A one parameter family of iterative methods for solving an operator equation with a bounded linear operator between Hilbert spaces is introduced and analyzed. The classical conjugate gradient method is included in the family as is a method due to Le Foll. The latter method is also studied without the assumption that the equation have a solution and convergence properties of the iterates are established in this setting. A general convergence theorem is given and a variety of known bounds for rates of convergence of the conjugate gradient method are extended to the entire class of methods.