IA ideas · Mathematics: analysis and approaches HL
Maths AA HL IA ideas (181)
At HL, Criterion E asks for mathematics commensurate with the HL course that is sophisticated and rigorous. For analysis and approaches that usually means proof, differential equations solved analytically, series, complex numbers or calculus beyond the SL techniques — used because the question needs them.
The ideas below all fit AA HL. Many are listed for SL too; for those, the HL route is in the “Course fit” and “Extending it for HL” sections of each idea page. Choose one where an HL technique does real work, not one where it is bolted on at the end.
At AA HL the extra mathematics often decides which idea to choose. Proof by induction suits a pattern you can generalise; a Maclaurin series suits an approximation you can test against exact values; complex numbers and De Moivre's theorem suit rotations, spirals and roots of unity; Bayes' theorem and continuous random variables suit probability questions; and differential equations solved by separating variables suit cooling, growth and motion with air resistance.
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 (32)
- 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
- 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
- Why do guitar frets get closer together? A geometric sequence on the fingerboardArt & music
- How fast does my heart rate recover after exercise?Health & biology
- Is home advantage in football disappearing?Statistics
- Infinite perimeter, finite area: designing your own snowflake curveFractals & chaos
- How big is my local lake? Estimating area with the trapezoidal ruleCalculus
- Is a lottery ticket ever worth buying? Expected value and rolloversProbability
- A sponge with no volume and infinite surface: the Menger spongeFractals & chaos
- 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
- 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
- Modelling the sun's height from a stick's shadowGeometry & Voronoi
- Do students who sleep less react more slowly?Statistics
- How fast does a speaker get quieter? Decibels, logarithms and distanceArt & music
- 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
Solid (104)
- 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
- Why does the ancient square-root method double the correct digits?Numerical methods
- 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
- What is a 'random' chord? Bertrand's paradox simulatedSimulation
- Where is the best seat in the cinema?Calculus
- How are letters drawn on a screen? Bézier curvesArt & music
- 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
- Estimating π by dropping needles: how fast does the estimate improve?Probability
- Does caffeine build up if you drink coffee every few hours?Differential equations
- Counting a hidden population: capture–recapture at schoolSurveys
- Why does a random game draw the Sierpiński triangle?Fractals & chaos
- How big is White's advantage in chess, and does it depend on skill?Games & puzzles
- 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
- Where does e come from? Compounding more and more oftenFinance
- Why is 355/113 such a good approximation to π?Pure maths
- How many primes are there below n? Testing the prime number theoremPure 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
- Fold a strip, unfold it: the dragon curveFractals & chaos
- How long does a bottle take to drain? Testing Torricelli's lawModelling
- Measuring the Earth with a stick and a partner schoolTrig & navigation
- Why does a piano have twelve notes per octave?Art & music
- Spreading beats evenly: Euclid's algorithm and musical rhythmArt & 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
- Cutting a cake fairly when people value it differentlyVoting & game theory
- Press cos again and again: when does fixed-point iteration converge?Numerical methods
- Does money buy a longer life? GDP and life expectancyStatistics
- Beyond Fibonacci: what ratio do other recurrence sequences approach?Pure maths
- 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
- The hidden angles of a hip roof: 3D trigonometryTrig & navigation
- Is the “hot hand” real? Streaks in my free throwsProbability
- How does a calculator work out sin(x)?Pure maths
- How far can you see from the top of a hill?Trig & navigation
- How many random numbers until the total passes 1? Finding e by chanceSimulation
- Why do a flute and a violin sound different on the same note?Art & music
- Which typing errors does a check digit catch? ISBN-10 versus ISBN-13Cryptography
- 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
- Patterns in the last digits of powersCryptography
- 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
- Why don't long jumpers take off at 45°?Sport
- How many attempts until the rare item? Drop rates and pity systemsGames & puzzles
