One challenge in improving our understanding of ice-stream dynamics is to develop models of the spatial and temporal transition from ice-sheet to ice-stream flow. We address this with a new, vertically integrated, higher-order formulation for ice-sheet dynamics that captures the leading-order physics of low aspect ratio, viscous fluid flow, regardless of the amount of slip at the bed. The theory introduces a parameter, λ, which approximates the ratio of the basal stress to the shear stress scale, providing a measure of the relative importance of sliding and internal deformation. Our model is able to simultaneously describe the dynamics of both a slow-moving sheet and rapidly flowing ice streams. To test the formulation, we apply a triple-valued sliding law as the basal boundary condition and obtain numerical solutions that can be compared with previous work. We investigate the sensitivity of flow regimes and shear margin width to parameter variation, deriving a scaling for the latter. We also consider a double-valued sliding law, which enforces a constant, low basal stress beneath the ice stream. Comparisons of the resultant stress fields illustrate the different stress balances that can maintain ice-stream flow.