Abstract: The families of motives of the title arise from classical one-variable hypergeometric functions. This talk will focus on the calculation of their Hodge numbers. I will describe an alternative proof of a conjectural formula for them from that of Fedorov. The proof is based on viewing the hypergeometric motive as the pure part of the cohomology of a hypersurface in a toric variety. The key fact is a description of the combinatorics of an associated polytope.