IA ideas · Mathematics: applications and interpretation HL
Maths AI HL IA ideas (181)
At HL, Criterion E asks for mathematics commensurate with the HL course that is sophisticated and rigorous. For applications and interpretation that often means graph theory and network algorithms, differential equations and Euler's method, Markov chains and matrices, or statistical inference beyond the SL tests.
All the ideas below fit AI HL. Where an idea is also listed for SL, the idea page shows what makes the HL version HL. The best AI HL explorations use technology heavily and still explain every step of the mathematics it carries out.
At AI HL the extra mathematics often decides which idea to choose. Matrices and Markov chains suit anything that moves between states, such as weather, brand loyalty or a board game; graph theory suits networks, from delivery rounds to timetables; differential equations with Euler's method or slope fields suit populations and epidemics; and the Poisson distribution, confidence intervals and non-linear regression suit data you collect yourself.
Next step: shortlist ideas with the free IA topic explorer, which filters starting points by course, interest and how easy the data is to get. For feedback on your own draft, the IA package (€39 once) gives you 5 AI feedback credits and every annotated exemplar in full.
Accessible (41)
- How far does a bouncing ball really travel?Modelling
- Breaking a cipher with letter frequencies and chi-squaredCryptography
- Why are camera f-stops 1.4, 2, 2.8, 4, 5.6?Art & music
- Where should you build your first settlement in Catan?Games & puzzles
- Same ballots, different winners: comparing voting methods in a class voteVoting & game theory
- Which mug keeps tea hot longest? Newton's law of cooling, comparedModelling
- How does the shape of the daylight curve change with latitude?Environment
- Is deforestation in [a country] slowing down?Environment
- Radioactive decay with dice: does chance produce an exponential curve?Probability
- How does bread dough rise? A logistic model in the kitchenDifferential equations
- Why aren't drinks cans the optimal shape?Calculus
- How fast does my heart rate recover after exercise?Health & biology
- Is home advantage in football disappearing?Statistics
- How much does a cold month cost my family?Environment
- Do more doctors mean fewer infant deaths?Health & biology
- Is crime seasonal? Testing monthly crime countsStatistics
- How big is my local lake? Estimating area with the trapezoidal ruleCalculus
- Is a lottery ticket ever worth buying? Expected value and rolloversProbability
- Can height predict lung capacity?Health & biology
- Estimating the area of a leaf with random pointsSimulation
- Is Moore's law still true?Modelling
- Is a longer mortgage ever the better deal?Finance
- Are NBA players getting taller?Statistics
- How long would it take to guess your password? Counting and logarithmsCryptography
- Does a pendulum's period really depend on the square root of its length?Modelling
- Does my phone battery really drain in a straight line?Modelling
- Has my pocket money kept up with inflation?Finance
- What is the best shape for a rain gutter?Calculus
- Are elite footballers more likely to be born early in the selection year?Statistics
- Can you beat people at rock–paper–scissors?Games & puzzles
- What is the cheapest way to hire coaches for a school trip?Optimisation
- Do we know how long we spend on our phones? Estimates versus screen-time dataSurveys
- Modelling the sun's height from a stick's shadowGeometry & Voronoi
- Do students who sleep less react more slowly?Statistics
- Are popular songs getting faster or shorter? Statistics on music dataArt & music
- How fast does a speaker get quieter? Decibels, logarithms and distanceArt & music
- Are our summers getting wetter or drier?Statistics
- How much does it cost to start saving ten years later?Finance
- What is the chance of hitting every green light on my route?Probability
- Do TV series really get worse over time?Statistics
- How fast do cars lose value? Modelling depreciation from listingsFinance
Solid (102)
- How many extra tickets should an airline sell?Probability
- Animating a logo with transformation matricesMatrices & Markov
- When might the Arctic be nearly ice-free in September?Environment
- Can statistics tell two authors apart?Statistics
- Modelling bacterial growth in the biology labHealth & biology
- What is the best angle to shoot a free throw?Sport
- Why do two nearly-equal notes 'wobble'? Beats and trigonometric identitiesArt & music
- Does Benford's law hold for the populations of the world's towns?Statistics
- Where is the best seat in the cinema?Calculus
- How are letters drawn on a screen? Bézier curvesArt & music
- How much does rain reduce bike-share use?Environment
- The birthday problem, extended: what about birthdays within a day?Probability
- Is BMI's “height squared” right? Evidence from OlympiansHealth & biology
- How do doctors estimate your skin area? Comparing body-surface-area formulasHealth & biology
- How fractal is a fern? Box-counting dimension from photographsFractals & chaos
- Will one phone brand take over? Brand loyalty as a Markov chainNetworks & graphs
- Estimating π by dropping needles: how fast does the estimate improve?Probability
- Modelling a bus journey between two stops from a phone's sensorsKinematics
- How few staff can a café roster? An integer programming rotaOptimisation
