  A plane curve is called strange if its tangent line at any smooth point
passes through a fixed point, called the strange point. In this paper, we study
$\mathbb{A}^1$-curves on the complement of a rational strange curve of degree
$p$ in characteristic $p$. We prove the connectedness of the moduli spaces of
$\mathbb{A}^1$-curves with given degree, classify their irreducible components,
and exhibit the inseparable $\mathbb{A}^1$-connectedness via the
$\mathbb{A}^1$-curves parameterized by each irreducible component. The key to
these results is the strangeness of all $\mathbb{A}^1$-curves. As an
application, in every characteristic we construct explicit covering families of
$\mathbb{A}^1$-curves, whose total spaces are smooth along large numbers of
cusps on each general fiber.
