The use of approximate projection methods for modeling low Mach number flows avoids many of the numerical complications associated with exact projection methods, but introduces additional design choices in developing a robust algorithm. In this paper we first explore these design choices in the setting of inviscid incompressible flow using several computational examples. We then develop a framework for analyzing the behavior of the different design variations and use that analysis to explain the features observed in the computations. As part of this work we introduce a new variation of the approximate projection algorithm that combines the advantages of several of the previous versions.