`Mating' describes a collection of operations that combine two complex polynomials to obtain a new dynamical system on a quotient topological 2-sphere. The dynamics of the mating are then dependent on the two polynomials and the manner in which the quotient space was defined, which can be difficult to visualize. In this article, we use Hubbard trees and finite subdivision rules as tools to examine quadratic matings with preperiodic critical points. In many cases, discrete parameter information on such quadratic pairs can be translated into topological information on the dynamics of their mating. The central theorems in this work provide methods for explicitly constructing subdivision rules that model nonhyperbolic matings. We follow with several examples and connections to the current literature.