Let F be a signature and R a strictly orthogonal rewrite system on ground terms of F. We give an effective proof of a bounding condition for R, based on a detailed analysis of how terms are transformed during the rewrite process, which allows us to give recursive bounds on the derivation lengths of terms. We give a syntactic characterisation of the Grzegorczyk hierarchy and a rewriting schema for calculating its functions. As a consequence of this, using results of Elias Tahhan-Bittar, it can be shown that, for n > 2, the derivation length functions for functions in Grzegorczyk class En belong to Grzegorczyk class En+1. We also give recursive bounds for the derivation lengths of functions defined by parameter recursion.