Do you know the name of the personnel/human relations manager? Address the letter to him or her and simply state that you understand that they are currently looking for a candidate for X position, and you would like to submit your enclosed resume/CV for consideration.
If they don't find a qualified candidate internally, they will hire from the outside, and you'll have a foot in the door. They will probably admire your initiative, as well, since you've already demonstrated good networking skills.
- For Teachers