The other day in my maths class the students were asked to find the average gradient for two points on a parabola. What they did was use the derivative to find the gradient at each point and then average those gradients.
Guess what? They all got the answer that I expected but none of them did it correctly (simple rise over run using two points). Well, I was a little surprised, but of course I wondered if the two "averages" were the same for parabolas.
The drawing below simply shows two points, A and B, that you can move around on the parabola (which can also be moved), and the average gradient (from A to B) always equals the average of the two gradients at A and B.
With a little algebra it can be shown that for a function of the form and two points with x-values of h and k, the average gradient = the average of the gradients = .

So the task, which can be done by someone with a knowledge of basic calculus, is to prove that both "averages" will work out to the same result.