I know that $\mathbb{A}_k^1$ is birationally isomorphic to $Z(y^2-x^3-x^2)$. Apparently the map is given by taking the slope of a line through the origin and mapping it to the other intersection. Geometrically, I can kind of see this since $y^2=x^3+x^2$ is a loop that passes through the origin.
I'm having difficulty making sense of this though. Is there a more algebraic way to do this, perhaps an explicit birational isomorphism?
The line of slope $s$ is parameterized by $(t, st)$. So set $x = t, y = st$ in the defining equation. This gives
$$s^2 t^2 = t^3 + t^2$$
hence either $t = 0$ or
$$s^2 = t + 1.$$
So the birational map sends $s$ to the point $(s^2 - 1, s(s^2 - 1))$.