This paper presents Span, a power saving technique for multi-hop ad hoc wireless networks that reduces energy consumption without significantly diminishing the capacity or connectivity of the network. Span builds on the observation that when a region of a shared-channel wireless network has a sufficient density of nodes, only a small number of them need be on at any time to forward traffic for active connections. Span is a distributed, randomized algorithm where nodes make local decisions on whether to sleep, or to join a forwarding backbone as a coordinator. Each node bases its decision on an estimate of how many of its neighbors will benefit from it being awake, and the amount of energy available to it. We give a randomized algorithm where coordinators rotate with time, demonstrating how localized node decisions lead to a connected, capacity-preserving global topology. Improvement in system lifetime due to Span increases as the ratio of idle-to-sleep energy consumption increases, and increases as the density of the network increases. For example, our simulations show that with a practical energy model, system lifetime of an 802.11 network in power saving mode with Span is a factor of two better than without. Span integrates nicely with 802.11 - when run in conjunction with the 802.11 power saving mode, Span improves communication latency, capacity, and system lifetime.