- Does caffeine build up if you drink coffee every few hours?Differential equations
- Are arrivals at the school canteen random? A Poisson model of a queueProbability
- Counting a hidden population: capture–recapture at schoolSurveys
- The cheapest healthy weekly shop: a linear programming diet problemOptimisation
- How big is White's advantage in chess, and does it depend on skill?Games & puzzles
- What is the shortest route to walk every corridor in my school?Networks & graphs
- How long is a coastline? Measuring fractal dimensionPure maths
- Linear or quadratic drag? Testing air resistance with falling coffee filtersDifferential equations
- The Collatz conjecture: what can we say about stopping times?Pure maths
- Exponential or logistic? Modelling my country's populationModelling
- How fast did COVID-19 spread at the start? Comparing doubling timesHealth & biology
- Are cricket batting scores geometrically distributed?Sport
- Crossing a river with a current: fastest or straightest?Kinematics
- How quickly does a swing die down? Exponential or linear decay?Kinematics
- Designing a working sundial for my latitudeGeometry & Voronoi
- How long does a bottle take to drain? Testing Torricelli's lawModelling
- Why does a piano have twelve notes per octave?Art & music
- How small must the step be? Euler's method against an exact solutionNumerical methods
- What is the fewest number of exam sessions my school needs?Networks & graphs
- How connected is my year group? Analysing a friendship networkNetworks & graphs
- Does money buy a longer life? GDP and life expectancyStatistics
- Measuring inequality: fitting Lorenz curves and computing the Gini coefficientFinance
- How often do big earthquakes happen? The Gutenberg–Richter lawStatistics
- Is a hanging chain a parabola? Catenary versus quadratic modelsModelling
- Is the “hot hand” real? Streaks in my free throwsProbability
- How much is being near a station worth?Statistics
- How big should a school poll be? Sampling from a known populationSurveys
- Why do a flute and a violin sound different on the same note?Art & music
- Modelling the Keeling curve: trend plus seasonsEnvironment
- Does the best team win? Knockout cups versus leaguesSimulation
- How long would a lake take to flush out a pollutant?Differential equations
- Should I lease or buy my first car?Finance
- Do left-handed tennis players have an advantage?Statistics
- The lifeguard problem: what is the fastest route to a swimmer?Calculus
- How high did the lift go? Integrating a phone's accelerometerKinematics
- Can you average a 1–5 rating? Analysing Likert-scale survey answersSurveys
- What interest rate are you really paying? Solving for the rate by iterationNumerical methods
- Why don't long jumpers take off at 45°?Sport
- How many attempts until the rare item? Drop rates and pity systemsGames & puzzles
- Can letter transitions tell languages apart? Markov chains for textMatrices & Markov
- A positive test result: how likely is it that you really have the condition?Health & biology
- What is the cheapest way to connect villages with fibre broadband?Networks & graphs
- The Monty Hall problem when the host is forgetfulProbability
- What speed minimises the total cost of a long drive?Optimisation
- One snake queue or a queue at every till? Simulating supermarket checkoutsSimulation
- Left, right or centre? Game theory for penalty kicksVoting & game theory
- Is a penalty shoot-out fair to the team that shoots second?Probability
- How accurate is the perspective in a Renaissance painting?Art & music
- Cutting shelves from planks with the least wasteOptimisation
- Does the wording change the answer? A split-sample survey experimentSurveys
- Does rain today predict rain tomorrow? A Markov chain for the weatherMatrices & Markov
- How far does a random walker get? Distance after n stepsSimulation
- Is solar power growing logistically?Environment
- What are the odds in a Risk battle?Games & puzzles
- How fast does news spread through a year group?Differential equations
- What price maximises profit at our school bake sale?Finance
- Turning votes into seats: D'Hondt, Sainte-Laguë and largest remainderVoting & game theory
- Will the sofa fit round the corner? The longest object through a corridorCalculus
- What is the best tilt for a solar panel at my home?Environment
- How fast is a sprinter at each moment? Differentiating 100 m splitsCalculus
- How many packs does it take to finish a sticker album?Probability
- Will I ever repay my student loan?Finance
- Why do sunflower seeds form spirals? The golden angleArt & music
- How much does a small edge on each point matter in tennis?Probability
- Is the golden ratio really everywhere? Testing a popular claimArt & music
- How many different games of noughts and crosses are there?Games & puzzles
- Can a sum of sine waves predict the tides?Environment
- Balancing traffic flow through a road network with linear equationsMatrices & Markov
- Planning the shortest delivery round: bounds for the travelling salesman problemNetworks & graphs
- Can you predict a tree's height from its trunk?Modelling
- Which Underground station is the most central?Networks & graphs
- How much vaccination is enough to stop measles?Health & biology
- What is the volume of an egg?Calculus
- Would a Voronoi diagram have pointed John Snow to the Broad Street pump?Geometry & Voronoi
