We present an investigation about the shape of the initial mass function (IMF) of early-type galaxies (ETGs), based on a joint lensing and dynamical analysis, and on stellar population synthesis models, for a sample of 55 lens ETGs identified by the Sloan Lens ACS (SLACS) Survey. We construct axisymmetric dynamical models based on the Jeans equations which allow for orbital anisotropy and include a dark matter halo. The models reproduce in detail the observed textit{HST} photometry and are constrained by the total projected mass within the Einstein radius and the stellar velocity dispersion ($sigma$) within the SDSS fibers. Comparing the dynamically-derived stellar mass-to-light ratios $(M_*/L)_{rm dyn}$, obtained for an assumed halo slope $rho_{rm h}propto r^{-1}$, to the stellar population ones $(M_*/L)_{rm pop}$, derived from full-spectrum fitting and assuming a Salpeter IMF, we infer the mass normalization of the IMF. Our results confirm the previous analysis by the SLACS team that the mass normalization of the IMF of high $sigma$ galaxies is consistent on average with a Salpeter slope. Our study allows for a fully consistent study of the trend between IMF and $sigma$ for both the SLACS and ATLAS samples, which explore quite different $sigma$ ranges. The two samples are highly complementary, the first being essentially $sigma$ selected, and the latter volume-limited and nearly mass selected. We find that the two samples merge smoothly into a single trend of the form $logalpha =(0.38pm0.04)timeslog(sigma_{rm e}/200,mathrm{km~s}^{-1})+(-0.06pm0.01)$, where $alpha=(M_*/L)_{rm dyn}/(M_*/L)_{rm pop}$ and $sigma_{rm e}$ is the luminosity averaged $sigma$ within one effective radius $R_{rm e}$. This is consistent with a systematic variation of the IMF normalization from Kroupa to Salpeter in the interval $sigma_{rm e}approx90-270,mathrm{km~s}^{-1}$.
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