In a usual modern physics class the Compton effect is used as the pedagogical model for introducing relativity into quantum effects. The shift in photon wavelengths is usually introduced and derived using special relativity. Indeed, this works well for explaining the effect. However, in the senior author's class one of the student coauthors of this paper, Sandeep Giri, asked what would happen if classical expressions for the electrons' momentum (m v) and kinetic energy [(1/2)mv[superscript 2]] were used. The first response given to the question was that the relevant energies were relativistic and hence this approach would not work. Further thought led to the realization that the electron receives only the difference in the energies of the incoming and outgoing photons. This left the initial conclusion in doubt and we began a serious look at what would the answer be. As a result of our analyses, we believe that the Compton effect provides the clearest pedagogical test for the need of relativity in the case of gamma ray scattering while allowing both the classical and relativistic results to explain the original x-ray results of Compton.