Given two $n$-polytopes $P$ and $Q$ in $\mathbf{R}^n$ we say that they are \textsl{translationally scissors congruent} if there exist decompositions $P = \bigcup_{i=1}^n P_i$ and $Q = \bigcup_{i=1}^n Q_i$ such that for $i\neq j$ $P_i \cap P_j$ and $Q_i \cap Q_j$ are measure-$0$ sets, and such that $P_i$ is a translation of $Q_i$ for all $i=1,\ldots,n$. We can analyze these using the translational scissors congruence group $\mathcal{P}(\mathbf{R}^n,\mathbf{R}^n)$, which is the free abelian group generated by $n$-polytopes in $\mathbf{R}^n$ modulo the relation that if $[P\cup Q] = [P] + [Q]$ if $P\cap Q$ has measure $0$, and that $[P] = [Q]$ if $P$ and $Q$ are translates of one another. It turns out that $\mathcal{P}(\mathbf{R}^n,\mathbf{R}^n)$ is an $\mathbf{R}$-vector space. in this talk we give an exposition of this result, together with some ruminations of further possible directions that the techniques of the proof can develop in.