We exploit the gravitational potential of massive cluster lenses to probe the emission-line properties of six z = 1 galaxies which appear as highly magnified luminous arcs. Using the Gemini Multi-Object Spectrograph (GMOS) integral field spectrograph together with detailed cluster lens models, we reconstruct the intrinsic morphologies and two-dimensional velocity fields in these galaxies on scales corresponds to 0.5 kpc (unlensed) at z = 1. Four of the galaxies have stable disc-like kinematics, whilst the other two resemble interacting or starburst galaxies. These galaxies lie close to the mean rest-frame I-band Tully–Fisher relation for nearby spirals suggesting a clear preference for hierarchical growth of structure. In the rest-frame B band, the observations suggest 0.5 ± 0.3 mag of brightening, consistent with increased star-formation activity at z = 1. However, the galaxies with stable disc kinematics have more slowly rising rotation curves than expected from galaxies with similar surface brightness in the local Universe. We suggest that this may arise because the distant galaxies have lower bulge masses than their local counterparts. Whilst this study is based on only six galaxies, the gain in flux and in spatial resolution achieved via gravitational magnification provides a much more detailed view of the high-redshift Universe than that possible with conventional surveys.