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Reductions of particular hypergeometric functions $_3F_2(a,a+1/3,a+2/3;p/3,q/3;\pm 1)$

We principally present reductions of certain generalized hypergeometric functions $_3F_2(\pm 1)$ in terms of products of elementary functions. Most of these results have been known for some time, but one of the methods, wherein we simultaneously solve for three alternating binomial sums, may be new. We obtain a functional equation holding for all three of this set of alternating binomial sums. Using successive derivatives, we show how related chains of $_3F_2(\pm 1)$ values may be obtained. It may be emphasized that we make no reliance on the WZ method for hypergeometric summation. Additional material on Pochhammer symbols and certain of their products is presented in an Appendix to supplement the pedagogical content of the paper.