Annotated exemplar · AA SL · deliberately mid-band
Drinks-can optimisation: a mid-band AA SL draft, annotated to show how to improve
Exemplar written by IB Math Revision for teaching — not a real student's IA, not moderated by the IB. Study it — don't reuse it: submitting or copying it is academic misconduct. The data and the student voice are illustrative, and the marks are our judgement. Schools check coursework with similarity software such as Turnitin.
A deliberately mid-band AA SL draft on the most popular calculus IA topic, the drinks can, with an annotation on every problem explaining how to fix it. It shows why a correct derivative is not enough: the draft's headline result rests on a comparison that is not like with like, and fixing it changes the answer.
How this exemplar would be marked
| Criterion | Mark | Why |
|---|---|---|
| A · Presentation | 3/4 | Organised and mostly coherent (introduction, method, result, conclusion), but the aim is vague, the introduction is padded with a copied definition, and the conclusion does not answer a precise question. |
| B · Mathematical communication | 2/4 | Some relevant, appropriate communication — the formulas, a table and a graph are there — but not mostly consistent: calculator notation runs through the key working, units are missing from the table and formulas, = is used for rounded values, and the graph is unlabelled. |
| C · Personal engagement | 2/3 | Significant but not outstanding: the student chose to test the textbook result against real cans they measured themselves, which is their own step beyond the exercise; but no decisions, predictions or further questions of their own drive the work. |
| D · Reflection | 1/3 | Limited, generic reflection only at the end; the surprising 16% result is never questioned, although questioning it would have exposed the main error. |
| E · Use of mathematics | 4/6 | Relevant AA SL calculus commensurate with the course, but only partially correct, with some knowledge and understanding: the derivative and the stationary point are right, but the minimum is not justified, rounding introduces an error, and the comparison with real cans is invalid because the volumes differ. |
| Total | 12/20 | About 12/20. The calculus is fine; the marks are lost in the aim, the checking and the reflection. Every annotation above is a fix a student can make in a second draft. |
Marks are our judgement of this teaching exemplar against the current criteria, explained criterion by criterion. They are not IB moderation results.
Excerpts with examiner annotations
Free sections are shown below with comments; the rest is in the full exemplar, available in the protected viewer with the IA package or a Pro plan.
Introduction
Aluminium is “a chemical element with the symbol Al and atomic number 13. It is a silvery-white, soft, non-magnetic and ductile metal”. Billions of drinks cans are made every year and they use a lot of aluminium, which is bad for the environment.
The aim of this IA is to find the optimal dimensions of a drinks can and see if real cans are optimal.
I chose this topic because we did optimisation in class and I drink a lot of cans.
A How to improve: “optimal” is never defined. Optimal for what — least metal, lowest cost, easiest to hold? Write an aim with an answer, e.g. “How much more aluminium does a standard 330 ml can use than the cylinder of least surface area with the same volume, and what explains the difference?”
A How to improve: the chemistry definition is copied (and not cited) and does nothing for the aim. Cut it; padding costs conciseness.
C How to improve: “we did it in class and I drink cans” is not engagement. Engagement is shown by what you do — measuring several cans yourself, weighing a lid against the body, making a prediction you then test.
Measurements and formulas
I measured two cans with a ruler.
| Can | Diameter | Height | Surface area |
|---|---|---|---|
| Standard can | 6.6 | 11.5 | 306.9 |
| Slim can | 5.8 | 14.6 | 318.9 |
The formulas for a cylinder are V=πr²h and SA=2πr²+2πrh. The volume is 330 ml so 330=πr²h and h=330/πr². Putting this into the surface area gives SA=2πr²+660/r. I drew this on Desmos:
B How to improve: units are missing from the table and from the formulas, and ml and cm³ are mixed without saying 1 ml = 1 cm³. Put units in every column heading and define r and h (in cm) before using them.
B How to improve: calculator notation (h=330/πr², which literally means 330r²/π) and no display equations. Write fractions properly, number the key equations, and use ≈ for rounded values.
B How to improve: the graph has no axis labels, units, sensible title or marked minimum. Label everything, choose a window that shows the minimum clearly, and refer to the figure in the text.
E How to improve: the measured cans are not checked against the model. A cylinder 6.6 cm wide and 11.5 cm tall holds much more than 330 cm³ (see the next sections), so the comparison later on is not like with like. Always test data against the assumptions of your model.
Finding the optimal can
In the full exemplar (about 1 page). The derivative set to zero, and the optimal radius and height. Open in the protected viewer
Comparing with real cans
In the full exemplar (about 0.5 page). Percentage extra material for the two real cans. Open in the protected viewer
Conclusion
In conclusion the optimal can has a radius of 3.74cm and a height of 7.51cm. Real cans are not optimal because they are taller and thinner, probably because they are easier to hold and look better for marketing. My results were quite accurate but I only measured two cans and there could be measuring errors with the ruler. In the future I could look at other shapes like cuboids or spheres.
D How to improve: the reflection is generic and only at the end (“measuring errors”, “only two cans”). Say how big the ruler error is and whether it could change your answer (it can't — the volume mismatch can), and reflect after each result, not just here.
A How to improve: the conclusion repeats rounded values that were computed with a rounding error and never answers “how much more metal, and why?” with evidence. Answer your aim precisely, with the key number.
C How to improve: grip and marketing are guesses. Test one: measure hand span in your class, or survey which can people prefer, and bring the result into the model as a constraint.
What would push it higher?
- Define “optimal” (least metal for 330 cm³) and write an aim with an answer and a prediction.
- Justify the minimum with the second derivative, and show algebraically that h = 2r at the optimum.
- Check your data against the model: measure the volume the can actually holds (fill it with water) before comparing.
- Go beyond the textbook: re-optimise with thicker ends, and test a real constraint such as grip or stacking.
- Reflect after each result — ask whether each number is believable, and what would change it.
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All 16 annotated exemplars, including three deliberately mid-band drafts.
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