The flight instructor and student had been practicing touch and go landings. During the attempted full stop landing to runway 36 (1,938 feet long by 50 feet wide, turf), the airplane touched down with "slight extra speed" approximately 100 feet beyond the approach end of the runway. The flight instructor then took the airplane controls and applied brakes; however, there was "no friction on the mowed dry grass." As the airplane approached the end of the runway, the instructor applied left rudder to avoid a ditch in front of the airplane. Subsequently, the right main landing gear went into the ditch and the airplane flipped over. Examination of the airplane revealed the vertical stabilizer and rudder were bent. No mechanical anomalies were reported with the airplane prior to the accident. The instructor reported they should have landed on the adjacent asphalt runway or performed a go-around after "over-shooting final" during the landing.