The Math AA HL IA is a mathematical exploration. The criteria are the same as SL, but HL work should normally show more rigorous mathematics, stronger justification and clearer sophistication in Criterion E.
Clear aim, logical order, readable graphs/tables and a conclusion that answers the research question.
Define variables, use correct notation, label axes, state units and explain what each formula means.
Make the context yours: your phone, canteen, container, chain, ramp, measurements or design decision.
Compare models, test assumptions, discuss errors and explain what the answer means in the real situation.
Use course-level mathematics accurately; for HL, add derivation, proof, model comparison or numerical depth.
A Math AA HL IA should show more than a difficult-looking formula. The mathematics needs to do real work: derive a model, compare it with data, optimise a design, or explain why a simpler model fails. HL is the right place for differential equations, parametric curves, hyperbolic functions, numerical summation, arc length and model comparison.
The attached guide gives three strong HL routes. Each has an observable system, a theory-driven model and a useful endpoint. That combination matters: the IA should not stop at confirming a known equation. It should use the equation to design a drain time, estimate cable length and sag, or compare ramp shapes for fastest motion.
The best HL work explains where the model comes from. Derive the differential equation, justify the catenary against a parabola, or explain the numerical method used to approximate travel time. Use real data to test the model, then include residuals or a model-comparison metric. End with a design calculation: a hole radius for a target drain time, a cable length for a chosen sag, or a fastest ramp shape under controlled assumptions.
This feels like a real engineering IA: a bottle becomes a drainage system, and the final answer can be a designed hole size for an exact emptying time.
Research question: Can a differential equation model how water height changes as a bottle drains, and what hole size gives a target draining time?
Make small holes near the bottom of identical bottles, fill each to the same starting height and record water height every few seconds as it drains. The theory predicts that water height decreases faster when the water level is high, giving a nonlinear curve. Start with a differential equation, separate variables, integrate, and fit the model to your data. The IA becomes excellent when you use the fitted constant to solve a design problem, such as finding the hole radius needed to drain in exactly 30 seconds.
IB topics used: differential equations, integration, model fitting, optimisation, nonlinear functions
Data to collect: Water height at fixed time intervals for one or more hole radii, with repeated trials and residual analysis.
Diagram to include: Bottle cross-section with water height h, hole radius r and a height-time curve bending downward.
A hanging chain looks like a parabola, but it is not one. The IA becomes a model duel: parabola versus catenary, with real coordinate data deciding the winner.
Research question: Does a hanging chain follow a catenary more accurately than a parabola, and how can the model predict cable length and sag?
Hang a chain, necklace, cable or string between two fixed points and photograph it straight-on with a ruler or grid. Extract coordinate points along the curve, fit both a quadratic model and a catenary model, then compare residuals or sum of squared errors. The catenary can then be used to estimate sag depth and cable length for a chosen span. This is strong HL mathematics because it uses hyperbolic functions and arc length, while still being observable and measurable.
IB topics used: hyperbolic functions, curve fitting, residuals, arc length, model comparison
Data to collect: Coordinate points from a photographed chain, fitted to parabolic and catenary models.
Diagram to include: Photo grid with chain points, overlaid parabola and catenary, plus a residual comparison.
This is the classic shortest-path trap: the straight ramp is shorter, but the curve that drops faster may win. That makes it instantly interesting.
Research question: Which ramp shape gives the shortest travel time for a marble: straight, circular or cycloid-like?
Build three ramps with the same start and end points: a straight ramp, a circular arc and a cycloid-like curve. Time a marble or small ball using video or a photogate. Mathematically, represent the curves, estimate speed from vertical drop and divide each path into small sections so travel time can be approximated by a sum. The IA becomes strong when theory and experiment are compared, then limitations such as rolling friction, rotational kinetic energy and ramp construction are evaluated carefully.
IB topics used: parametric equations, numerical methods, energy, summation, optimisation
Data to collect: Travel time for three ramp shapes with identical start and end points, repeated and compared with numerical predictions.
Diagram to include: Three ramp curves between the same endpoints, with section lengths and vertical drop marked.
Choose the investigation where you can collect the cleanest data, not the one with the most dramatic title. A high-scoring IA usually has one clear independent variable, one meaningful dependent variable, enough repeats, a sensible model and an evaluation that admits what the data cannot prove. Before starting, confirm your exact subject guide, safety rules and teacher expectations.
