes are there below n? Testing the prime number theoremAA SLAA HLAI SLSolid
- Why is the main body of the Mandelbrot set a cardioid?AA HLAmbitious
- Why does 1 + 1/4 + 1/9 + … equal π²/6?AA HLAmbitious
- How does RSA encryption work, and how long would it take to break a small key?AA HLAA SLAmbitious
- How long is a coastline? Measuring fractal dimensionAA SLAA HLAI SLAI HLSolid
- How does a calculator work out sin(x)?AA HLAA SLSolid
Art, music & design
- Is the golden ratio really everywhere? Testing a popular claimAI SLAI HLAA SLAA HLSolid
- Why does a piano have twelve notes per octave?AA SLAA HLAI SLAI HLSolid
- Why do a flute and a violin sound different on the same note?AA SLAA HLAI SLAI HLSolid
- Symmetry in Islamic geometric art: which wallpaper groups appear?AA HLAA SLAmbitious
- How are letters drawn on a screen? Bézier curvesAA HLAA SLAI HLSolid
- How accurate is the perspective in a Renaissance painting?AA SLAI SLAA HLAI HLSolid
- Why do sunflower seeds form spirals? The golden angleAA SLAA HLAI SLAI HLSolid
- What curve do straight lines make in string art?AA SLAA HLSolid
- Why are camera f-stops 1.4, 2, 2.8, 4, 5.6?AA SLAI SLAA HLAI HLAccessible
- Why do guitar frets get closer together? A geometric sequence on the fingerboardAA SLAI SLAA HLAccessible
- Why do two nearly-equal notes 'wobble'? Beats and trigonometric identitiesAA SLAA HLAI SLAI HLSolid
- Spreading beats evenly: Euclid's algorithm and musical rhythmAA SLAA HLSolid
- How fast does a speaker get quieter? Decibels, logarithms and distanceAA SLAI SLAA HLAI HLAccessible
- Are popular songs getting faster or shorter? Statistics on music dataAI SLAI HLAA SLAccessible
Games & puzzles
- How long does a game of Snakes and Ladders last?AI HLAA HLAmbitious
- Which Monopoly squares are landed on most often?AI HLAA HLAmbitious
- How big is White's advantage in chess, and does it depend on skill?AI SLAI HLAA HLSolid
- Can you beat people at rock–paper–scissors?AI SLAI HLAA SLAA HLAccessible
- What are the chances of rolling a Yahtzee in one turn?AA SLAA HLAI SLAI HLSolid
- How many attempts until the rare item? Drop rates and pity systemsAA SLAA HLAI SLAI HLSolid
- How many positions does a Rubik's Cube have?AA HLAA SLAmbitious
- How many different games of noughts and crosses are there?AA SLAA HLAI SLAI HLSolid
- How many shuffles does it take to mix a deck of cards?AI HLAA HLAI SLAmbitious
- What are the odds in a Risk battle?AA SLAA HLAI SLAI HLSolid
- Where should you build your first settlement in Catan?AA SLAI SLAA HLAI HLAccessible
Optimisation & linear programming
- What is the cheapest way to hire coaches for a school trip?AA SLAI SLAA HLAI HLAccessible
- The cheapest healthy weekly shop: a linear programming diet problemAI SLAI HLAA SLSolid
- How many of each? Planning a charity bake sale with linear programmingAA SLAI SLAccessible
- Cutting shelves from planks with the least wasteAA SLAA HLAI SLAI HLSolid
- How few staff can a café roster? An integer programming rotaAI SLAI HLAA SLSolid
- Where should a shared mast go? Minimising total distance to villagesAA HLAI HLAA SLAmbitious
- What speed minimises the total cost of a long drive?AA SLAI SLAA HLAI HLSolid
- Square or hexagonal? Packing tins into a boxAA SLAA HLAI SLSolid
Simulation & Monte Carlo methods
- One snake queue or a queue at every till? Simulating supermarket checkoutsAI SLAI HLAA SLAA HLSolid
- Estimating the area of a leaf with random pointsAA SLAI SLAA HLAI HLAccessible
- How far does a random walker get? Distance after n stepsAA HLAA SLAI HLSolid
- Why do buses come in bunches? A simulation of a bus routeAI HLAI SLAA HLAmbitious
- How many random numbers until the total passes 1? Finding e by chanceAA HLAA SLSolid
- Does the best team win? Knockout cups versus leaguesAI SLAI HLAA SLAA HLSolid
- What is a 'random' chord? Bertrand's paradox simulatedAA HLAA SLSolid
