 The paper introduces a model for fluidized granular media that involves a scalar velocity field on a 1D lattice and collisions that conserve momentum but not energy. This leads to non-linear fluctuating hydrodynamics equations for the macroscopic velocity and temperature fields, which replicate the behavior of shear modes in a granular fluid. The model predicts an essential part of the peculiar behavior of granular fluids, such as the instability of homogeneous cooling states at large sizes or in elasticity. Additionally, the paper computes the exact shape of long-range spatial correlations, which have the physical consequence of modifying the cooling rate even far from instability, due to momentum conservation. This article was authored by Alessanta, Amanacorda, Aprodos, and others.