In this article we study a renormalization scheme with which we find all semi-regular continued fractions of a number in a natural way. We define two maps, T^ _slow and T^ _fast: these maps are defined for (x,y)∈[0,1], where x is the number for which a semi- regular continued fraction representation is developed by T^ _slow according to the parameter y. The set of all possible semi-regular continued fraction representations of x are bijectively constructed as the parameter y varies. The map T^ _fast is a “sped up" version of the map T^ _slow, and we show that T^ _fast is ergodic with respect to a probability measure which is mutually absolutely continuous with Lebesgue measure. In contrast, T^ _slow preserves no such measure, but does preserve an infinite, σ-finite measure mutually absolutely continuous with Lebesgue measure. Furthermore, we generate a sequence of substitutions which generate a symbolic coding of the orbit of y under rotation by x modulo one. In the last section we highlight how our scheme can be used to generate semi-regular continued fractions explicitly for specific continued fraction algorithms such as Nakada’s α-continued fractions.