This article presents a closed form analytical solution for one-dimensional transient heat conduction in a material where the thermal conductivity varies linearly through the thickness but the thermal diffusivity is held constant. This solution is used to validate the results from finite-difference and finite-element approximations that account for this variation at the element level. This was motivated by a suggested limitation on the minimum time step used in the commercial finite-element software code ABAQUS for quadratic elements. Good agreement was found between the analytical and numerical approximations, indicating that conventional numerical techniques may be sufficiently robust to analyze heat conduction problems in functionally graded materials without the use of special elements. The minimum time step constraint was found to be unnecessary for a convective boundary condition for the one-dimensional elements and property variation used in this study.