The shifting strategy, introduced by Hochbaum and Maass, and independently by Baker, is a unified framework for devising polynomial approximation schemes to NP-Hard problems. This strategy has been used to great success within the computational geometry community in a plethora of different applications; most notably covering, packing, and clustering problems. In this paper, we revisit the shifting strategy in the context of the streaming model and develop a streaming-friendly shifting strategy. When combined with the shifting coreset method introduced by Fonseca et al., we obtain streaming algorithms for various graph properties of unit disc graphs. As a further application, we present novel approximation algorithms and lower bounds for the unit disc cover (UDC) problem in the streaming model, for which currently no algorithms are known.