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

Does my phone battery really drain in a straight line?

AI SLAI HLAA SL Accessible Also in: Statistics

Research question

Is my phone's battery percentage over a day of normal use better described by a linear, piecewise-linear or exponential model, and can the best model predict when the phone will die?

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Why it makes a good exploration

Phones show a percentage and an estimated time left — but how do they estimate it? Logging your own battery and testing models against the phone's prediction is personal, easy to collect and surprisingly subtle.

The mathematics you'll need

Linear and exponential regression
Piecewise functions
Residual analysis and R²
Using a model to predict and comparing with reality

Modelling the data, step by step

This idea usually needs exponential or piecewise models. See it worked step by step, with a criterion tip at every step: Exponential y = a e^(kx) + c and y = a·bˣ · Piecewise models · 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

Record battery percentage every 15 minutes on three days with similar use (screenshots of the battery graph work), noting screen-on periods.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

Describe your usage pattern and data collection honestly.
Fit linear and exponential models to day one.
Notice where the fit breaks (heavy use, idle) and build a piecewise model.
Predict day two's shut-down time and compare with what happened.
Reflect on why batteries do not discharge evenly (percentage is estimated, not measured).

Pitfalls that cost marks

Changing usage between days so the models are not comparable.
Using R² alone to pick a model.
Treating the percentage as an exact measurement.

Showing personal engagement

Compare your model's predicted time left with the phone's own estimate.
Run a controlled test (video playing, screen at fixed brightness) to separate usage from the battery itself.
Compare two phones or an old and a new battery.

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	Linear and exponential regression; Piecewise functions
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	Linear and exponential regression; Piecewise functions
AI HL	Fits, but add an HL technique	Linear and exponential regression; Piecewise functions

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: changing usage between days so the models are not comparable; and using R² alone to pick a model. Say how each one affects your answer.

Use of mathematics (E)

SL: linear and exponential regression; piecewise functions; residual analysis and R²; using a model to predict and comparing with reality, each explained in your own words and carried out correctly.

How modelling explorations reach the top bands in general: Modelling with functions IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Investigate the voltage–charge curve of lithium-ion cells and explain why the last 20% often falls faster.

Extending it for HL

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

See a complete IA, marked

Should I lease or buy my first car? (AI SL) is a complete modelling 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?

Linear and exponential regression, piecewise functions, residual analysis and R² and using a model to predict and comparing with reality.

Where can I get data for this IA?

Record battery percentage every 15 minutes on three days with similar use (screenshots of the battery graph work), noting screen-on periods. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Does my phone battery really drain in a straight line?” filled in: the question, the maths and the data sources.

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