What is the origin of the use of expanded notation in Math? My students want to know why they would ever use it.

It helps you understand how our base 10 system works. For example:

4,625 = 4*1000 + 6*100 + 2*10 + 5*1

Or stated another way:

4 * 10^3 + 6*10^2 + 2*10^1 + 5*10^0

That may be a little ahead of your students, but it lays the foundation for working in other bases (Base 8, for example). You can also show how when doing simple operations like additon and subtraction you add or subtract digits of similar rank. Finally, it can help you do multiplication in your head - see:

Expanded notation (http://mathmojo.com/interestinglessons/expanded_notation/expanded_notation.html)

4000 600 40 2 is in expanded form so that is important