- Will the pension pot last? Monte Carlo savings with random returnsAI SLAI HLAmbitious
Cryptography & number theory
- Which typing errors does a check digit catch? ISBN-10 versus ISBN-13AA HLAA SLSolid
- Breaking a cipher with letter frequencies and chi-squaredAI SLAA SLAI HLAA HLAccessible
- Finding the key length of a Vigenère cipher with the index of coincidenceAA HLAI HLAmbitious
- How long would it take to guess your password? Counting and logarithmsAA SLAI SLAA HLAI HLAccessible
- Agreeing a secret in public: the Diffie–Hellman key exchangeAA HLAmbitious
- Which numbers are the sum of two squares?AA HLAA SLAmbitious
- Patterns in the last digits of powersAA SLAA HLSolid
- Perfect numbers and Mersenne primes: proving Euclid's ruleAA HLAmbitious
- Encrypting with matrices: when can a Hill cipher be undone?AI HLAA HLAmbitious
Numerical methods & error analysis
- Why does the ancient square-root method double the correct digits?AA SLAA HLSolid
- Trapezoidal rule or Simpson's rule? Areas under the normal curveAA SLAA HLAI SLSolid
- When does Newton–Raphson fail? Cycles, divergence and basins of attractionAA HLAA SLAmbitious
- How small must the step be? Euler's method against an exact solutionAA HLAI HLSolid
- Press cos again and again: when does fixed-point iteration converge?AA SLAA HLSolid
- How many terms do you need? Fast and slow series for πAA HLAmbitious
- What interest rate are you really paying? Solving for the rate by iterationAI SLAI HLAA SLSolid
- When the computer gets it wrong: rounding error and cancellationAA HLAA SLAI HLAmbitious
Fractals & chaos
- Infinite perimeter, finite area: designing your own snowflake curveAA SLAI SLAA HLAccessible
- From steady to chaotic: the logistic map's period doublingAA HLAI HLAmbitious
- How fractal is a fern? Box-counting dimension from photographsAI SLAA SLAI HLAA HLSolid
- Why does a random game draw the Sierpiński triangle?AA SLAA HLAI SLSolid
- Inside or out? Proving the escape radius for Julia setsAA HLAmbitious
- Fold a strip, unfold it: the dragon curveAA SLAA HLSolid
- A sponge with no volume and infinite surface: the Menger spongeAA SLAI SLAA HLAccessible
- A fractal shortcut for the travelling salesman: ordering stops along a Hilbert curveAI HLAA HLAmbitious
Matrices, transformations & Markov chains
- Does rain today predict rain tomorrow? A Markov chain for the weatherAI HLAA HLSolid
- Animating a logo with transformation matricesAI HLAA HLSolid
- Projecting a country's age structure with a Leslie matrixAI HLAmbitious
- Can letter transitions tell languages apart? Markov chains for textAI HLAA HLAI SLSolid
- Gambler's ruin: how long until someone runs out?AI HLAA HLAmbitious
- How does a game draw 3D on a flat screen? Projection matricesAI HLAA HLAmbitious
- Balancing traffic flow through a road network with linear equationsAI HLAI SLAA HLSolid
- Ranking teams that haven't all played each other: the Colley matrixAI HLAA HLAmbitious
Trigonometry, navigation & surveying
- How wide is the river? Measuring a distance you can't crossAA SLAI SLAccessible
- How far can you see from the top of a hill?AA SLAI SLAA HLSolid
- How far off do you end up? Navigating a route by bearingsAA SLAI SLAccessible
- What is the best angle to sail upwind?AA HLAA SLAI HLAmbitious
- Measuring the area of an irregular field with trianglesAA SLAI SLAccessible
- Measuring the Earth with a stick and a partner schoolAA SLAI SLAA HLSolid
- Designing an accessible ramp for a real entranceAA SLAI SLAccessible
- The hidden angles of a hip roof: 3D trigonometryAA SLAI SLAA HLSolid
Kinematics & physics-style modelling
- How high did the lift go? Integrating a phone's accelerometerAA SLAA HLAI HLSolid
- Estimating g from a video of a falling ballAA SLAI SLAccessible
- Why is a rolling marble slower than a sliding block?AA SLAI SLAA HLSolid
- Crossing a river with a current: fastest or straightest?AA HLAI HLAA SLSolid
- The path of a bike valve: parametric equations of a cycloidAA HLAI HLAmbitious
