Scaled momentum distributions for the strange hadrons K 0 S and Λ/Λ were measured in deep inelastic ep scattering with the ZEUS detector at HERA using an integrated luminosity of 330 pb −1 . The evolution of these distributions with the photon virtuality, Q 2 , was studied in the kinematic region 10 < Q 2 < 40000 GeV 2 and 0.001 < x < 0.75, where x is the Bjorken scaling variable. Clear scaling violations are observed. Predictions based on different approaches to fragmentation were compared to the measurements. Leading-logarithm parton-shower Monte Carlo calculations interfaced to the Lund string fragmentation model describe the data reasonably well in the whole range measured. Nextto-leading-order QCD calculations based on fragmentation functions, FFs, extracted from e + e − data alone, fail to describe the measurements. The calculations based on FFs extracted from a global analysis including e + e − , ep and pp data give an improved description. The measurements presented in this paper have the potential to further constrain the FFs of quarks, anti-quarks and gluons yielding K 0 S and Λ/Λ strange hadrons.