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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.