Just for fun. My piano teacher asked me to harmonize a folk tune on the spot; I gave him this. Note the chromatically descending bass.
Yes that is a cute trick to give that inane tune a bit of body. Your teacher is really wicked ;-) But you rose to the occasion. It is very good to be able to invent harmonizations on the spot.
Hm, yes, now that you mention it, the 4th sure does sound like that. Which reminds me of a really enjoyable show, Piano Puzzler: http://www.npr.org/templates/story/stor ... Id=4464231 (main page) http://www.npr.org/rss/rss.php?id=4464231 (Podcast feed) Were you saying my harmonization sounds Rach-y? I could see that.