We consider an Euler-Maruyama type approximation method for a stochastic differential equation (SDE) with a discontinuous drift and regular diffusion coefficient. The method regularizes the drift coefficient within a certain class of functions and then the Euler-Maruyama scheme for the regularized scheme is used as an approximation. This methodology gives two errors, the first is the error of regularization of the drift coefficient within a given class of parametrized functions and the second is the error of the regularized Euler-Maruyama scheme. After an optimization procedure with respect to this parameter we obtain various rates, which improve other known results.