C172 FLT INSTRUCTOR AND STUDENTS HAVE ROUGH RUNNING ENG DURING TKOF FROM KAND.

Narrative

THE DAY BEGAN WITH A FLT FROM GREENVILLE; SC; TO ZZZ; INSTRUCTING STUDENTS IN A C172. WE LANDED IN ZZZ TO TRADE OUT STUDENTS AND REST FROM THE HEAT. AFTER A 20 MIN REST; WE GOT BACK IN THE C172; STARTED IT UP; AND THEN TAXIED TO RWY 23. A RUN-UP HAD BEEN DONE EARLIER THAT DAY SO THERE WAS NO NEED TO DO ONE AT THIS TIME. THE TKOF WENT AS NORMAL UNTIL WE REACHED MIDFIELD. AT THIS TIME THE ENG BEGAN TO CUT OUT. THE PLANE WOULD NOT CLB OR STAY IN LEVEL FLT. THIS CAUSED THE PLANE TO DSND ABNORMALLY. I MADE THE BEST POSSIBLE DECISION (WITH MY EXPERIENCE) BY PUTTING THE FLAPS IN; AND MANEUVERING THE PLANE INTO A SLIP. I ATTEMPTED TO LAND ON WHAT WAS REMAINING OF THE RWY. I FELT THAT THIS WAS SAFER THAN LNDG IN THE TREES THAT LIE AHEAD. I LANDED AT A HIGH SPD AND WAS UNABLE TO STOP BEFORE HITTING 2 RWY LIGHTS AND PROCEEDING OFF THE RWY INTO THE GRASS. AFTER MAKING SURE THAT EVERYONE WAS OK; I TAXIED OUT OF THE GRASS TO THE NEAREST TXWY AND SHUT THE PLANE DOWN. I INSPECTED THE PLANE AND NOTICED THAT THE R LNDG GEAR HAD BEEN STRUCK BY THE RWY LIGHTS. I THEN PROCEEDED TO TELL THE NEAREST MECH AND ARPT MGR WHAT OCCURRED. THE MECH'S INSPECTION INCLUDED HIS OBSERVATION THAT THE CUTTING OUT OF THE ENG WAS DUE TO A STUCK VALVE.