abstract: Curve fragments are (bi-)Lipschitz images of compact subsets of R. They have a surprising connection to differentiability of Lipschitz functions on metric measure spaces, in terms of Alberti representations - decompositions of the measure into curve fragments. In this talk I describe the duality between Alberti representations and modulus (a central tool in Sobolev analysis on metric spaces), the arising curve fragment-wise differentiable structure, and its connection with Lipschitz differentiability spaces.