IB Mathematics Internal Assessment
Calculus is the tool students reach for most and leave doing nothing most often. Here is what separates a derivative that carries an argument from one that just decorates the page, at SL and at HL.
A great many explorations differentiate something, set it to zero and report a maximum or minimum. That is one routine step, and on its own it sits comfortably inside the course. The marks for the mathematics come from what surrounds the step: why that function, what the answer means, and whether you trust it.
Before you commit to a calculus topic, ask what the derivative or integral tells you that you could not have seen otherwise. If the answer is "the best value", ask how you would know it was the best in the real situation, not just on paper.
A cost, a time limit, a material limit, or a rule from a sport you play. The interesting part is not finding the stationary point. It is deciding which constraint matters and what happens when it changes. Vary a parameter and track how the optimum moves.
Cooling, charging, filling, decay, growth in a plant or a culture. You measure, you model, and then the rate of change becomes the story: when is it fastest, and does your model predict that moment or miss it? Data that disagrees with the model is a gift for reflection.
Total distance from a speed record, total energy from a power trace, the volume of an object with a curved profile. Integrating is only interesting when you must decide how to represent the shape, and then compare your answer with a physical or numerical check.
Where an exact integral is awkward or impossible, you can compare numerical approaches, look at how their errors behave as you refine them, and judge which one suits the job. That is a real choice between methods, which is exactly where the mathematics scores.
Mostly an HL direction. Build the equation from a physical or biological assumption, solve it analytically or numerically, then check it against reality. The assumption behind the equation is where the thinking lives, so spend your words there.
A graphing tool hands you the derivative, the integral and the answer. If the page shows the output but not your reasoning about why the method applies, the reader cannot tell whether you understood any of it. Use technology, but show the decisions you made and the checks you ran.
Students pick a neat polynomial because it is pleasant to differentiate, not because it describes anything. The "optimum" is then an artefact of the function you chose. Justify the function before you use calculus on it.
A maximum at 3.7 metres, a total of 412 units. Does it make sense in the real situation? Is it stable if your inputs are a little wrong? Without a sanity check and a comment on sensitivity, the result is just a number.
Do not abandon it. Find the one step where the mathematics is routine and ask what a sceptical reader would want to know there. Usually the fix is a justification before the calculus and a check after it, plus letting one assumption fail on purpose so you can see what it does to the answer.
Not sure your calculus is doing enough?
Send me the question and the function you plan to differentiate or integrate. I read real submitted explorations every year, and I will tell you honestly whether the calculus is carrying the exploration, while there is still time to change it.
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Written by a serving IB Diploma and Career-related Programme Coordinator and Head of Mathematics, who reads internal assessments across every subject group every year. If you then want the whole draft reviewed properly against all five criteria, that is the paid one, and it is refunded if it does not name at least three specific things to fix.
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