Building on work of Ghys and Sergiescu, D. Calegari V. Kleptsyn, and I. Liousse independently proved that if $f$ is a piecewise linear homemorphism of the circle, mapping rationals to rationals and having slopes which are powers of a single integer, then $f$ has a rational rotation number. He asked if his result could be strengthened by replacing ``integer'' by ``rational.'' We prove that this is not the case and exhibit a counterexample with slopes which are powers of $3/2$ where the rotation number is $\sqrt{2}-1$. We use this to prove that the Thompson-like group $F_{3/2}$ does not embed into $F$. This is joint work with Jim Belk and James Hyde.