Ah, at last someone else remembers the first part about Grandpa's wooden leg. We didn't always sing it, but it can be used to extend the song a bit (giving it another leg to stand on, so to speak). It's unfortunate that some versions say incorrectly, "...for a duck may be somebody's mother. He lives..." etc. A mother, he lives?? Come on, now. I always heard and sang, "a duck may be somebody's brother. He lives..." Technically, that is also incorrect graqmmatically since a brother would be a drake, not a duck. The REALLY correct line would be to sing "mother" and "she."
Though I doubt if anyone cares
About worrying over split hairs