Show them the rational parametrization of the unit circle. Then generalize the method to other conics with an obvious rational point like x^2 + y^2 = 5 and x^2 - 2y^2 = 1 and discuss why it doesn't work on x^2 + y^2 = 3 (i.e., how would you show this conic has no rational points at all). Finally, discuss the parametrization of the integral points on x^2 - 2y^2 = 1, where the story is quite different from the case of rational points.

Show them the rational parametrization of the unit circle. Then generalize the method to other conics with an obvious rational point like $x^2 + y^2 = 5$ and $x^2 - 2y^2 = 1$ and discuss why it doesn't work on $x^2 + y^2 = 3$ (i.e., how would you show this conic has no rational points at all). Finally, discuss the parametrization of the integral points on $x^2 - 2y^2 = 1$, where the story is quite different from the case of rational points.