We investigate the interaction between matter and causal dynamical triangulations (CDT) in the context of two-dimensional quantum gravity. We focus on the Ising model coupled to CDT, contrasting this with Liouville gravity and the relation to the Knizhnik-Polyakov-Zamolodchikov (KPZ) formula. We analytically demonstrate that the scaling dimensions of fields on CDT align with those on a fixed lattice by evading the KPZ relation. This is achieved through adapting the Duplantier-Sheffield framework to CDT, emphasising the one-dimensional nature of CDT and its description via stochastic differential equations.