  Given $M_\varphi$, a fibered 3-manifold with boundary, we show that the
translation distance of the monodromy $\varphi$ can be bounded above by the
complexity of an essential surface with non-zero slope. Furthermore we prove
that the minimal complexity of a surface with non-zero slope in $M_{\varphi^n}$
tends to infinity as $n\to\infty$. Additionally, we show that an infinite
family of fibered hyperbolic knots has translation distance bounded above by
two, satisfying a conjecture by Schleimer which postulates that this behavior
should hold for all fibered knots.
