Is there a way to record my iPad screen for free?
There used to be some free apps, but they stopped working because the devs didn't have time to update them to work with the latest changes to airplay. 

If you are willing to pay a bit of money you can get apps which turn your Mac into an Airplay server. You can then mirror your iPad's screen to your Mac, and record it there.


Reflector is $12.99
AirServer is $14.99
X-Mirage is $16.00


Check them out, and work out which one is best for you.

EDIT:

X-Mirage has a 7-day free trial, so I think it's the closest you'll get to a "free" solution.