- area of the part of the plane through a circle enclosed by the circle?!

Thanks, that's nice.
My epiphany came well into my 20s at a trig point one summer in the Black Forest where I noticed the sun was setting somewhat north of West. It was my uncle who explained it to me.

Wow - so what's the general rule?!

Yes.
I didn't realise I'd squacketed.

- and then there's the cruelty of requiring a GCSE pass from these students when only 22% manage to do so

mis-use of 'countless'?
Numerous?
Many?

Graph v Double number line

It's also interesting that one can use 'local' reasoning (eg the y-coord of the middle red dot is halfway between 0 and 3; the first dot's is halfway between 0 and 1½; etc) or 'general' reasoning (eg the y-coords of the red dots are 3 times those of the green dots).

Yes, that's nice - extendable in small or larger ways.
[it would be a kindness if you labelled the point (3,4).]

But here's a strange thing. I've lived in my current house for many years, but it was only yesterday that it dawned on me that the numbers of my house and of two local buses form an AP!

Sure - in which case, can one initiate them into thinking in a more mathematical way?

Yes. Interesting.

Yes, stories and drawings can also help: 17 bags of 241 sweets minus 16 bags of 241 sweets, etc.

There's an interesting tension there; the hope was that the larger numbers would discourage students from multiplying.

Yes, it would be worth trying. Or insist they only work mentally, perhaps adding that the tasks have been contrived to work that way. Or, as a last resort, make the structural property explicit as it might be quite unfamiliar to them.