A new model for biological growth is introduced that couples the geometry of an organism (or part of the organism) to the flow and deposition of material. The model has three dynamical variables (a) a Riemann metric tensor for the geometry, (b) a transport velocity of the material and (c) a material density. While the model was developed primarily to determine the effects of geometry (i.e. curvature and scale changes) in two-dimensional systems such as leaves and petals, it can be applied to any dimension. Results for one dimensional systems are presented and compared to measurements of growth made on blades of grass and corn roots. It is found that the model is able to reproduce many features associated with botanical growth.