FATIGUED BY A LONG DAY OF TRAINING; INSTRUCTOR AND STUDENT PLT HAVE NMAC WHILE PRACTICING GROUND REFERENCE MANEUVERS AT A STUDENT TRAINING AREA.

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EVENT TRANSPIRED WHILE INSTRUCTING GND REF MANEUVERS TO A PRIMARY STUDENT DURING DAYLIGHT CONDITIONS. AFTER CLRING THE AREA OF TFC; I DEMONSTRATED S-TURNS; TURNING TO THE L (SBOUND). AFTER THE DEMONSTRATION; THE STUDENT FLEW TO THE NE TO REPOS THE ACFT TO A SIMILAR STARTING SPOT. AS WE FLEW E TO W TO ARRIVE OVER THE ROAD AND BEGIN THE MANEUVER; I WAS FURTHER EXPLAINING SOME COMMON STUDENT ERRORS AND SCANNING FOR TFC. THE STUDENT BEGAN TO EXECUTE THE MANEUVER. APPROX AFTER TURNING 90 DEGS; FLYING S; WITH THE REDUCTION OF OUR BANK ANGLE WE WERE SURPRISED WITH AN UP-CLOSE AND PERSONAL VIEW OF THE SIDE OF WHAT APPEARED TO ME TO BE A PIPER MODEL SINGLE ENG ACFT. WE WERE CLOSE ENOUGH IN PROX FOR ME TO DETERMINE THAT THIS WAS NOT ONE OF THE TRAINING ACFT FROM ZZZ FIELD; WHERE I TEACH. THE OTHER ACFT WAS TRAVELING S TO N AND THE EVASIVE MANEUVER THE OTHER ACFT TOOK WAS A BANK TO THE R TO AVOID COLLISION. THE CONTRIBUTING FACTORS TO THE NEAR COLLISION WERE THE PIC'S WORKLOAD OF TEACHING AND (EVIDENTLY) INEFFICIENT SCAN OF TFC. AN ADDITIONAL CONTRIBUTING FACTOR WS THE INSTRUCTOR'S DECREASING ALERTNESS FROM HAVING TAUGHT FOR 7 HRS PREVIOUS TO THIS FLT (4.0 HRS FLT; 3.0 HRS GND).