We combine theories of scattering for linearized water waves and flexural waves in thin elastic plates to characterize and achieve control of water wave scattering using floating plates. This requires manipulating a sixth-order partial differential equation with appropriate boundary conditions of the velocity potential. Making use of multipole expansions, we reduce the scattering problem to a linear algebraic system. The response of a floating plate in the quasistatic limit simplifies, considering a distinct behavior for water and flexural waves. Unlike for similar studies in electromagnetics and acoustics, scattering of gravity-flexural waves results in a nonvanishing scattering cross-section in the zero-frequency limit, dominated by its zeroth-order multipole. Potential applications lie in floating structures manipulating ocean water waves.