IB Maths IA · modelling · AA & AI · SL & HL
IA modelling, step by step
You have your data. Now what? This is the complete workflow for a modelling exploration, with a fully worked example for every model students use — every number shown and checked — and a helper that walks through the same steps with your own data.
The modelling workflow, step by step
Plot the data first
Draw a labelled scatter graph before any calculation. Look at the shape: straight, curved, one turning point, repeating, levelling off? Look for outliers and ask whether they are errors or real. Say what you see in words.
Choose candidate models from the shape and the context
Use both. The shape suggests a family (line, quadratic, exponential…); the context says what must be true — an asymptote at room temperature, a ceiling, a period of a year, a line through the origin. Pick two or three candidates and justify each before fitting.
Fit each model
Show at least one fit in full by hand (points and simultaneous equations, completing the square, the least-squares sums, or a linearisation with logs). Then use technology for the rest and say exactly what it did.
Check the fit
Tabulate the residuals (data − model) and plot them against x. A good model leaves residuals that look random; a pattern (a curve, a fan) means the model misses something. Report SSR, and R² or r where it applies.
Compare the models
Put the candidates side by side: SSR or R², the number of parameters, the residual plots, and whether the parameters make sense. The best model is the simplest one that captures the shape and makes sense in context — not automatically the one with the highest R².
Test on points you did not use
Keep one or two points back (or collect new ones), fit on the rest and predict the held-out points. A model that predicts well on new data is far more convincing than one that fits its own data closely.
State the domain and the limitations
Give the domain where the model is valid, say what happens outside it, and discuss extrapolation. Name the assumptions (constant conditions, measurement error) and how they limit your conclusions.
Reflect
What do the parameters mean in the context? Do they agree with theory or other data? How sensitive is the result to your choices (which points, which asymptote, where you split)? What would you do next?
Each step feeds a criterion: choosing and justifying models (C and E), fitting and checking them (E), comparing, testing and discussing limitations (D), and explaining it all clearly (A and B). See Criterion E: use of mathematics and Criterion D: reflection.
Which model? Start from the shape and the context
| Model (worked example) | Shape clue | Context clue | Courses |
|---|---|---|---|
| Straight line y = mx + c | The points lie close to a straight line with no bend. | A constant rate: cost per item, energy per litre, distance at steady speed. | AA SL, AA HL, AI SL, AI HL |
| Quadratic: three points, completing the square, regression | One turning point (a single peak or trough) and roughly symmetrical sides. | Projectiles and anything thrown or sprayed (constant acceleration). | AA SL, AA HL, AI SL, AI HL |
| Sinusoidal y = a sin(b(x − c)) + d | The data rise and fall repeatedly, with peaks the same height and troughs the same depth. | Anything driven by the Earth's rotation or orbit: daylight, temperature, tides. | AA SL, AA HL, AI SL, AI HL |
| Logarithmic y = a ln x + b | Steep at first, then flattening out, but with no sign of a ceiling. | Learning and practice: speed or score against time spent. | AA SL, AA HL, AI HL |
| Reciprocal y = a/x + b and a/(x − h) + k | y falls steeply at first, then levels off towards a horizontal line (or rises towards one). | Time for a fixed distance or task at different speeds or rates. | AA SL, AA HL, AI SL, AI HL |
| Exponential y = a e^(kx) + c and y = a·bˣ | Equal steps in x multiply y (or y − c) by roughly the same factor. | Cooling or warming towards a surrounding temperature (Newton's law of cooling). | AA SL, AA HL, AI SL, AI HL |
| Power y = a·xⁿ (log–log) | The curve passes through (or heads towards) the origin, with no asymptote other than the axes. | Physics laws with a power: pendulum period (n = ½), gravity (n = −2), falling distance (n = 2). | AA SL, AA HL, AI SL, AI HL |
| Logistic y = L/(1 + C e^(−kx)) | An S-shape: slow start, steep middle, then levelling off. | Growth with a limit: plant height, a population with limited food, adoption of a product. | AI HL |
| Cubic and polynomials (and overfitting) | A cubic: one rise and one fall with an asymmetric shape, or a point of inflection. | Volume-type relationships (a box, a container) where x³ appears naturally. | AA SL, AA HL, AI SL, AI HL |
| Piecewise models | A sudden change of gradient (a kink) or a change of shape part-way through. | A physical change of state (heating then boiling, melting). | AA SL, AA HL, AI SL, AI HL |
Not sure between two? That is normal: fit both and compare them properly.
