IA ideas · Mathematics: analysis and approaches HL
Maths AA HL IA ideas (115)
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.
Accessible (23)
- How far does a bouncing ball really travel?AA SLAI SLAA HLAI HLAccessible
- Why are camera f-stops 1.4, 2, 2.8, 4, 5.6?AA SLAI SLAA HLAI HLAccessible
- Where should you build your first settlement in Catan?AA SLAI SLAA HLAI HLAccessible
- Which mug keeps tea hot longest? Newton's law of cooling, comparedAA SLAA HLAI SLAI HLAccessible
- How does the shape of the daylight curve change with latitude?AA SLAI SLAA HLAI HLAccessible
- Radioactive decay with dice: does chance produce an exponential curve?AA SLAI SLAA HLAI HLAccessible
- How does bread dough rise? A logistic model in the kitchenAA SLAA HLAI SLAI HLAccessible
- Why aren't drinks cans the optimal shape?AA SLAI SLAA HLAI HLAccessible
- How fast does my heart rate recover after exercise?AA SLAI SLAA HLAI HLAccessible
- Is home advantage in football disappearing?AI SLAI HLAA SLAA HLAccessible
- How big is my local lake? Estimating area with the trapezoidal ruleAI SLAI HLAA SLAA HLAccessible
- Is a lottery ticket ever worth buying? Expected value and rolloversAA SLAA HLAI SLAI HLAccessible
- Is Moore's law still true?AA SLAI SLAA HLAI HLAccessible
- Is a longer mortgage ever the better deal?AI SLAI HLAA SLAA HLAccessible
- Are NBA players getting taller?AI SLAI HLAA SLAA HLAccessible
- Does a pendulum's period really depend on the square root of its length?AA SLAI SLAA HLAI HLAccessible
- What is the best shape for a rain gutter?AA SLAI SLAA HLAI HLAccessible
- Are elite footballers more likely to be born early in the selection year?AI SLAI HLAA SLAA HLAccessible
- Can you beat people at rock–paper–scissors?AI SLAI HLAA SLAA HLAccessible
- Modelling the sun's height from a stick's shadowAA SLAI SLAA HLAI HLAccessible
- Do students who sleep less react more slowly?AI SLAI HLAA SLAA HLAccessible
- How much does it cost to start saving ten years later?AI SLAI HLAA SLAA HLAccessible
- What is the chance of hitting every green light on my route?AI SLAI HLAA SLAA HLAccessible
Solid (68)
- How many extra tickets should an airline sell?AA SLAA HLAI SLAI HLSolid
- When might the Arctic be nearly ice-free in September?AA SLAA HLAI SLAI HLSolid
- Modelling bacterial growth in the biology labAA SLAA HLAI SLAI HLSolid
- What is the best angle to shoot a free throw?AA SLAA HLAI SLAI HLSolid
- Does Benford's law hold for the populations of the world's towns?AI SLAI HLAA HLSolid
- Where is the best seat in the cinema?AA SLAA HLAI SLAI HLSolid
- How are letters drawn on a screen? Bézier curvesAA HLAA SLAI HLSolid
- The birthday problem, extended: what about birthdays within a day?AA SLAA HLAI SLAI HLSolid
- Is BMI's “height squared” right? Evidence from OlympiansAI SLAI HLAA SLAA HLSolid
- How do doctors estimate your skin area? Comparing body-surface-area formulasAA SLAA HLAI SLAI HLSolid
- Estimating π by dropping needles: how fast does the estimate improve?AA HLAA SLAI SLAI HLSolid
- Does caffeine build up if you drink coffee every few hours?AA SLAA HLAI SLAI HLSolid
- How big is White's advantage in chess, and does it depend on skill?AI SLAI HLAA HLSolid
- How long is a coastline? Measuring fractal dimensionAA SLAA HLAI SLAI HLSolid
- Linear or quadratic drag? Testing air resistance with falling coffee filtersAA HLAI HLSolid
- The Collatz conjecture: what can we say about stopping times?AA SLAA HLAI SLAI HLSolid
- Where does e come from? Compounding more and more oftenAA SLAA HLSolid
- 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
- Exponential or logistic? Modelling my country's populationAA SLAA HLAI SLAI HLSolid
- How fast did COVID-19 spread at the start? Comparing doubling timesAA SLAA HLAI SLAI HLSolid
- Are cricket batting scores geometrically distributed?AI SLAI HLAA HLAA SLSolid
- Designing a working sundial for my latitudeAA SLAA HLAI SLAI HLSolid
- How long does a bottle take to drain? Testing Torricelli's lawAA SLAA HLAI SLAI HLSolid
- Why does a piano have twelve notes per octave?AA SLAA HLAI SLAI HLSolid
- What is the fewest number of exam sessions my school needs?AI HLAA HLAI SLSolid
- Does money buy a longer life? GDP and life expectancyAI SLAI HLAA SLAA HLSolid
- Beyond Fibonacci: what ratio do other recurrence sequences approach?AA HLAA SLSolid
- Measuring inequality: fitting Lorenz curves and computing the Gini coefficientAA SLAA HLAI SLAI HLSolid
- How often do big earthquakes happen? The Gutenberg–Richter lawAI SLAI HLAA SLAA HLSolid
- Is a hanging chain a parabola? Catenary versus quadratic modelsAA SLAA HLAI SLAI HLSolid
- Is the “hot hand” real? Streaks in my free throwsAI SLAI HLAA SLAA HLSolid
- How does a calculator work out sin(x)?AA HLAA SLSolid
- Why do a flute and a violin sound different on the same note?AA SLAA HLAI SLAI HLSolid
