The k-Schur functions are symmetric functions introduced by Lapointe, Lascoux, and Morse in the theory of Macdonald polynomials. I conjectured and Thomas Lam proved, that the k-Schur functions are the Schubert basis of Bott's realization of the homology of the affine Grassmannian. I will present a new construction of the k-Schur functions that is analogous to Knutson and Miller's construction of Schubert polynomials for flag varieties as multidegrees of matrix Schubert varieties. This is ongoing joint work with my student Cory Brunson.