Hi,

Oops, I see I automatically changed the spelling to match the etymology.

Yes, it's building up from 1/8th=Pin, 1/4=Firkin, so if you have the Kilderkin verse it's the next one 1/2=Kilderkin. And next of course you reach the whole barrel that the refrain is talking about.
I guess for a short version, one could just to the three fractions and the refrain.

You're welcome.
(For extra credit, the pupil may sing Barley Mow from Nipperkin up to Tun.)
                                                                  Mysha