Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

How does the shape of the daylight curve change with latitude?

AA SLAI SLAA HLAI HL Accessible Also in: Modelling, Geometry & Voronoi

Research question

When daylight hours through the year are modelled with sinusoids for cities at different latitudes, how does the amplitude depend on latitude, and is the sinusoidal model equally good everywhere?

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Why it makes a good exploration

Modelling daylight with a sine curve is common; comparing many latitudes and modelling the amplitude itself as a function of latitude makes it original.

The mathematics you'll need

Sinusoidal models: amplitude, period, phase, vertical shift
Fitting with technology
Modelling amplitude against latitude
Residuals near the poles

Modelling the data, step by step

This idea usually needs a sinusoidal model. See it worked step by step, with a criterion tip at every step: Sinusoidal y = a sin(b(x − c)) + d · Choosing and comparing models.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

Where the data comes from

Day lengths on the 1st and 15th of each month for 6–8 cities from timeanddate.com.

timeanddate.com sun calculator — Sunrise, sunset and day length for any city and date.

A possible outline

Explain why day length varies.
Fit sinusoids for each city.
Plot amplitude against latitude and model it.
Test where the sinusoid fails (high latitudes).
Reflect on the astronomy behind the shape.

Pitfalls that cost marks

Only one city (too common).
Mixing time zones and daylight saving time.
Wrong period (use 365.25 days).

Showing personal engagement

Include your home city and relatives' cities.
Predict the amplitude for an unseen city.
Explain polar day and night with your model.

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	Sinusoidal models: amplitude, period, phase, vertical shift; Fitting with technology
AA HL	Fits, but add an HL technique	Sinusoidal models: amplitude, period, phase, vertical shift; Fitting with technology
AI SL	Good fit	Sinusoidal models: amplitude, period, phase, vertical shift; Fitting with technology
AI HL	Fits, but add an HL technique	Sinusoidal models: amplitude, period, phase, vertical shift; Fitting with technology

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

Reflection (D)

For this idea, start with the weak points: only one city (too common); and mixing time zones and daylight saving time. Say how each one affects your answer.

Use of mathematics (E)

SL: sinusoidal models: amplitude, period, phase, vertical shift; fitting with technology; modelling amplitude against latitude; residuals near the poles, each explained in your own words and carried out correctly.

How environment explorations reach the top bands in general: Environment & climate IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Compare your amplitude–latitude model with the astronomical formula using the sunrise equation.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL environment exploration adds.

See a complete IA, marked

When is the sea warm enough for our swimming club? A sinusoidal model of sea temperature (AI SL) explores a question like this one, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AI SL, AA HL and AI HL. We rate it Accessible: a good first extended piece of maths, with room to go deeper.

What mathematics does this IA use?

Sinusoidal models: amplitude, period, phase, vertical shift, fitting with technology, modelling amplitude against latitude and residuals near the poles.

Where can I get data for this IA?

Day lengths on the 1st and 15th of each month for 6–8 cities from timeanddate.com. Checked sources and tools: timeanddate.com sun calculator.

Turn this idea into your IA

The planner opens with “How does the shape of the daylight curve change with latitude?” filled in: the question, the maths and the data sources.

Plan this IA free

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