We develop the semiclassical theory of the sawtooth maps. These are piecewise linear maps on the torus which are completely chaotic in the range of parameters we study. The cat maps are embedded in this family as a set of measure zero and show up semiclassically when the, in general, approximate semiclassical periodic orbit sum becomes exact. the usefulness of this model for several purposes is discussed, including the testing of the validity of the semi-classical Green function in the time domain, and as a simple model to study the effects of classical partial barriers in phase space on quantum transport. We demonstrate by means of the nearest neighbour statistics a Poisson to Wigner like spectral transition as the partial barriers to transport become less effective. We also demonstrate using this statistic the remarkably rapid transition from the nongeneric quantum cat map to generic quantum behaviour.