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Mathematics: Analysis and ApproachesMathematics AA ExplorationSL

Modelling the spread of a rumour through a school using logistic growth

A genuinely engaged exploration built on the student's own data, with correct core mathematics. The marks are held back by notation that isn't always defined, a graph missing units and a measure of fit, and a reflection that identifies a real limitation without going on to refine the model.

~2400 words · Typically 12–20 pages; length is judged through conciseness, not a word count.

Estimated

12–16

out of 20

Decent
AI estimate — not an official IB grade.

Draft history

How this commentary developed, draft by draft — a record you can show your teacher.

Draft 1

20 Aug 2026

10/20
A 2/4B 2/4C 3/3D 1/3E 2/6

~2100 words

3/3 checklist done

14/20+4
A 2→3/4B 2/4C 3/3D 1→2/3E 2→4/6

~2400 words

0/3 checklist done

Focus on these first

The highest-impact changes, in order.

1

Finish the graph and the notation

Label axes with units, show the fitted parameters and a measure of fit, and define every symbol on first use.

Mathematical communication (Criterion B)

2

Show the parameter-fitting

Demonstrate how r, K and A were found (e.g. linearisation or least squares) rather than stating the values.

Use of mathematics (Criterion E)

3

Turn the limitation into a refinement

You spotted that a constant rate is unrealistic — now propose and justify a better model.

Reflection (Criterion D)

Comments on your text

Working wellProblemSuggestion

Click a highlighted passage to see its comment.

This exploration models how a rumour spreads through a school of 1,200 students using the logistic growth equation. I chose this topic because I noticed how quickly a piece of news travelled through my own year group1, and I wanted to see whether a mathematical model could capture it. The logistic model was solved to give N(t) = K / (1 + Ae^(-rt)), and the parameters were estimated by fitting the curve to data I collected over ten days2. A graph of the fitted logistic curve against my collected data points3 is shown in Figure 1. On reflection, the model assumes a constant spreading rate, which is unrealistic because interest in the rumour faded over time4.

Criterion breakdown

Criterion A

Presentation

3 / 4

GoodMedium confidence

How to improve

  • Trim the repetition in the modelling section to improve conciseness

To reach 4/4

Make the exploration more concise and ensure every section contributes directly to the aim.

Criterion B

Mathematical communication

2 / 4

DecentMedium confidence

How to improve

  • Define every symbol the first time it is used (K, A, r)
  • Label the graph axes with units and show the fitted parameters

To reach 3/4

Define all notation on first use and present graphs with labelled axes, units and the fitted parameters, so the mathematics is fully communicated.

Graphs, tables and figures

Image 1 · Scatter of data with a fitted logistic curve

Figure 1 — Logistic model vs collected data

Modelling section

Needs improvement
  • Label both axes with units (number of students, days)
  • Add the fitted parameter values and a measure of fit (R² or residuals)
  • Comment in the text on where the model fits well and where it doesn't

Criterion C

Personal engagement

3 / 3

GoodHigh confidence

How to improve

  • Keep this personal voice going into the reflection

Criterion D

Reflection

2 / 3

DecentMedium confidence

How to improve

  • Propose how to adapt the model (e.g. a time-varying rate)
  • Reflect on what the quality of the fit tells you about the assumptions

To reach 3/3

Extend the reflection: having found the constant-rate assumption unrealistic, propose and justify a refinement to the model.

Criterion E

Use of mathematics

4 / 6

DecentMedium confidence

How to improve

  • Show the parameter-fitting method, not just the fitted values
  • Verify the solution against a boundary condition

To reach 5/6

Demonstrate the parameter-fitting (e.g. linearisation or least squares) and verify the model, rather than stating the fitted values.

What's already working

  • Genuine personal engagement with the student's own data
  • Correct solution of the logistic differential equation
  • A relevant model matched to a real situation
  • A clear aim and logical structure

Revision Checklist

0 / 3 done