IA ideas · Mathematics: analysis and approaches SL
Maths AA SL IA ideas (174)
Analysis and approaches SL explorations score well on Criterion E when they use calculus, functions, sequences or probability that genuinely answer the question — correctly, and explained so a classmate could follow. You do not need HL mathematics; you need SL mathematics used with understanding.
Every idea below has been checked as a fit for AA SL: the core technique is in the SL course, or can be reached with technology you explain. Start with an Accessible idea if this is your first extended piece of maths; the Ambitious ones suit students who want to stretch into mathematics beyond the syllabus.
Good AA SL explorations usually rest on one technique doing real work: differentiation to optimise a shape or a cost, integration for an area or a volume, an exponential or sine model for growth or a repeating pattern, sequences and series for repeated change, or a probability worked out exactly and then checked by simulation. Matrices, graph theory and χ² tests are not in the AA course; you can still use them if you explain them clearly as new mathematics.
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 (55)
- Designing an accessible ramp for a real entranceTrig & navigation
- How far does a bouncing ball really travel?Modelling
- Breaking a cipher with letter frequencies and chi-squaredCryptography
- Which function best describes the arch of a famous bridge?Modelling
- 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
- How many of each? Planning a charity bake sale with linear programmingOptimisation
- 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
- Estimating g from a video of a falling ballKinematics
- 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
- How much does a cold month cost my family?Environment
- Do more doctors mean fewer infant deaths?Health & biology
- Does light really follow an inverse-square law?Modelling
- 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
- Can height predict lung capacity?Health & biology
- Which method measures a building's height most accurately?Geometry & Voronoi
- 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
- Does my phone battery really drain in a straight line?Modelling
- Contract or SIM-only? The real cost of a new phoneFinance
- Has my pocket money kept up with inflation?Finance
- Where should a pool ball hit the cushion? The geometry of bank shotsSport
- What is the best shape for a rain gutter?Calculus
- Are elite footballers more likely to be born early in the selection year?Statistics
- How wide is the river? Measuring a distance you can't crossTrig & navigation
- 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
- Why do stopping distances grow so fast? Modelling the Highway Code tableModelling
- Are our summers getting wetter or drier?Statistics
- Measuring the area of an irregular field with trianglesTrig & navigation
- 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
- Walk, cycle, bus or car? Travel to school and distanceSurveys
- How fast do cars lose value? Modelling depreciation from listingsFinance
- How far off do you end up? Navigating a route by bearingsTrig & navigation
Solid (104)
- How many extra tickets should an airline sell?Probability
- When might the Arctic be nearly ice-free in September?Environment
- Can statistics tell two authors apart?Statistics
- 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
- 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
- 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
- Counting a hidden population: capture–recapture at schoolSurveys
- Why does a random game draw the Sierpiński triangle?Fractals & chaos
- The cheapest healthy weekly shop: a linear programming diet problemOptimisation
- How long is a coastline? Measuring fractal dimensionPure maths
- 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
- 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 much is being near a station worth?Statistics
- How big should a school poll be? Sampling from a known populationSurveys
- 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
- Patterns in the last digits of powersCryptography
- The lifeguard problem: what is the fastest route to a swimmer?Calculus
- How high did the lift go? Integrating a phone's accelerometerKinematics
- 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
- Why is a rolling marble slower than a sliding block?Kinematics
- 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 the wording change the answer? A split-sample survey experimentSurveys
- 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
- Turning votes into seats: D'Hondt, Sainte-Laguë and largest remainderVoting & game theory
- 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
- Will I ever repay my student loan?Finance
- 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
- Trapezoidal rule or Simpson's rule? Areas under the normal curveNumerical methods
- Can you predict a tree's height from its trunk?Modelling
- How much vaccination is enough to stop measles?Health & biology
- What is the volume of an egg?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 (15)
- When should a Formula 1 driver pit? Modelling tyre degradationSport
- Why do flight paths curve on the map?Geometry & Voronoi
- Symmetry in Islamic geometric art: which wallpaper groups appear?Art & music
- Where should a shared mast go? Minimising total distance to villagesOptimisation
- Getting honest answers: the randomised response techniqueSurveys
- Designing a smooth roller-coaster drop with piecewise functionsCalculus
- 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 much should you bid in a sealed-bid auction?Voting & game theory
- When should you stop looking? The 37% rule for choosing the bestProbability
- Which numbers are the sum of two squares?Cryptography
- How strongly is global temperature linked to ln(CO₂)?Environment
- When does Newton–Raphson fail? Cycles, divergence and basins of attractionNumerical methods
AA SL mathematics that works well in an IA
- Optimisation with differentiation (maxima, minima, rates of change)
- Areas and definite integrals, including the trapezoidal rule
- Exponential, logarithmic and trigonometric models fitted to data
- Arithmetic and geometric sequences and series
- Binomial and normal distributions; regression and correlation
Revise the techniques: Applications of differentiation questions · Integration questions · Exponential log · Sequences and series questions · Normal distribution questions. How AA SL differs from the other courses: AA vs AI, SL vs HL.
Less common types of IA that suit AA SL: Optimisation & linear programming · Numerical methods & error analysis · Trigonometry, navigation & surveying · Kinematics & physics-style modelling · Fractals & chaos · Simulation & Monte Carlo methods.
Frequently asked questions
What is a good Maths AA SL IA topic?
A question you care about that needs SL calculus, functions or probability to answer: optimising a real object, modelling data with an exponential or trigonometric function, or a probability question you can test. Keep the maths at a level you can explain fully.
Can AA SL students use statistics in their IA?
Yes. Correlation, regression and the normal and binomial distributions are in AA SL. Statistical tests such as chi-squared are in the AI course; you may use them if you explain them, but make sure the exploration still shows AA-level mathematics.
Is calculus required for an AA SL IA?
No. It is often the most natural fit, but a strong exploration built on sequences, trigonometry or probability can reach the top of Criterion E if the mathematics is relevant, correct and well understood.
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