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!

## Rational points on the folium.

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!