- Where should my town put its next defibrillator? A Voronoi analysisGeometry & Voronoi
- Do school catchment areas match the Voronoi diagram of the schools?Geometry & Voronoi
- Who lives furthest from a supermarket? Food access with Voronoi diagramsGeometry & Voronoi
- Who answers online surveys? Measuring non-response biasSurveys
- Where should the 125 ml line go on a glass?Calculus
- Where should a winger shoot from? Maximising the angle to goalSport
- Will women's Olympic times ever match men's?Sport
- Is there a limit to how fast humans can run 100 m?Sport
- What are the chances of rolling a Yahtzee in one turn?Games & puzzles
Ambitious (38)
- Are Atlantic hurricanes becoming more frequent?Environment
- How much of cycling effort goes on air resistance? A coast-down testDifferential equations
- Ranking teams that haven't all played each other: the Colley matrixMatrices & Markov
- How likely is a voting paradox? Rock–paper–scissors preferencesVoting & game theory
- The path of a bike valve: parametric equations of a cycloidKinematics
- Where should I aim on a dartboard?Sport
- Can a simple rating system predict football results?Sport
- Is the euro–dollar exchange rate a random walk?Finance
- When should a Formula 1 driver pit? Modelling tyre degradationSport
- How many flights does it take to get anywhere?Networks & graphs
- Gambler's ruin: how long until someone runs out?Matrices & Markov
- Why do flight paths curve on the map?Geometry & Voronoi
- Encrypting with matrices: when can a Hill cipher be undone?Cryptography
- How many shuffles does it take to mix a deck of cards?Games & puzzles
- Can mathematics predict a battle? Lanchester's laws and Iwo JimaDifferential equations
- Projecting a country's age structure with a Leslie matrixMatrices & Markov
- From steady to chaotic: the logistic map's period doublingFractals & chaos
- Which Monopoly squares are landed on most often?Games & puzzles
- How does Google rank pages? PageRank on a small websiteNetworks & graphs
- How does a game draw 3D on a flat screen? Projection matricesMatrices & Markov
- Where should a shared mast go? Minimising total distance to villagesOptimisation
- Can the Lotka–Volterra model explain the lynx and hare cycles?Differential equations
- Getting honest answers: the randomised response techniqueSurveys
- Will the pension pot last? Monte Carlo savings with random returnsSimulation
- Designing a smooth roller-coaster drop with piecewise functionsCalculus
- When the computer gets it wrong: rounding error and cancellationNumerical methods
- What is the best angle to sail upwind?Trig & navigation
- How fast would flu spread through my school? An SIR modelDifferential equations
- When should you stop looking? The 37% rule for choosing the bestProbability
- How long does a skydiver take to reach terminal velocity?Differential equations
- How long does a game of Snakes and Ladders last?Games & puzzles
- A fractal shortcut for the travelling salesman: ordering stops along a Hilbert curveFractals & chaos
- How strongly is global temperature linked to ln(CO₂)?Environment
- Finding the key length of a Vigenère cipher with the index of coincidenceCryptography
- Which airport is really closest? Voronoi diagrams on a curved EarthGeometry & Voronoi
- Voronoi diagrams in a grid city: Euclidean or taxicab distance?Geometry & Voronoi
- Should bigger hospitals get bigger regions? Weighted Voronoi diagramsGeometry & Voronoi
- Why do buses come in bunches? A simulation of a bus routeSimulation
AI HL mathematics that works well in an IA
- Graph theory: minimum spanning trees, Chinese postman, travelling salesman bounds
- Differential equations with slope fields, Euler's method and coupled systems
- Markov chains, transition matrices and steady states
- Poisson distributions, confidence intervals and further hypothesis tests
- Log-log and semi-log linearisation of models
Revise the techniques: Graph theory · Differential equations · Markov chains · Normal poisson · Slope fields. How AI HL differs from the other courses: AA vs AI, SL vs HL.
Less common types of IA that suit AI HL: Matrices, transformations & Markov chains · Simulation & Monte Carlo methods · Survey-based investigations · Kinematics & physics-style modelling · Optimisation & linear programming · Numerical methods & error analysis.
Frequently asked questions
What makes an AI HL IA different from an AI SL one?
Only Criterion E: at HL the top bands expect HL-level mathematics used with sophistication and rigour. A regression line and a chi-squared test alone are SL-level.
Are graph theory IAs good for AI HL?
Yes. Real networks (a school, a transport map, a delivery round) give you algorithms to run, bounds to calculate and decisions to reflect on, all at HL level.
Can an AI HL IA use calculus?
Yes. Differential equations, Euler's method and coupled systems are in AI HL, and modelling with them is a strong route to the top of Criterion E.
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.
Your plan is kept in this browser. A free account keeps it with your drafts on any device: Join free · Sign in
- Free IA self-check: guided questions give a rough band for each criterion. No upload.
- The A–E checklist as a PDF, free by email.
- Have a draft? AI feedback against A–E (Criterion A free), or a one-to-one review with Pete (€100).
By course: AA SL · AA HL · AI SL · AI HL · 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 · All 239 ideas