According to my calculations Jupiter's moons do not follow Kepler's 3rd law - Why is that? I'm asked to collect data on the orbital properties of the four Galilean satellites of Jupiter and show that they obey the same scaling as in Kepler's 3rd law. 

My approach for moon Io:

Online, I found that the distance from Io to Jupiter is 422,000km ~= 0.00282089577 AU
which is 

$2.82\times10^{-3} AU$

Io's orbit around Jupiter is 1.77 Earth days; 1.77/365 ~= $ 4.85\times10^{-3}\, \mathrm{Earth\ years}$

By Kepler's 3rd law, $P^2(\mathrm{Earth\ years}) = a(AU)^3$ 

Thus

$(4.85\times10^{-3})^2 = (2.82\times10^{-3})^3$ which is obviously not true even just by looking at the powers of 10 after distributing the exponent on each side.
Kepler's third law is that $R^3/P^2$ is a constant. However it is not a universal constant; it depends on the mass of the body that is being orbited.
$$\frac{R^3}{P^2} \simeq \frac{GM}{4\pi^2},$$
where $M$ is the mass of the orbiting body (assuming that $M\gg$ the mass of the moons).

Depending on your level of sophistication, you could try plotting $R^3$ against $P^2$, or better, $\log R$ against $\log P$, to demonstrate consistency with Kepler's third law.