Flight Instructor reports his student became inadvertently airborne while doing fast taxi maneuvers to illustrate the landing attitude. The instructor elected to continue the takeoff as stopping seemed risky; resulting in the Tower canceling the takeoff clearance of another aircraft.

Narrative

I was conducting nose-high; fast-taxi maneuvers to illustrate the landing attitude to my student. To accomplish this; we had conducted three passes down Runway 12R/30L; turning 180 degrees at the ends; with Tower clearance. I had informed Tower that we may become momentarily airborne. After the third pass; we repositioned to Runway 30R as instructed; for a final fast-taxi pass; our intent was to then depart Runway 12L in left closed traffic for pattern work. I transferred control to my student; and asked that the she repeat the maneuver. She used sufficient power to become airborne; and I noted that she did not immediately retard the throttle; and we were climbing slowly. I took control at a moment when I could not see the 12L end of the runway. By the time I yawed the aircraft to give me sufficient visibility; I determined that continuing in a normal climb would be safer than attempting to land on the remaining runway. As we climbed out; Tower commented 'That was not the fast taxi we expected.' I apologized and stated that I did not believe we could have stopped safely in the runway remaining. I noted Tower instructing another aircraft to hold position; and attempted unsuccessfully to observe his position on the ground. Upon landing; Ground Control informed me of a possible pilot deviation and asked me to call the Tower. When I called; Tower explained that the other aircraft had already been issued a takeoff clearance. I do not know whether that aircraft had already begun moving; but it was clearly a potentially dangerous situation. Lessons for me include noting how quickly a student can convert a fast-taxi into a fully-developed takeoff and climb; and to be ready to assume control more quickly in that situation. Further; I believe that it would be better to request a takeoff clearance from Tower; and notify them that we may simply fast-taxi the length of the runway and exit at the far end; to cover the possibility of the fast-taxi becoming a departure.