Mathematical modelling in IB Maths AI & AA
A mathematical model turns a real-world situation (money, populations, temperature, motion) into an equation you can solve. Decide what kind of change it is, write the model with its variables and units, find the parameters, solve, then say what the answer means and whether the model is believable. Modelling runs through the whole AI course and through AA questions on growth, decay and rates of change, and the set-up earns marks as well as the final number.
The method Modelling in 5 steps
- Choose the type. Does the quantity change by a fixed amount (linear), by a fixed percentage (exponential), go up and down in a cycle (sinusoidal), or level off at a limit (logistic)?
- Define the variables and units. “Let V be the value in euros, t years after purchase.”
- Find the parameters from the information given, or by regression on your GDC when you have data.
- Solve the question that was asked, with your GDC where it is allowed.
- Interpret and test. Answer in a sentence in context, and say where the model stops being reasonable.
Worked example A car that loses value
A car bought for €18 000 loses 12% of its value each year. When is it first worth less than €9000?
- A fixed percentage each year, so the model is exponential.
- Let V be the value in euros, t years after purchase.
- Keeping 88% each year: V = 18 000 × 0.88t.
- 18 000 × 0.88t < 9000 gives 0.88t < 0.5, so t > ln 0.5 ÷ ln 0.88 ≈ 5.42.
- The car is first worth less than half its price during its sixth year. The model assumes the same 12% every year; real cars often lose more in the first year, so check it against real prices before trusting it.
Practise next: exponential models, linear models and sinusoidal models questions, each with worked examples and mark schemes.
Why it matters
- The AI syllabus has modelling with linear, quadratic, exponential, direct and inverse variation, cubic and sinusoidal functions, and AA tests exponential growth and decay and rates of change.
- HL Paper 3 wraps modelling inside a story you've never seen — the marker is checking whether you can spot the right type of relationship before you touch the GDC.
- Many Maths IAs (the Internal Assessment, 12 to 20 pages) build a model from data, and the same skills (choosing a model, testing it and saying where it fails) earn marks there.
Examiner signals
-
“A model for … is given by …”When you're given the model, the marks are for using it correctly — units, initial conditions, domain. Read every clause of the modelling assumption.
-
“Comment on the validity of your model …”This appears in almost every IA and Paper 3. Look for extrapolation, unrealistic assumptions (e.g. unbounded exponential growth) and mismatched units. Two easy marks for a sensible paragraph.
-
Financial questionsAlways identify: (i) formula (simple, compound, continuous), (ii) unit of time, (iii) whether the rate is nominal or effective. IB loves catching students who forget monthly-vs-annual compounding.
Interactive fluency practice for Mathematical Modelling for IB Maths AI & AA
A dedicated fluency-first practice set for this skill — thoughtful questions, instant feedback, and worked solutions that build understanding rather than reflexes — is queued up next. Drop your email to be the first to know when it goes live (no spam, one email when it's ready).
FAQ
How do I decide which formula to use for a real-world problem?
Ask: is the quantity changing by a fixed amount (linear / arithmetic), a fixed proportion (exponential / geometric), or driven by a rate that depends on the current state (differential equation)? Most IB modelling questions use one of these three.
What's the biggest mistake students make in Paper 3 modelling questions?
Skipping the interpretation step. If you compute V ≈ €5634.13 and stop, you lose the last mark. Always add a sentence: “so after 3 years Priya's investment is worth €5634.13”. This is exactly what mark schemes reward.
Can I use a GDC for every modelling question?
Not in AA Paper 1, which is calculator-free. On every AI paper and on AA Papers 2 and 3 you can, and you are expected to. Even with a GDC, write down the model you set up: if the final answer is wrong, the model is what earns method marks.