IB Math AA SL Internal Assessment — the complete guide
The AA SL exploration is a 12–20 page investigation worth 20 marks and 20% of your grade. The good news for SL students: Criterion E is marked against the SL syllabus, so calculus, exponential modelling or regression done correctly and precisely can earn the full 6 marks. The trap is choosing a topic so simple that the mathematics never reaches course level.
This page shows what "at the level of the course" means for AA SL, gives six topics that get there, and walks through two complete explorations.
What Criterion E expects at AA SL
For AA SL, "commensurate with the level of the course" means mathematics from the AA SL syllabus used in a real investigation — not a GCSE-level calculation dressed up, and not HL content copied without understanding. Precision and correctness, not difficulty, separate 5/6 from 6/6 at SL.
| Clearly at AA SL level | Differentiation and optimisation with a constraint; definite integration for area or volume; exponential and logarithmic modelling (fitting k from data); simple first-order differential equations; trigonometric modelling of periodic data; binomial and normal distributions; correlation and least-squares regression with interpretation. |
|---|---|
| Borderline — add a layer | A single regression line with r; one optimisation with no verification; a geometric series for compound interest. Each becomes solid SL work when you compare two models, verify the optimum with the second derivative, or quantify the error. |
| Not enough on its own | Pythagoras, area and perimeter, percentage change, basic probability trees, descriptive statistics (mean, median) with no inference. Use them as supporting tools only. |
For the full level descriptors of every criterion, see the IB Math IA criteria and rubric; for the 6-mark criterion in depth, see Criterion E — Use of Mathematics.
Six IA topic ideas at AA SL level
Each one has a research question you could use tomorrow, the mathematics that makes it course-level, and where the data comes from. None of them appears in the general topic list, so they are not the ones every examiner has seen.
Modelling the bounce height of a ball
How accurately does a geometric sequence predict the height of successive bounces, and what is the coefficient of restitution of a tennis ball?
Geometric sequences and sums, exponential functions, logarithms to linearise the data, percentage error.
Slow-motion phone video; a classic SL topic made personal by the ball you choose and the surface you test.
The optimal angle for a free kick or basketball shot
At what launch angle does a ball travel furthest for a given speed, and does the answer change when the target is above the launch point?
Trigonometry, quadratic functions, differentiation to maximise range, parametric reasoning.
Derive the range formula yourself rather than quoting it — that is the Criterion E difference.
Modelling daylight hours in Dubai with a sine curve
How well does a sinusoidal model predict sunrise-to-sunset time across a year, and where does it fail?
Trigonometric functions (amplitude, period, phase shift), fitting parameters from data, residual analysis.
Public sunrise/sunset tables give 365 data points; the "where does it fail" question drives reflection.
Does caffeine leave the body exponentially?
Using published half-life data, how long after a coffee is the caffeine below a chosen threshold, and how does repeated dosing accumulate?
Exponential decay, logarithms, geometric series for repeated doses, limits of a series.
Easy to relate to your own routine; the repeated-dose model is where the SL maths becomes interesting.
The area of an irregular shape — a lake or a leaf
How closely do the trapezium rule and a fitted polynomial estimate the area of a real lake outline, compared with a GIS value?
Integration, numerical integration (trapezium rule), polynomial fitting, error analysis as the number of strips increases.
Trace the outline from a map; compare three methods and discuss convergence.
Is the normal distribution a good model for exam scores?
Do my school's mock-exam scores follow a normal distribution, and what proportion of students would be expected above each grade boundary?
Normal distribution, z-scores, mean and standard deviation from grouped data, comparison of expected and observed frequencies.
Anonymised class data from a teacher; interpreting tails and skew is the reflection material.
Two complete AA SL explorations, summarised
These are two of the 16 worked IA examples on this site, chosen because their core technique sits squarely at AA SL level. Here you get the idea, the mathematics and the reflection; the full page shows every calculation step.
Worked example 1: Modelling Newton's Law of Cooling
"How accurately can Newton's Law of Cooling predict the temperature of coffee over a 60-minute period?"
Mathematical concepts used
- First-order separable differential equation: dT/dt = −k(T − Ta)
- Separation of variables and integration to find T(t)
- Exponential decay model fitting using experimental data
- Least-squares curve fitting; R² as a goodness-of-fit measure
What the answer means — and how to reflect on it
The answer to the original question falls straight out of the model: solve 60 = 20 + 75e−0.04463t to find t ≈ 14 minutes, so the drink is at a comfortable temperature about a quarter of an hour after pouring. State this in plain language in your conclusion — the examiner wants to see the mathematics return to the real question.
