IA idea · Trigonometry, navigation & surveying

Designing an accessible ramp for a real entrance

AA SLAI SL Accessible Also in: Optimisation

Research question

For an entrance at your school, what ramp layouts (straight, dog-leg, switchback) meet your country's maximum gradient and landing rules, and which uses the least space or material?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

A practical design problem with real constraints. Gradients, angles and lengths are SL trigonometry, and comparing layouts adds a clear optimisation.

The mathematics you'll need

Gradient, angle of elevation and length
Trigonometry for slope lengths
Area and volume of material
Comparing layouts under constraints
Piecewise length functions

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Measure the rise at a real entrance and the available space; look up your country's building regulations for ramps and cite them.

GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Measure the site and state the rules.
Design a straight ramp and check it fits.
Design alternatives with landings.
Compare length, footprint and material.
Reflect on comfort and the rule's purpose.

Pitfalls that cost marks

Rules quoted without a source.
Confusing gradient (ratio) with angle.
Ignoring landings and handrails.

Showing personal engagement

Use a real entrance that needs a ramp.
Talk to someone who uses a wheelchair about comfort (with care).
Present your design to the school.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Gradient, angle of elevation and length; Trigonometry for slope lengths
AA HL	Not a natural fit	The mathematics is mainly from the AI course; at AA HL the exploration would need an AA-level approach (calculus, proof or probability theory) to reach the top of Criterion E.
AI SL	Good fit	Gradient, angle of elevation and length; Trigonometry for slope lengths
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

Level: Accessible. A good first extended piece of maths, with room to go deeper. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Take the measurements yourself, choose the place, and compare at least two methods. Explaining why you chose each position or angle is engagement an examiner can see.

Reflection (D)

Analyse measurement error: how much does a small error in an angle change the answer, which method is most robust, and would you trust the result for the decision you are making? For this idea, start with: rules quoted without a source — say how it affects your answer.

Use of mathematics (E)

SL: Right-angled trigonometry, sine and cosine rules, bearings and 3D trigonometry applied correctly to your own measurements, with an error analysis.

HL: Error propagation with derivatives, vectors in three dimensions, spherical geometry derived rather than quoted, or an optimisation of where to measure from.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Optimise a switchback layout to fit a given footprint.

Extending it for HL

Use derivatives to find how sensitive the result is to each measured angle, and choose the measuring position that minimises the error.

See a complete IA, marked

Our annotated exemplar Is my 07:40 bus late because of the rain? (AI SL) asks a different question, but shows how a complete trig & navigation exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the 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. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).