I learned this 35 years ago to the tune of Clementine, youre using black sheep, my spouse uses another tune, but what funny is I learned it not with the 'i' endings, but with the dictionary form (sometimes called base-3) 'u' forms.

u tsu ru tte bu mu nu nde ku ite, gu ite

without the shi shite as that had been learned well ahead of the lesson adding ta/te forms.

I just think it's interesting how readily a little ditty tune helps people with recall, regardless of the actual tune.