Light aircraft instructor pilot reported an NMAC with a military jet in SAV airspace.

Narrative

I was performing stall training with my student when I scanned the horizon and saw an aircraft in front of us. When I realized we were approaching head-on; I started altering course to the right; but the other aircraft; a military fighter jet; was not altering course and approaching us too quickly for a turn to have saved us. I nose-dived into an emergency descent to avoid a collision and they kept flying straight above us. I could see the jet very close and clearly. My boss; the owner of our flight school; received a call from Savannah Approach that we should not be flying south of their space without transponders; but we were not breaking any rules or regulations; we were not close to any federal airways; and that area; Class E airspace; has been our practice area for over 7 years now. I feel that if ATC noticed there was a problem; they could have directed the jet to divert in a more safe way. The jet had to have seen me too late; but they should have been looking and flying slower if possible below 10;000 feet MSL. I saw the jet because as I always teach; 90% of my attention is out the windows looking for traffic in VFR conditions.