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We study a simple numerical scheme based on a new type of Finite Element Method (FEM) to solve 1D Shallow Water Equation (SWE). In the new scheme, the surface elevation variable is approximated by a linear continuous base function (P1) and the velocity potential variable is approximated by one-dimensional discontinuous linear non-conforming base function (P1NC). Here, we implement the P1-P1NC finite element pair to solve the 1D SWE on a structured grid, whereas the the Runge Kutta method is adopted for time integration. We verified the resulting scheme by conducting several simulations such as a standing wave simulation, and a propagation of an initial hump over sloping bathymetry. The resulting scheme free from numerical damping error, conservative and both standing wave and shoaling phenomena are well simulated.