The fun thing for me about teaching maths is that you keep "discovering" new things - for me this is after about twenty years of teaching high school maths. The discoveries are not always new to the world but are new to me and my students. And the ideas often come from questions that my students ask me.
So this discovery came about when students were getting the right answer but doing it the wrong way. The average gradient for two points on a curve does not usually equal the average of the gradient at each of the points, but for parabolas (and linear equations) it does. I'm not saying this doesn't happen for other equations, but I also haven't investigated too much.
Finding the average of the two gradients ([math]2ah + b[/math] and [math]2ak + b[/math]) is pretty easy to simplify, but simplifying the expression for the gradient between the two points ([math]h[/math], [math]ah^2 + bh + c[/math]) and ([math]k[/math], [math]ak^2 + bk + c[/math]) is a little more complicated. For these expressions the [i]x[/i]-values of the points are [i]h[/i] and [i]k[/i].
