IB Maths AI HL Paper 3: modelling questions, GDC technique and a worked task

IB Maths AI HL Paper 3 is a 1 hour 15 minute, 55-mark paper with two compulsory extended questions, and a graphing calculator is required. Each question is a modelling investigation: you are given a context and usually some data, then asked to build or choose a model, compute with technology, interpret what the numbers mean, and judge where the model breaks down. The students who do well translate carefully between the context and the maths in both directions, use the calculator efficiently, and write assumptions and limitations that are specific to the situation.

What is AI HL Paper 3?

Older sources say one hour; the official May 2026 timetable gives 1 hour 15 minutes. For how AI's papers compare with AA's, see IB Maths AA vs AI exams.

How is AI Paper 3 different from Papers 1 and 2?

Papers 1 and 2 ask many questions, each on a recognisable topic. Paper 3 stays in one context for a long time and asks you to make decisions: which model, which variables, what the result means, whether it is believable. Expect to:

• work from a table of data or a description to a mathematical model;
• fit, compare or refine models (linear, exponential, logistic, power, sinusoidal, or a model with a fixed asymptote);
• use the model to predict, then comment on how reliable the prediction is;
• change an assumption and see how the conclusion changes.

Interpretation carries marks of its own. A correct number with no sentence about what it means in the context often earns only part of what's available.

How do you read the context?

Before computing anything, spend a couple of minutes pulling the maths out of the words:

• Variables and units. What is the independent variable (time in minutes? years since 2010?) and what is being modelled? Write them down with units.
• Fixed facts. A room temperature, a population ceiling, a starting value. These usually become parameters or asymptotes in the model.
• Constraints. Can the quantity be negative? Is there a maximum? Does the model only need to work over a certain interval?
• The direction of the question. Read every part. If the last part asks for a long-term prediction, the model you choose early on needs sensible long-term behaviour.

A quick sketch of the scatter plot or of the expected shape is worth the time: model-choice marks usually come from matching the shape to the context.

How do you choose and justify a model?

Justify a model with reasons from the context and from the data, not from "it fits well" alone.

Evidence from the data that a model fits:

• Linearisation. If ln y against x is close to a straight line, an exponential model is reasonable; if ln y against ln x is straight, a power model is.
• Residuals. Residuals with a pattern (positive, then negative, then positive) mean the model has the wrong shape, even when the fit statistic looks high.
• Sum of squared residuals and R². Useful for comparing models on the same data, but not on their own a reason to accept a model whose long-term behaviour is impossible.

How do you use the calculator efficiently?

• Enter data once. Put the data in lists, then build transformed lists from them (for example a list defined as ln(L2 − 20)) rather than retyping values.
• Store regression results. Most calculators can paste or store the regression equation straight into a graph function. Retyping rounded coefficients introduces errors into every later answer.
• Graph the model over the scatter plot. It is the quickest check that you have the right model and haven't swapped the lists.
• Use the solver or graph intersection for "when does the model reach…" questions, then write the equation you solved.
• Write the maths, not the keystrokes. Write the equation and the result; leave out calculator menu names.

Settings matter too: radian mode for sinusoidal models in calculus, statistics diagnostics switched on (TI-84) so r and r² appear, and full precision kept between parts. Our guide to GDC mistakes in IB maths covers these model by model.

How do you write assumptions and limitations that earn marks?

Good limitations name a specific feature of this model and this context, and say what effect it has.

Other strong angles: extrapolating beyond the range of the data, a parameter that is assumed constant but probably isn't, a discrete quantity (people, cars) modelled with a continuous function, and a small or unrepresentative sample.

How should you split the time?

75 minutes for 55 marks is about 1.4 minutes per mark. Divide the time between the two questions in proportion to their marks, and switch on time: the opening parts of the second question are usually easier marks than the final part of the first. Within a question:

• 2–3 minutes reading the whole question and sketching;
• enter data carefully once (a typo costs more time than anything else on this paper);
• keep interpretation sentences short but always present;
• if one part depends on a value you couldn't get, pick a sensible value, say that you are using it, and carry on. Follow-through can still earn the method marks.

A worked mini Paper 3 task: cooling tea

This is an original task, shorter than a real question. A mug of tea is left in a room at 20 °C. Its temperature T (°C) is recorded t minutes later:

(a) Why not a simple exponential model?

A model T = ab^t with 0 < b < 1 tends to 0 °C as t increases. The tea cannot cool below the room temperature of 20 °C, so this model must fail for large t.

(b) Linearising a better model

Try T = 20 + Ae^−kt. Then T − 20 = Ae^−kt, and taking natural logs:

ln(T − 20) = ln A − kt.

So if the model is right, ln(T − 20) plotted against t is a straight line with gradient −k and intercept ln A.

(c) Finding the parameters on the GDC

Enter t and T as lists, create a third list ln(T − 20), and run a linear regression of ln(T − 20) on t:

ln(T − 20) = −0.0811t + 4.2537, with r ≈ −1.00.

So k = 0.0811 and A = e^4.2537 = 70.4. The model is T = 20 + 70.4e^−0.0811t.

Use the stored intercept, not a rounded one: e^4.25 = 70.1, which is already wrong at 3 significant figures.

(d) Using the model

When is the tea at 50 °C? Solve 20 + 70.4e^−0.0811t = 50:

e^−0.0811t = 30/70.4, so t = ln(70.4/30)/0.0811 = 10.5 minutes (3 s.f.).

In context: the model predicts the tea reaches 50 °C about ten and a half minutes after it was poured, which fits the data (51.3 °C at 10 minutes).

(e) Comparing with the calculator's exponential regression

Running the GDC's exponential regression on t and T directly gives T = 86.7 × 0.952^t.

The second model fits the data reasonably over 20 minutes, but its residuals show the wrong shape and its long-term prediction is impossible. The first model is better on both the data and the context.

(f) Assumptions and limitations

• The room temperature is assumed constant at 20 °C throughout.
• Predictions beyond 20 minutes are extrapolation; the model has only been checked against data up to t = 20.
• The model ignores how the tea is cooled (stirring, a lid, adding milk), which would change k.

Notice the balance: most of the arithmetic was done by the calculator in a few seconds, and most of the marks were for deciding what to compute and saying what it meant. That's AI Paper 3 in miniature. For the slips that cost AI students marks across all three papers, see Maths AI common mistakes.