Tsiolkovsky rocket equation

Hello,
I am trying to understand the Tsiolkovsky rocket equation.
I am looking at this right now:
http://www.math24.net/rocket-motion.html

they said that the momentum of the rocket itself is:
p1 = (m-dm)(v+dv)

and the momentum of the gas was:
p2 = dm(v-u)

Here is the problem- u is relative to the rocket.
In my opinion p2 should be:
dm( (v + dv) -u )
because the speed of the rocket has changed too

Where is my mistake?

Thank you,
Marina