2026-06-12 Updated 2026-09-05 12 min read IBretake Editorial Team

Math AA HL IA ideas: differential equations, catenaries and fastest paths

Make the assumptions visible in an HL model

An HL exploration can gain depth by testing a model's assumptions, not merely adding more advanced notation. For a simple population model dP/dt = kP, the assumption is that the growth rate per individual stays constant. The solution P(t) = P(0)e^kt is mathematically useful, but unlimited exponential growth may be implausible over a long interval.

State the units of t, P and k.
Explain how k would be estimated from legitimate data.
Check whether the same estimate describes different time intervals.
Compare an alternative only if you can explain its mathematics and assumptions.

Check what the parameter means numerically

With an illustrative k = 0.1 per year, the model's doubling time is ln(2)/0.1, approximately 6.93 years. This is a consequence of the assumed model—not a forecast for a real population. Discuss when the assumption could fail and what evidence would reveal the failure.

For a simpler starting point, see SL model comparison. The feasibility checklist and evaluation guide help keep the investigation explainable and genuinely your own.

IA weighting and scoring criteria

Grade weighting: 20% of the final subject grade Total marks: 20 marks

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.

A Presentation · 4 marks

Clear aim, logical order, readable graphs/tables and a conclusion that answers the research question.

B Mathematical communication · 4 marks

Define variables, use correct notation, label axes, state units and explain what each formula means.

C Personal engagement · 3 marks

Make the context yours: your phone, canteen, container, chain, ramp, measurements or design decision.

D Reflection · 3 marks

Compare models, test assumptions, discuss errors and explain what the answer means in the real situation.

E Use of mathematics · 6 marks

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.

What the IA is trying to do: For Math AA HL, the IA is a mathematical exploration worth 20% of the final subject grade. It still uses the same idea of personal mathematical inquiry, but the mathematics should be richer, better justified and more sophisticated than an SL treatment.

What a strong IA in this subject shows

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.

Three go-to Math AA HL IA ideas

1. Water draining from a bottle using differential equations

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.

Core model

\frac{d h}{d t} = - k \sqrt{h}

\int h^{- \frac{1}{2}} d h = - k \int d t

2 \sqrt{h} = - k t + C

h (t) = {(A - B t)}^{2}

A_{hole} = π r^{2}

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.

Guide rating: Grade potential: very high; difficulty: moderate; data quality: high.

2. Catenary curve of a hanging chain or cable

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.

Core model

y = a x^{2} + b x + c

y = a cosh (\frac{x}{a}) + c

SSE = \sum_{i = 1}^{n} {(y_{i} - {\hat{y}}_{i})}^{2}

L = \int_{x_{1}}^{x_{2}} \sqrt{1 + {(\frac{d y}{d x})}^{2}} d x

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.

Guide rating: Grade potential: very high; difficulty: moderate; data quality: very high.

3. Fastest ramp shape for a rolling marble

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, compare the ideal sliding point-mass speed with the solid-sphere rolling model, 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.

Core model

x (θ) = a (θ - sin θ)

y (θ) = a (1 - cos θ)

v_{point} = \sqrt{2 g Δ h} (ideal sliding comparison)

v_{sphere} = \sqrt{\frac{10}{7} g Δ h} (solid sphere, rolling without slipping)

T \approx \sum_{i = 1}^{n} \frac{Δ s_{i}}{v_{i}}

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.

Guide rating: Grade potential: high; difficulty: challenging; data quality: medium-high.

How to choose from these ideas

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.

Sources checked

Editorial review

Reviewed against the cited official IB subject and assessment sources. This article provides planning examples, not examiner advice or ready-to-submit coursework.

See something outdated? Email support@ibretake.com so the editorial team can check and correct it.

This guide is educational planning content, not official IB assessment advice. Always follow your current subject guide, teacher instructions, school safety rules and coordinator deadlines.

Math AA HL IA ideas: differential equations, catenaries and fastest paths

Make the assumptions visible in an HL model

Check what the parameter means numerically

IA weighting and scoring criteria

What a strong IA in this subject shows

Three go-to Math AA HL IA ideas

1. Water draining from a bottle using differential equations

2. Catenary curve of a hanging chain or cable

3. Fastest ramp shape for a rolling marble

How to choose from these ideas

Sources checked

Editorial review

More IA idea articles