- Why is a rolling marble slower than a sliding block?Kinematics
- 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
- 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
- Square or hexagonal? Packing tins into a boxOptimisation
- How many times can you fold a sheet of paper?Modelling
- Why does Pascal's triangle make a fractal?Pure maths
- 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 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
- Is a satellite dish really a parabola, and where is its focus?Geometry & Voronoi
- What price maximises profit at our school bake sale?Finance
- Secret Santa: how often does someone draw their own name?Voting & game theory
- Who should pay what for a shared taxi? The Shapley valueVoting & game theory
- How fast does a sledge go? Acceleration, friction and slopeKinematics
- 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
- What curve do straight lines make in string art?Art & music
- Finding formulas for 1ᵏ + 2ᵏ + … + nᵏPure maths
- 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
- Trapezoidal rule or Simpson's rule? Areas under the normal curveNumerical methods
- What is the volume of an egg?Calculus
- 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 (45)
- Why does 1 + 1/4 + 1/9 + … equal π²/6?Pure maths
- 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
- Agreeing a secret in public: the Diffie–Hellman key exchangeCryptography
- 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
- 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
- How many terms do you need? Fast and slow series for πNumerical methods
- Symmetry in Islamic geometric art: which wallpaper groups appear?Art & music
- Inside or out? Proving the escape radius for Julia setsFractals & chaos
- Can mathematics predict a battle? Lanchester's laws and Iwo JimaDifferential equations
- From steady to chaotic: the logistic map's period doublingFractals & chaos
- Why is the main body of the Mandelbrot set a cardioid?Pure maths
- Which Monopoly squares are landed on most often?Games & puzzles
- How does Google rank pages? PageRank on a small websiteNetworks & graphs
- Perfect numbers and Mersenne primes: proving Euclid's ruleCryptography
- 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
- Designing a smooth roller-coaster drop with piecewise functionsCalculus
- The product of a regular polygon's diagonalsPure maths
- When the computer gets it wrong: rounding error and cancellationNumerical methods
- How does RSA encryption work, and how long would it take to break a small key?Pure maths
- How many positions does a Rubik's Cube have?Games & puzzles
- What is the best angle to sail upwind?Trig & navigation
- How fast would flu spread through my school? An SIR modelDifferential equations
- How much should you bid in a sealed-bid auction?Voting & game theory
- 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
- Which numbers are the sum of two squares?Cryptography
- How strongly is global temperature linked to ln(CO₂)?Environment
- Finding the key length of a Vigenère cipher with the index of coincidenceCryptography
- Voronoi diagrams in a grid city: Euclidean or taxicab distance?Geometry & Voronoi
- When does Newton–Raphson fail? Cycles, divergence and basins of attractionNumerical methods
- Why do buses come in bunches? A simulation of a bus routeSimulation
AA HL mathematics that works well in an IA
- Differential equations by separation of variables
- Proof by induction or contradiction
- Maclaurin series and approximation
- Complex numbers and De Moivre's theorem
- Integration by parts, substitution and volumes of revolution
- Bayes' theorem and continuous random variables
Revise the techniques: Differential equations · Proof by induction · Maclaurin series · Complex numbers · Volumes revolution. How AA HL differs from the other courses: AA vs AI, SL vs HL.
Less common types of IA that suit AA HL: Cryptography & number theory · Numerical methods & error analysis · Fractals & chaos · Optimisation & linear programming · Voting, fairness & game theory · Simulation & Monte Carlo methods.
Frequently asked questions
What makes an AA HL IA different from an SL one?
Only Criterion E: the top bands expect HL-level mathematics that is sophisticated and rigorous. The other four criteria are the same.
Should an AA HL IA be a proof?
It doesn't have to be. Proof-based explorations suit AA HL well, but modelling and optimisation IAs can reach the top of E too if they use HL techniques such as differential equations or series with rigour.
Is a differential equations IA a good choice for AA HL?
Often, yes: setting up a DE from stated assumptions, solving it analytically and testing it against data uses HL mathematics naturally and gives plenty to reflect on.
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