A function $f\colon S(\mathscr{X})\to\mathbb{R}$ is called piecewise linear if there is a subdivision of $S(\mathscr{X})$ such that the restriction of $f$ to each edge of the subdivison is affine. We call such an $f$ integral $\Gamma$-affine if there is a subdivision such that each edge $e$ has length in $\Gamma$, such that $f|_{e}$ has integer slopes, and such that $f(v)\in\Gamma$ for each vertex $v$ of the subdivision.