- How fast does a sledge go? Acceleration, friction and slopeAA SLAI SLAA HLSolid
- How quickly does a swing die down? Exponential or linear decay?AA SLAA HLAI HLAI SLSolid
- Modelling a bus journey between two stops from a phone's sensorsAA SLAI SLAI HLSolid
Voting, fairness & game theory
- Same ballots, different winners: comparing voting methods in a class voteAA SLAI SLAA HLAI HLAccessible
- How likely is a voting paradox? Rock–paper–scissors preferencesAA HLAI HLAmbitious
- Cutting a cake fairly when people value it differentlyAA SLAA HLSolid
- Left, right or centre? Game theory for penalty kicksAA SLAI SLAA HLAI HLSolid
- Who should pay what for a shared taxi? The Shapley valueAA SLAI SLAA HLSolid
- Turning votes into seats: D'Hondt, Sainte-Laguë and largest remainderAI SLAA SLAI HLSolid
- Secret Santa: how often does someone draw their own name?AA HLAA SLSolid
- How much should you bid in a sealed-bid auction?AA HLAA SLAmbitious
Survey-based investigations
- Do we know how long we spend on our phones? Estimates versus screen-time dataAI SLAA SLAI HLAccessible
- Does the wording change the answer? A split-sample survey experimentAI SLAI HLAA SLSolid
- Getting honest answers: the randomised response techniqueAA HLAI HLAA SLAmbitious
- How big should a school poll be? Sampling from a known populationAI SLAI HLAA SLSolid
- Can you average a 1–5 rating? Analysing Likert-scale survey answersAI SLAI HLSolid
- Walk, cycle, bus or car? Travel to school and distanceAI SLAA SLAccessible
- Who answers online surveys? Measuring non-response biasAI SLAI HLSolid
- Counting a hidden population: capture–recapture at schoolAI SLAA SLAI HLAA HLSolid
No ideas match those filters — try fewer.
How to choose between them
- Pick two or three that connect to something you already care about — a sport you play, a subject you enjoy, a decision your family is making.
- For each, check the course-fit table on the idea page: is the core mathematics in your course (AA or AI, SL or HL)?
- Check you can really get the data in the time you have: open the sources, or browse the free data sets.
- Change the question until it is yours: a different place, dataset, comparison or constraint.
- Take your shortlist to your teacher before you collect anything.
Then plan it with the timeline and planning template.
Frequently asked questions
What makes a good Maths IA topic?
A precise question with an answer, about something you genuinely care about, that needs mathematics at the level of your course, and that you can get the data for in the time you have. A narrow question explored in depth scores better than a broad one.
Are these IA ideas free?
Yes. All 239 ideas, across 24 topics, are free to read: each has a research question, the mathematics labelled AA/AI and SL/HL, data sources we have checked, an outline, the pitfalls that cost marks and ways to show personal engagement.
Which IA ideas suit AA and which suit AI?
Both courses can explore almost any context; what differs is the mathematics. AA explorations often lean on calculus, functions and proof, and AI explorations on statistics, modelling and technology. Filter the list by course, or open the AA SL, AA HL, AI SL and AI HL collections.
Can I copy a research question from this list?
Use it as a starting point, not a finished question. Change the place, the data or the comparison until it is yours, and agree it with your teacher before you collect anything. Your IA must be your own work, with any tools you used acknowledged.
Start your IA plan, free
Answer the planning questions: your question, the maths, the data and milestones back from your deadline. Then take it to your first teacher meeting.
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- Free IA self-check: guided questions give a rough band for each criterion. No upload.
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