Internal Assessment · Maths AA · Maths AI · 7 min read
What a full IB maths exploration looks like: an annotated example
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Students starting the maths exploration usually have two problems. They don't know what a finished one looks like, and they think it needs very advanced maths to score well. Neither is true in the way they fear. A good exploration is a clear piece of mathematical thinking about something you care about, using maths you actually understand, written so a classmate could follow it.
Below is a complete example, section by section: an exploration of basketball free throws using video analysis. Each section shows an excerpt of what the student would write, then why it works in terms of the criteria.
Read this first. The topic, data and wording are invented for this guide. They show the shape of a strong exploration. Don't copy them, and don't reuse the topic just because the work is done here. Personal engagement is assessed, and your teacher authenticates your work. Use it to see what's expected, then build your own. If you're still looking for a topic, start with the maths IA topics guide.
The criteria, and what they mean in practice
Both Maths courses (AA and AI, SL and HL) use the same five criteria, 20 marks in total:
| Criterion | Marks | In practice |
|---|---|---|
| A Presentation | 4 | Clear structure: aim, development, conclusion. Easy to follow, nothing irrelevant |
| B Mathematical communication | 4 | Correct notation, defined variables, labelled graphs, units, no calculator syntax |
| C Personal engagement | 3 | Your decisions, curiosity and ownership are visible throughout, not just in an anecdote |
| D Reflection | 3 | Critical thinking about results, method and limitations, throughout |
| E Use of mathematics | 6 | Maths relevant to your course level, correct, and well understood |
These are the criteria for the current courses, examined until November 2028. The revised courses, first examined in May 2029, mark the exploration on four new criteria, so if you are in a later cohort, check which ones your teacher is using. The maths IA criteria guide goes through each criterion in more detail.
The usual guidance is roughly 12–20 double-spaced pages. That's a lot less space than it sounds once graphs and tables are in.
1. Introduction and aim
I've played basketball for six years, and my free-throw percentage has been stuck at around 60% for two seasons. My coach keeps saying "more arc". I wanted to find out whether that's actually true for me: is there a release angle that minimises how far my shots miss the centre of the hoop, and if so, what is it?
Aim: to model the path of my free throws mathematically and use my own shots to find the release angle that minimises my miss distance.
Why this works: the question comes from a real situation and a real disagreement ("is my coach right?"). The aim is one sentence and mathematical: model, then optimise. That's personal engagement (C) and presentation (A) in the first paragraph, without any fluff.
2. Collecting the data
I filmed 40 free throws in slow motion (120 frames per second) from a tripod placed side-on to the line of the shot, 6 m away, at the height of my release point. A 1.00 m strip of tape on the wall behind me gave Tracker (free video-analysis software) the scale. For each shot I recorded the release angle θ from the video, and a friend measured the horizontal distance d (cm) between where the ball crossed rim height and the centre of the hoop.
I tried to vary my release angle deliberately, from flat to very high, because if I only took my normal shot I couldn't find an optimum.
Why this works: the method is short but explains the decisions that affect the maths: why side-on (distortion), why a scale reference, why the angle was varied on purpose. A continuous measure (d in cm) rather than "in or out" gives the analysis much more to work with. More on filming in the video analysis guide.
3. Modelling one shot
I first checked that a free throw really follows a parabola. For one shot, Tracker gave 21 positions (x, y) in metres, with x measured horizontally from the release point and y measured from the floor.
A quadratic regression on these points gives
y = −0.251x² + 1.256x + 2.113, with R² = 0.9996.
The release angle is the angle of the tangent at x = 0. Since dy/dx = −0.502x + 1.256, the gradient at release is 1.256, so
θ = arctan(1.256) ≈ 51.5°.
For a projectile without air resistance, the coefficient of x² should equal −g ⁄ (2v²cos²θ). Solving with g = 9.81 m s⁻² gives a release speed of v ≈ 7.1 m s⁻¹, which is realistic for a free throw. At the hoop (x = 4.20 m) the model gives y ≈ 2.96 m, just below rim height (3.05 m): this shot was slightly short, which matches what the video shows.
Why this works: the student doesn't just fit a curve. They justify why a quadratic (projectile physics), extract something meaningful from it (release angle from the derivative, release speed from the coefficient), and check the model against reality (the shot really was short). That's use of mathematics (E) and reflection (D) in the same section. Notation is correct: no "x^2", no "R2=0.99".
4. Finding the best angle
Repeating this for all 40 shots gave 40 pairs (θ, d).
The data is clearly not linear: shots that were too flat and too steep both missed by more. I fitted a quadratic model:
d(θ) = 0.387θ² − 39.69θ + 1028.5, with R² = 0.95.
To find the minimum, I set the derivative to zero:
d′(θ) = 0.774θ − 39.69 = 0 ⟹ θ ≈ 51.3°.
Since d″(θ) = 0.774 > 0, this is a minimum. The model predicts a minimum miss distance of about 11 cm at a release angle of 51.3°.
Why this works: the choice of model is explained by the shape of the data, the calculus is correct and verified (second derivative), and the result is stated in context, with units. For AA students this is a natural place for calculus. An AI student might instead compare several regression models (quadratic, absolute-value, piecewise) and justify the choice with residuals. Both are valid. Use the maths of your course.
5. Reflection and conclusion
My optimum release angle of about 51° is higher than my normal shot. Over the season my usual release was around 46–47°, where the model predicts a miss of roughly 18–22 cm. So my coach was right, and the model even tells me by how much.
But the model has limits I need to be honest about:
- It's my data, not a law of physics. The optimum depends on my height, release speed and technique. A taller player releases closer to the hoop and would have a different optimum.
- Angle isn't the only thing that changes. When I shot higher, I probably also changed my speed, so part of what the model attributes to angle may be speed. A better design would also measure release speed from the video for every shot.
- The quadratic is a convenient shape, not a theory. It fits well within 40–62°, but I shouldn't extend it outside the range I tested.
- Measurement error. At 120 fps the ball moves about 6 cm between frames, so release angles have an uncertainty of roughly ±1–2°. That's small compared with the 22° range I tested, but it means "51.3°" is really "about 50–52°".
I'm now practising at about 51°. In six weeks I'll film another 40 shots to see if the model's prediction holds.
Why this works: reflection (D) isn't a paragraph at the end saying the project was interesting. It questions the model, the data and the method, and it's specific: ±1–2°, speed as a confounder, range limits. The final line shows the maths had a real consequence for the student, which is strong personal engagement (C).
6. Bibliography
Cite the software (Tracker), any physics or maths sources you used for the projectile model, and any AI tool you used, following the IB's rules (see using AI honestly).
What made this exploration work
- A real question with a real answer. Not "the maths of basketball", but "what angle minimises my misses?"
- Own data, collected so the maths could work: side-on camera, a scale, deliberate variation, a continuous measurement.
- Maths that's justified, not decorative: a quadratic because of physics, a derivative because the question is about optimising, a model check against the video.
- Honest reflection throughout, not just at the end.
Common ways explorations lose marks
- A topic summary with some textbook maths, but no question of your own.
- Advanced maths you clearly don't understand. It costs marks on E and on communication.
- Calculator notation ("x^2", "1.2E-3", "R2") instead of proper mathematical notation.
- Graphs without labels, units or captions.
- Personal engagement as a single anecdote in the introduction, then nothing.
- Reflection only in the final paragraph.
Next steps
- How to film motion and use Tracker
- AA vs AI: exams and the exploration criteria
- What a full science IA looks like
- Plan your timeline with the free IA Planner template