Worked examples, every step shown
Each page takes one small, clearly labelled example data set and fits the model by hand and with technology — every sum, every substitution, the residual table, R² and a graph — plus the key sequences for the TI-84 Plus CE, TI-Nspire and Casio fx-CG50, the Desmos syntax, how to write it up and a Criterion E checklist.
- Straight line y = mx + cA line is the simplest model and the one to beat.
- Quadratic: three points, completing the square, regressionThree ways to fit y = ax² + bx + c to the same data: solve three simultaneous equations through three points, build the vertex form y = a(x − h)² + k from the turning point, and use least-squares regression.
- Sinusoidal y = a sin(b(x − c)) + dFor anything that repeats — daylight, tides, temperatures, a Ferris wheel — find a, b, c and d by hand from the maximum, minimum and period, convert to the calculator's form a sin(bx + c) + d, then refine the fit by minimising the sum of squared residuals.
- Logarithmic y = a ln x + bWhen something keeps growing but more and more slowly — skills, learning, diminishing returns — try y = a ln x + b.
- Reciprocal y = a/x + b and a/(x − h) + kInverse relationships — time against speed, cost per person against group size — curve towards two asymptotes.
- Exponential y = a e^(kx) + c and y = a·bˣGrowth or decay by the same factor in each step — cooling, depreciation, radioactive or drug decay — is exponential.
- Power y = a·xⁿ (log–log)When y is proportional to a power of x — area and length, period and length, metabolic rate and mass — plot ln y against ln x.
- Logistic y = L/(1 + C e^(−kx))S-shaped growth — plants, epidemics, app downloads, the spread of news — starts almost exponential and levels off at a ceiling L.
- Cubic and polynomials (and overfitting)Adding terms to a polynomial always lowers the sum of squared residuals on the data you fitted — until a curve of degree n − 1 passes through every point.
- Piecewise modelsSome data follow one rule and then another: water heats steadily and then boils, a tariff changes at a threshold, a runner speeds up for the finish.
- Choosing and comparing modelsA good modelling IA does not just fit a curve — it argues for one.
Model my data: the same steps with your numbers
Paste your x and y values, pick a model, and the helper shows the whole play-by-play with your numbers — parameters, working, residual table, R², SSR and a graph — and compares several models, including a test on held-out points. It runs in your browser: nothing is uploaded or stored.
Use it to learn and to check, not to copy. Do the working yourself, justify every choice in your own words, cite the tool if your school's policy asks you to, and never paste its output into your IA.
How much maths to show at SL and HL
- At SL, fitting with a GDC or Desmos is expected; what earns Criterion E marks is showing one fit by hand, explaining what the technology did, and using the results correctly.
- At HL, examiners look for more sophistication and rigour: derive what you use (least squares by calculus, linearisation with the laws of logarithms), refine iteratively and explain it, test on unused data, or build the model from a differential equation.
- At both levels, the comparison, the interpretation of the parameters, the domain and the limitations are Criterion D reflection — often where modelling IAs lose most marks.
Our summary of the current criteria (exams up to 2028), not the IB's wording. How the IA differs between AA and AI, SL and HL.
Technology
Desmos, GeoGebra and spreadsheets all fit these models; the worked pages give the exact syntax. For graphs, tables and how to describe technology so it earns marks, see using Desmos, GeoGebra and Excel in your IA. For a complete modelling exploration marked criterion by criterion, see the annotated exemplars.
Frequently asked questions
What does “modelling” mean in a Maths IA?
Choosing a mathematical function to describe real data, finding its parameters, testing how well it fits and predicts, and interpreting it in the context. The quality of the justification and the reflection matters as much as the fit.
How many models should I compare in my IA?
Usually two or three well-chosen candidates, each justified by the data's shape and the context. Comparing many models without reasons reads as trial and error.
Can I use a GDC or Desmos to fit my model?
Yes — technology is expected. Show one fit by hand so the examiner can see you understand the method, then say clearly which tool did the rest and what it minimised.
Is R² enough to choose a model?
No. R² rises as you add parameters and is not directly comparable between models fitted in transformed units. Use residual plots, SSR on the original data, the number of parameters, the context, and a test on points not used for fitting.
Can I use the model-my-data helper for my IA?
Use it to learn and check the method, then do the working yourself and explain every choice in your own words. If you use it, acknowledge it as your school's policy requires, and never paste its output into your IA.
Free: the IA checklist an examiner uses
Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.
While you wait for the email: read how to choose and compare models →