STUDENT PLT OF SMA SEL TRAINING ACFT DEVIATED FROM TURNING TO ASSIGNED HDG AFTER TKOF DUE TO A PROBLEM WITH THE ACFT DIRECTIONAL GYROSCOPE.

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I WAS ON AN INSTRUCTIONAL FLT WITH A STUDENT PLT. THE STUDENT HAS OVER 40 HRS. HE WAS OPERATING ON THE RADIO. I RECALL BEING CLRED FOR TKOF BUT NOT A CERTAIN HDG. AFTER TKOF; I RECALL US BEING GIVEN A HDG OF 010 DEGS TO FLY. OUR DIRECTIONAL GYROSCOPE WAS PLACARDED 'INOP;' THEREFORE; WE WERE MAINTAINING HDG BY OUR MAGNETIC COMPASS. THIS CAUSED SOME CONFUSION FOR MY STUDENT. I REALIZED HE WASN'T TURNING AS SOON AS I WANTED; SO I PROMPTED HIM TO START THE TURN. HIS FIRST REACTION WAS TO TURN IN THE OPPOSITE DIRECTION. I PROMPTED HIM; WAS AGAIN WHEN HE HAD TURNED TO ABOUT A HDG OF 140 DEGS. AT THAT POINT HE STARTED HIS TURN TOWARD 010 DEGS. I THEN RECALL DEP ASKING OUR HDG. MY STUDENT REPLIED WITH HIS ALT. I TOLD HIM 'YOUR HDG.' HE REPLIED '010 DEGS; N NUMBER.' I THEN SAW ACR ACFT AIRLINES APPROX 1 1/2 MI AHEAD AT THE 12:15-12:30 POS; CLBING OUT AWAY FROM US. I THOUGHT TO MYSELF HE WAS OFF RWY 3. I DIDN'T THINK HE WAS A PROBLEM BECAUSE DEP DIDN'T ADVISE US OF TFC THAT I COULD RECALL. I AM FAMILIAR WITH THE OPS OF THE ARPT. NORMALLY; WHEN WE DEPART OFF OF 12R OR 12L; TWR WILL ADVISE US TO TURN PRIOR TO REACHING RWY 3/21. MY STUDENT OR MYSELF DON'T RECALL TWR ADVISING US OF THAT. THE RED LINE IS MY APPROX GND TRACK.