Why is October the 10th Month of the Year?

The other day my second grader comes home from school and asks:

"Why is October the 10th month of the year when 'Octo' means eight?"

Hmmm...good question.  I had a response that I thought was right, but it turns out I was wrong, proving that we all need to double check when we're not quite sure.

My initial response was that October is named after the Roman emperor Octavius.  I probably picked this up as a misconception somewhere that has some reasonableness behind it.  July is named after Juilus Caesar and August is named after Augustus.  I took a Roman history course in college and remember discussing Octavius, but obviously don't remember the details.  Octavius IS Augustus.  Augustus was born Gaius Octavius.

If October is named after Octavius, where does the name come from?  It comes from 'Octo' meaning eight.  The original Roman year had 10 months, starting with March.  In this calendar, October was the eighth month of the year.  When January and February were added, October became the tenth month of the year, but retained the name October.

I may have my Ph.D. in astrophysics, but my second grader (and pre-schooler) teach me new things every day!!!