- Modelling the Keeling curve: trend plus seasonsAA SLAA HLAI SLAI HLSolid
- How long would a lake take to flush out a pollutant?AA HLAI HLSolid
- Do left-handed tennis players have an advantage?AI SLAI HLAA HLSolid
- The lifeguard problem: what is the fastest route to a swimmer?AA SLAA HLAI SLAI HLSolid
- Why don't long jumpers take off at 45°?AA SLAA HLAI SLAI HLSolid
- How many attempts until the rare item? Drop rates and pity systemsAA SLAA HLAI SLAI HLSolid
- A positive test result: how likely is it that you really have the condition?AA HLAA SLAI SLAI HLSolid
- The Monty Hall problem when the host is forgetfulAA SLAA HLAI SLAI HLSolid
- How many times can you fold a sheet of paper?AA SLAA HLSolid
- Why does Pascal's triangle make a fractal?AA HLAA SLSolid
- Is a penalty shoot-out fair to the team that shoots second?AA SLAA HLAI SLAI HLSolid
- How accurate is the perspective in a Renaissance painting?AA SLAI SLAA HLAI HLSolid
- Is solar power growing logistically?AA SLAA HLAI SLAI HLSolid
- What are the odds in a Risk battle?AA SLAA HLAI SLAI HLSolid
- How fast does news spread through a year group?AI SLAI HLAA SLAA HLSolid
- Is a satellite dish really a parabola, and where is its focus?AA SLAA HLSolid
- What price maximises profit at our school bake sale?AA SLAI SLAA HLAI HLSolid
- Will the sofa fit round the corner? The longest object through a corridorAA SLAA HLAI SLAI HLSolid
- What is the best tilt for a solar panel at my home?AA SLAI SLAA HLAI HLSolid
- How fast is a sprinter at each moment? Differentiating 100 m splitsAA SLAA HLAI SLAI HLSolid
- How many packs does it take to finish a sticker album?AA SLAA HLAI SLAI HLSolid
- What curve do straight lines make in string art?AA SLAA HLSolid
- Finding formulas for 1ᵏ + 2ᵏ + … + nᵏAA HLAA SLSolid
- Why do sunflower seeds form spirals? The golden angleAA SLAA HLAI SLAI HLSolid
- How much does a small edge on each point matter in tennis?AA SLAA HLAI SLAI HLSolid
- Is the golden ratio really everywhere? Testing a popular claimAI SLAI HLAA SLAA HLSolid
- How many different games of noughts and crosses are there?AA SLAA HLAI SLAI HLSolid
- Can a sum of sine waves predict the tides?AA SLAA HLAI SLAI HLSolid
- What is the volume of an egg?AA HLAI HLAA SLSolid
- Where should the 125 ml line go on a glass?AA HLAI HLSolid
- Where should a winger shoot from? Maximising the angle to goalAA SLAA HLAI SLAI HLSolid
- Will women's Olympic times ever match men's?AI SLAI HLAA SLAA HLSolid
- Is there a limit to how fast humans can run 100 m?AA SLAA HLAI SLAI HLSolid
- What are the chances of rolling a Yahtzee in one turn?AA SLAA HLAI SLAI HLSolid
Ambitious (24)
- Why does 1 + 1/4 + 1/9 + … equal π²/6?AA HLAmbitious
- How much of cycling effort goes on air resistance? A coast-down testAA HLAI HLAmbitious
- Where should I aim on a dartboard?AI SLAI HLAA HLAmbitious
- Can a simple rating system predict football results?AI SLAI HLAA HLAmbitious
- Is the euro–dollar exchange rate a random walk?AI SLAI HLAA HLAmbitious
- When should a Formula 1 driver pit? Modelling tyre degradationAA SLAI SLAA HLAI HLAmbitious
- Why do flight paths curve on the map?AA SLAA HLAI SLAI HLAmbitious
- How many shuffles does it take to mix a deck of cards?AI HLAA HLAI SLAmbitious
- Symmetry in Islamic geometric art: which wallpaper groups appear?AA HLAA SLAmbitious
- Can mathematics predict a battle? Lanchester's laws and Iwo JimaAI HLAA HLAmbitious
- Why is the main body of the Mandelbrot set a cardioid?AA HLAmbitious
- Which Monopoly squares are landed on most often?AI HLAA HLAmbitious
- How does Google rank pages? PageRank on a small websiteAI HLAA HLAmbitious
- Can the Lotka–Volterra model explain the lynx and hare cycles?AI HLAA HLAmbitious
- Designing a smooth roller-coaster drop with piecewise functionsAA SLAA HLAI SLAI HLAmbitious
- The product of a regular polygon's diagonalsAA HLAmbitious
- How does RSA encryption work, and how long would it take to break a small key?AA HLAA SLAmbitious
- How many positions does a Rubik's Cube have?AA HLAA SLAmbitious
- How fast would flu spread through my school? An SIR modelAI HLAA HLAmbitious
- When should you stop looking? The 37% rule for choosing the bestAA HLAA SLAI HLAmbitious
- How long does a skydiver take to reach terminal velocity?AA HLAI HLAmbitious
- How long does a game of Snakes and Ladders last?AI HLAA HLAmbitious
- How strongly is global temperature linked to ln(CO₂)?AI SLAI HLAA SLAA HLAmbitious
- Voronoi diagrams in a grid city: Euclidean or taxicab distance?AI SLAI HLAA HLAmbitious
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.
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.
Free: the IA checklist an examiner uses
Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it.
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 · All 153 ideas