For critical reflection, test the assumptions. Repeat the experiment with a lid, or with a metal cup instead of ceramic, and compare the fitted values of k. If k changes, you have shown that the constant is really a property of the whole cup-and-room system, not of the liquid alone. Fit the model to the full 30 readings using logarithms (plot ln(T − 20) against t; the gradient is −k) and compare with the two-point value — if they differ, say which you trust and why. An extension that compares Newton's single-exponential model with a two-term model is the kind of independent thinking that lifts Criteria C and D to full marks.
See every calculation step for this exploration, explained line by line →Worked example 2: Optimizing Cylindrical Packaging
"What radius minimizes the surface area of an aluminium can holding exactly 330 cm³ of liquid?"
Mathematical concepts used
- Surface area and volume of a cylinder: SA = 2πr² + 2πrh, V = πr²h
- Constraint substitution to form a single-variable function
- First derivative test to locate the minimum
- Second derivative to confirm it is a minimum (not maximum)
What the answer means — and how to reflect on it
Why is the real can taller? Because the model only counts area. Real cans have thicker metal in the ends than in the side (so the ends cost more per cm²), they must fit a hand and a cup-holder, they must stack efficiently, and marketing research shows people perceive taller cans as holding more. Re-run the optimisation with the ends weighted twice as heavily as the side — the optimal ratio changes — and you have an extension that shows exactly how a model can be improved.
In your reflection, resist the temptation to say the manufacturer is wrong. The strongest conclusion is that the optimisation gives a baseline, and the gap between baseline and reality measures the cost of the other constraints. Quantify it: compute the surface area of the real can and of the optimal can and express the difference as a percentage of extra metal. That single number turns a textbook exercise into an exploration.
See every calculation step for this exploration, explained line by line →How the AA SL IA is marked — the five criteria
20 marks in total, identical for every course except the level descriptors of Criterion E. Your teacher marks it; the IB moderates a sample.
| Criterion | Marks | What full marks look like |
|---|---|---|
| A — Presentation | 4 | Coherent, well organised and concise. Clear sections, numbered figures, 12–20 pages. |
| B — Mathematical communication | 4 | Correct notation, variables defined with units, graphs and tables labelled, consistent throughout. |
| C — Personal engagement | 3 | Your own question, your own data or decisions, independent thinking that is visible in every section. |
| D — Reflection | 3 | Critical reflection on results, assumptions, limitations and extensions — throughout, not just at the end. |
| E — Use of mathematics | 6 | Relevant mathematics at SL level, correct, precise, with thorough understanding shown. |
| Total | 20 | 20% of the final grade |
How to start your AA SL IA this week
AA SL IA — word count, format and other questions
Can an AA SL IA use HL mathematics?
You may, but it rarely helps. HL techniques used without full understanding cost marks in Criterion E ("understanding shown") and often in B. Examiners would rather see SL calculus done precisely and verified than a second-order ODE copied from a website.
What is the most common reason AA SL students score 3/6 on Criterion E?
Mathematics that is correct but below course level — descriptive statistics, percentage calculations or a single straight-line fit. The fix is usually one extra layer: fit a second model and compare, differentiate to find and verify an optimum, or integrate rather than approximate.
Is there a word count for the IB Math IA?
No — the IB sets a page limit, not a word count. The exploration should be 12–20 pages including graphs and tables but excluding the bibliography and appendices. Most strong IAs land at 14–18 pages and roughly 2,500–4,000 words of prose; if you are over 20 pages you lose the "concise" descriptor in Criterion A, so the maximum there drops to 3/4.
What format should the IA be in?
A typed document (Word, Google Docs or LaTeX) exported to PDF, with page numbers, a title that states the research question, headed sections, numbered figures and tables, and an equation editor for every piece of mathematics. Double spacing is not required; 11–12 pt text is normal. Raw data and repetitive calculations go in an appendix.
How much of my grade is the IA worth?
The exploration is worth 20% of the final grade for every IB Mathematics course (AA SL, AA HL, AI SL and AI HL). The remaining 80% comes from the written papers. It is marked by your teacher out of 20 and then moderated by the IB.
Can I use technology — a GDC, Excel, Desmos, Python?
Yes, and for most explorations you should. Technology is expected for data fitting, graphs and repetitive calculation, but Criterion E rewards the mathematics you show: formula → substitution → result, with technology used to check or to extend. "My calculator gave r = 0.81" with no formula earns little; showing the formula, one worked calculation and then the technology output earns full credit.
How many drafts can my teacher look at?
The IB allows the teacher to give written feedback on one draft. They may discuss ideas and answer questions throughout, but only one full draft receives written comments, so make that draft as complete as possible — including the bibliography and appendix.
Want a second pair of eyes on your AA SL IA?
Mr Ahmad has guided IB Mathematics: Analysis and Approaches SL students in Dubai and online from topic choice to final draft — including the honest "is this at course level?" conversation that saves marks in Criterion E.
IA tutoring — how it works →