I just found this on iTunes U. (Click 'New Theories Revel the Natural Numbers') It outlines the basic theory and major questions in Partition Theory. Enjoy it!
My students are studying implicit differentiation, we look at the folium of Descartes:
They found the equation of the tangent at
For their homework I want to have them look at another point, but I wanted it to be rational, I almost gave them the origin... But that's good. The expression given by implicit differentiation is undefined there. (Though, one can find the slope of the two tangents using limits.)
Next I searched for rational points between 0 and 3. I'm confidant that there are only two x values that give at least one rational y value these are 36/217 and 216/217. (I used the rational root theorem to show this.)
Sadly, these numbers are too messy to use in a homework! Especially when they are mad at me for giving a very tough problem the other day.
Each of these x values has 3 corresponding y values, what are they?
Well, it's time to look for another relation... I want something that will let them find more than one tangent line at a given x value. Oh, and it must be rational... And pretty!