Skyhawk 172/Cutlass 172
Flight Crew Flight Instructor; Flight Crew Commercial
Flight Crew Last 90 Days 88; Flight Crew Total 538; Flight Crew Type 101
Situational Awareness; Communication Breakdown
Conflict NMAC; Deviation / Discrepancy - Procedural Published Material / Policy
Flight Crew Took Evasive Action
Procedure; Human Factors
While conducting a student training flight under Part 91; I had asked my student to track the 270 radial to the ZZZ VOR/DME. The entire flight was conducted under positive control and we were receiving traffic advisories from ZZZ Approach. I was scanning outside looking for traffic; and upon reaching the ZZZ VOR/DME; a BE36 approached from our 2 o'clock position; crossing south to north; coming within what I can approximate as 50 ft. or so. My initial reaction was to climb and bank; similar to a chandelle maneuver in order to avoid a collision. After resuming level flight now on a 270 heading; I asked approach if they had issued a traffic advisory; assuming I may have missed it. Approach had told me they had indeed not; and was not seeing any traffic in my area. I continued to keep the traffic in visual contact as they proceeded north; evidently unaware they had almost been in a mid-air collision. I then resumed my instructional responsibilities as normal and proceeded back to ZZZ1 for touch-and-goes.It is possible I became distracted as a result of trying to relay information to my student; but also that the incident aircraft was not equipped with a properly reporting transponder. Whatever the reasoning; I do believe this could be resolved in the future by equipping the facility with the ability to read ADS-B sources; as the other aircraft in the near collision was reporting ADS-B data according to my cellular device; allowing me to view their registration number. It also could be prevented by equipping the aircraft I was using with an ADS-B in receiver; to view said data.
Instructor Pilot reported taking evasive action to avoid another aircraft while conducting a training flight.