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Mathematics: Applications and InterpretationMathematics AI ExplorationSL

Modelling the cooling of a cup of coffee with Newton's law of cooling

A practical, well-motivated exploration using the student's own measurements, with correct use of an exponential model. It is held back by a graph without units or a measure of fit, and a reflection that identifies a modelling assumption without testing it.

~2300 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

~2050 words

3/3 checklist done

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

~2300 words

0/3 checklist done

Focus on these first

The highest-impact changes, in order.

1

Finish the graph

Label axes with units, show the fitted parameters, and add a measure of fit (R² or residuals).

Mathematical communication (Criterion B)

2

Show the fitting

Demonstrate how the decay constant was found (e.g. by linearising with logarithms) and verify the model.

Use of mathematics (Criterion E)

3

Test your assumption

Check the constant-room-temperature assumption against your data and propose a refinement.

Reflection (Criterion D)

Comments on your text

Working wellProblemSuggestion

Click a highlighted passage to see its comment.

This exploration models how a cup of coffee cools over time using Newton's law of cooling. I chose this because I always wonder how long to wait before my coffee is drinkable1. I measured the temperature of a cup of coffee every minute for thirty minutes2. A graph of the measured temperatures and the fitted exponential model is shown in Figure 13. The decay constant was found by fitting the model to the measured data4. The model assumes the room temperature stays constant, which may not be exactly true5.

Criterion breakdown

Criterion A

Presentation

3 / 4

GoodMedium confidence

How to improve

  • Trim the data-collection description to improve conciseness

To reach 4/4

Make the exploration more concise so every section serves the aim directly.

Criterion B

Mathematical communication

2 / 4

DecentMedium confidence

How to improve

  • Label axes with units (°C against minutes)
  • Add a measure of fit, such as R² or residuals

To reach 3/4

Present the graph with labelled axes, units and a measure of fit so the mathematics is fully communicated.

Graphs, tables and figures

Image 1 · Scatter of data with a fitted exponential curve

Figure 1 — Measured temperature and fitted cooling model

Modelling section

Needs improvement
  • Label both axes with units (°C, minutes)
  • Add R² or residuals and the fitted parameters
  • Comment on where the model under- or over-predicts

Criterion C

Personal engagement

3 / 3

GoodHigh confidence

How to improve

  • Keep this personal angle in the reflection

Criterion D

Reflection

2 / 3

DecentMedium confidence

How to improve

  • Test the assumption against your data
  • Suggest and justify a refinement to the model

To reach 3/3

Test the constant-room-temperature assumption and propose a justified refinement to the model.

Criterion E

Use of mathematics

4 / 6

DecentMedium confidence

How to improve

  • Show the fitting method (e.g. linearising with logarithms)
  • Verify the model against a known point

To reach 5/6

Demonstrate the parameter-fitting (e.g. linearise with logarithms) and verify the model rather than stating the result.

What's already working

  • Strong personal engagement with a practical question
  • The student's own measured data
  • Correct use of an exponential model
  • A clear aim and structure

Revision Checklist

0 / 3 done