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

Modelling the Keeling curve: trend plus seasons

AA SLAA HLAI SLAI HL Solid Also in: Modelling

Research question

Can monthly Mauna Loa CO₂ be modelled as a smooth trend (quadratic or exponential) plus a sine wave with a one-year period, and how well does a model fitted to 1960–2010 predict the last decade?

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Why it makes a good exploration

The Keeling curve rises and wiggles: the wiggle is the Northern Hemisphere's plants breathing. Separating the two parts is a proper modelling problem with long-running, freely available data, and testing predictions on recent years keeps it honest.

The mathematics you'll need

Polynomial and exponential trend models
Sinusoidal functions: amplitude, period, phase
Fitting a combined model (subtract the trend, then fit the residuals)
Out-of-sample testing and residual analysis

Modelling the data, step by step

This idea usually needs sinusoidal, exponential or polynomial models. See it worked step by step, with a criterion tip at every step: Sinusoidal y = a sin(b(x − c)) + d · Exponential y = a e^(kx) + c and y = a·bˣ · Cubic and polynomials (and overfitting) · 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

Download monthly mean CO₂ at Mauna Loa from NOAA GML.

NOAA Global Monitoring Laboratory — Mauna Loa CO₂ — Monthly and annual mean atmospheric CO₂ since 1958 (the Keeling curve), CSV.

Real data for this idea

Primary source: NOAA Global Monitoring Laboratory: Mauna Loa CO₂: Mauna Loa monthly and annual means. Download the monthly mean CSV (year, month, average). Licence: Free to use; cite the dataset, its version and the date you downloaded it.
Readable chart: Our World in Data: Global atmospheric CO₂ concentration. Use it to see the pattern; download your numbers from the primary source or the chart's own download. Cite as: Our World in Data, ‘Global atmospheric CO₂ concentration’, CC BY 4.0. Original source: NOAA Global Monitoring Laboratory, Trends in Atmospheric Carbon Dioxide (2026).

NOAA revises recent values, so write down the file's version date and the day you downloaded it. Links checked October 2026; we link to the data and never copy it. More free data sets.

Ready to use: CO₂ at Mauna Loa (yearly) · CO₂ at Mauna Loa (monthly) in the IA data bank — cited, licensed for reuse, and one click to open in the grapher.

A possible outline

Explain the two processes behind the curve.
Fit trend models to 1960–2010 and compare.
Fit a sinusoid to the detrended data.
Predict 2011–present and measure the error.
Reflect on why the trend model may not hold into the future.

Pitfalls that cost marks

Fitting everything at once with no explanation.
Using a period other than 12 months without justification.
Treating a good fit as proof of a mechanism.

Showing personal engagement

Explain why the seasonal amplitude is larger in the north.
Compare with a Southern Hemisphere station.
Predict when 450 ppm will be passed and state your uncertainty.

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	Polynomial and exponential trend models; Sinusoidal functions: amplitude, period, phase
AA HL	Good fit	Polynomial and exponential trend models; Sinusoidal functions: amplitude, period, phase
AI SL	Good fit	Polynomial and exponential trend models; Sinusoidal functions: amplitude, period, phase
AI HL	Good fit	Polynomial and exponential trend models; Sinusoidal functions: amplitude, period, phase

Level: Solid. Needs some independent work beyond class examples. 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: fitting everything at once with no explanation; and using a period other than 12 months without justification. Say how each one affects your answer.

Use of mathematics (E)

SL: polynomial and exponential trend models; sinusoidal functions: amplitude, period, phase; fitting a combined model (subtract the trend, then fit the residuals); out-of-sample testing and residual analysis, 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

Test whether the seasonal amplitude itself has changed over time.

Extending it for HL

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

See a complete IA, marked

When can we launch? Modelling the tide at our sailing club's slipway (AA SL) is a complete environment exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Polynomial and exponential trend models, sinusoidal functions: amplitude, period, phase, fitting a combined model (subtract the trend, then fit the residuals) and out-of-sample testing and residual analysis.

Where can I get data for this IA?

Download monthly mean CO₂ at Mauna Loa from NOAA GML. Checked sources and tools: NOAA Global Monitoring Laboratory: Mauna Loa CO₂, Our World in Data and NOAA Global Monitoring Laboratory — Mauna Loa CO₂.

Turn this idea into your IA

The planner opens with “Modelling the Keeling curve: trend plus seasons” filled in: the question, the maths and the data sources.

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