We consider approximate solutions of the following  Fredholm integral equation $$x (s) - \int_0^1 \kappa (s, t) x (t) d t  = f(s), \; s \in [0, 1],$$ using projection methods. It is assumed that the kernel $\kappa$ is smooth. Approximating space is chosen to be a piecewise polynomial space with respect to a uniform partition of $[0, 1]$ and the projection is either the orthogonal projection or the interpolatory projection at Gauss points onto this space. Orders of convergence of the approximate solution using the  modified projection method  and its iterated version are available in literature. However, in practice, it is necessary to replace all the integrals by a numerical quadrature formula giving rise to the discrete versions of these methods. We obtain the orders of convergence in the discrete methods and specify a choice of numerical quadrature which preserves the orders of convergence in the original methods.