IA ideas library
153 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).
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
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
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