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

IA idea · Differential equations & dynamics

How does bread dough rise? A logistic model in the kitchen

AA SLAA HLAI SLAI HL Accessible Also in: Modelling, Health & biology

Research question

Does the volume of rising bread dough follow a logistic curve, and how do the growth rate and maximum volume change with temperature?

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Why it makes a good exploration

Yeast grows quickly and then runs out of sugar and space — a textbook logistic process you can measure with a jar, a ruler and a timer. Changing the temperature gives a real comparison.

The mathematics you'll need

Logistic function and its parameters
Fitting by technology and interpreting parameters
Comparing with exponential growth early on
HL: the logistic DE and its solution

Modelling the data, step by step

This idea usually needs exponential or logistic models. See it worked step by step, with a criterion tip at every step: Exponential y = a e^(kx) + c and y = a·bˣ · Logistic y = L/(1 + C e^(−kx)).

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Where the data comes from

Put equal dough portions in straight-sided jars at 2–3 temperatures and record height every 10 minutes for 3 hours (time-lapse helps).

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

A possible outline

Explain why growth should be logistic.
Collect heights at different temperatures.
Fit logistic models; compare with exponential for early data.
Compare fitted growth rates across temperatures.
Reflect on collapse after the peak and measurement from the meniscus.

Pitfalls that cost marks

Jars with sloping sides (height is not proportional to volume).
Inconsistent dough portions.
Fitting only the early part of the curve.

Showing personal engagement

Bake with a family recipe and explain its proving time using your model.
Predict the best proving temperature and test it.
Compare fresh and dried yeast.

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	Logistic function and its parameters; Fitting by technology and interpreting parameters
AA HL	Fits, but add an HL technique	Logistic function and its parameters; Fitting by technology and interpreting parameters
AI SL	Good fit	Logistic function and its parameters; Fitting by technology and interpreting parameters
AI HL	Fits, but add an HL technique	Logistic function and its parameters; Fitting by technology and interpreting parameters

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: jars with sloping sides (height is not proportional to volume); and inconsistent dough portions. Say how each one affects your answer.

Use of mathematics (E)

SL: logistic function and its parameters; fitting by technology and interpreting parameters; comparing with exponential growth early on, each explained in your own words and carried out correctly.

HL adds: the logistic DE and its solution, each explained in your own words and carried out correctly.

How differential equations explorations reach the top bands in general: Differential equations & dynamics IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Estimate the doubling time of the yeast population from the early exponential phase.

Extending it for HL

This idea already has HL mathematics in it: the logistic DE and its solution. Make that the centre of the exploration and use it with full rigour.

See a complete IA, marked

Is it cheaper to leave the heating on low overnight? A two-temperature model of my bedroom (AI HL) is a complete differential equations 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 Accessible: a good first extended piece of maths, with room to go deeper.

What mathematics does this IA use?

Logistic function and its parameters, fitting by technology and interpreting parameters, comparing with exponential growth early on and hL: the logistic DE and its solution.

Where can I get data for this IA?

Put equal dough portions in straight-sided jars at 2–3 temperatures and record height every 10 minutes for 3 hours (time-lapse helps). Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “How does bread dough rise? A logistic model in the kitchen” filled in: the question, the maths and the data sources.

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