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

How much does a cold month cost my family?

AI SLAI HLAA SL Accessible Also in: Statistics, Finance

Research question

How strongly is my home's monthly gas (or electricity) use related to the average outside temperature, and at what temperature does heating effectively switch off?

Make it yours, then agree it with your teacher.

Next step: plan this IA free — the planner opens with this question filled in. For feedback on your own draft, the IA package (€39 once) gives you 5 AI feedback credits and every annotated exemplar in full.

Why it makes a good exploration

Energy analysts use “heating degree days” to predict demand. Using your own family's bills and local weather data gives a model with a clear physical interpretation and a practical answer.

The mathematics you'll need

Scatter plots and linear regression
Piecewise linear models (heating on/off)
Interpreting gradient and intercept
Residuals and unusual months

Modelling the data, step by step

This idea usually needs piecewise or straight-line models. See it worked step by step, with a criterion tip at every step: Piecewise models · Straight line y = mx + c · 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

Monthly meter readings or bills (with family permission) and monthly mean temperature from the nearest Met Office station.

Met Office historic station data — Monthly max/min temperature, rainfall, frost days and sunshine for 37 UK stations, some from 1853.

Real data for this idea

Primary source: Met Office historic station data: Historic station data. Choose the station nearest you; the table gives monthly maximum and minimum temperature, rainfall and sun hours. Licence: Check the terms on the Met Office page.
Optional tool: Historical weather API. Daily weather for any location: a quick way to get daily values the monthly station tables lack. Data CC BY 4.0; the free service is for non-commercial use only.

Links checked October 2026; we link to the data and never copy it. More free data sets.

A possible outline

Explain the physics of heat loss.
Match bills to monthly temperatures.
Fit linear and piecewise models.
Estimate the base temperature and cost per degree.
Reflect on holidays, occupancy and insulation changes.

Pitfalls that cost marks

Bill periods that do not match months.
Using temperatures from a distant station without comment.
Ignoring price changes when using costs.

Showing personal engagement

Use your own home's data.
Estimate the saving from lowering the thermostat by 1 °C.
Compare with a neighbour's home (with permission).

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	Scatter plots and linear regression; Piecewise linear models (heating on/off)
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	Scatter plots and linear regression; Piecewise linear models (heating on/off)
AI HL	Fits, but add an HL technique	Scatter plots and linear regression; Piecewise linear models (heating on/off)

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: bill periods that do not match months; and using temperatures from a distant station without comment. Say how each one affects your answer.

Use of mathematics (E)

SL: scatter plots and linear regression; piecewise linear models (heating on/off); interpreting gradient and intercept; residuals and unusual months, 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

Build a heating-degree-day model and compare it with your piecewise model.

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

What mathematics does this IA use?

Scatter plots and linear regression, piecewise linear models (heating on/off), interpreting gradient and intercept and residuals and unusual months.

Where can I get data for this IA?

Monthly meter readings or bills (with family permission) and monthly mean temperature from the nearest Met Office station. Checked sources and tools: Met Office historic station data.

Turn this idea into your IA

The planner opens with “How much does a cold month cost my family?” filled in: the question, the maths and the data sources.

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