Many reduced models of the water wave equations, in particular the Green-Naghdi equations, can be presented as a "projection" of the shallow water equations onto a set of admissible functions. We will see why we use quotation marks for this property, and how to use it to its advantage for many numerical and modeling problems: entropic stability, boundary conditions, balanced schemes, high order schemes. Finally, we propose a generalization of the model, keeping the "projection" structure, allowing to answer several practical questions such as the coupling with other reduced model or the optimization of the dispersion relation.