“Discrete polynomial blending” is a term used to define a certain discretized version of curve blending whereby one approximates from the “sum of tensor product polynomial spaces” over sparse grids. We combine the theory of Boolean sum methods with dual bases connected to the Bernstein basis to construct a new basis and quasi-interpolant for discrete blending. Our blended element has geometric properties similar to that of the Bernstein-Bézier tensor product surface patch, and rates of approximation that are comparable with those obtained in tensor product polynomial interpolation.Our approximation scheme is suitable for approximating bivariate functions, and for the multivariate degree reduction of polynomials in the Bernstein basis.