So here’s the question:

And here’s the answer:

Angle TAP is a right angle, because PTN is an equilateral triangle (all sides equal) and it’s half-way along, so bisects the angle at P and must do so perpendicularly. OP can be calculated from Pythagorus (sqrt(90^2 – 40^2) = 80.62. AT is 20cm long and is the hypotenuse of a right triangle ATP, so AP = sqrt(40^2 – 20^2) = 30.64. We know OP and AP and tan(OAP) = OP/AP = 2.327 and angle OAP = arctan(2.327) = 1.165 radians = 66.75 degrees.

The question is two applications of Pythagorus’ theorem, one of SOHCAHTOA and one of the properties of equilateral triangles. The question points towards the solution by a) asking you to calculate angle TAP and then calculating AP. In the context of GCSE maths, you can only calculate AP if angle TAP is a right-angle. They don’t do the general version of Pythagorus. That’s a clue right there.

What makes it difficult is that the prompts in the question only take you half-way there. To get angle OAP, you need its sin, cos or tan, and you can’t read those off from the question. Because OA is

*not*as long as ON. (Following the hint that TON is an isosceles triangle will take you up the garden path.) We know AP so we need OA or OP. OPT is a right triangle with two known lengths (PT and OT), so we calculate OP. This gives us the Opposite and the Adjacent of angle OAP, and that’s its tangent. Now find the arctan on your calculator.

It’s the need for sustained reasoning, for spotting the false starts, and for solving the problem of the missing bit of information, that makes this a difficult question. It’s not the maths that’s hard - this is Year 8 at most - but the ability to perform sustained reasoning and problem-solving.

Most people can’t do that, anymore than most people can run five-minute miles or deadlift 200+ lbs. So there’s two things here: the first is to sort out the young people who show some aptitude for it, so they can pointed to subjects where it is needed; the second is how to design a syllabus and examination that gives the rest of the world something useful. Even if you can’t deadlift 200lbs (I can’t) you can still be taught useful exercises. Even if you can’t conduct a chain of reasoning, you can still be taught to do basic numeracy, estimation, ratios and comparisons.

My memory is that, one year after doing O-level maths, and so half-way through an OND in engineering, I and everybody else on the course looked at an O-level paper and realised it was trivial compared to what we had learned since. How had we ever thought it was hard? That was when the O-level included calculus, and most of us knew about “imaginary numbers” and had done ever since learning the formula for solving a quadratic equation. Back then the maths teachers used to say they thought that including complex arithmetic in the O-level was only a couple of years away. Well, we’re regressed a lot since then.

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