Date: 202109
Local time of day: 1201-1800
Location: ZZZ.Airport, US
Flight conditions: VMC
Weather elements / visibility: 20
Light: Daylight
Aircraft: PA-28 Cherokee/Archer/Dakota/Pillan/Warrior
Operator: FBO
Operating under: Part 91
Flight plan: None
Mission: Training
Flight phase: Landing
Airspace: Class G ZZZ
ATC / advisory: CTAF ZZZ
Second aircraft: Skyhawk 172/Cutlass 172
Reporter organization: Personal
Reporter function: Instructor
Reporter qualification: Flight Crew Multiengine; Flight Crew Flight Instructor; Flight Crew Commercial; Flight Crew Instrument
Human factors: Other / Unknown; Workload; Communication Breakdown; Confusion; Distraction
Anomaly: Conflict NMAC
Detector: Person Flight Crew
When detected: In-flight
Result: Flight Crew Took Evasive Action
Contributing factors: Human Factors
Primary problem: Human Factors

Narrative: My student and I flew to ZZZ1 to work on takeoffs and landings. Our intention was touch and go's remaining in left closed traffic. We entered the pattern at ZZZ1 on the 45 deg left traffic Runway XX. We chose this runway as another Aircraft Y reported a 7 mile straight in approach to Runway XX. Winds at the time were reported as calm. Once we were established and reported on the left downwind Runway XX; the straight in airplane reported 3 mile final. We planned to follow this airplane on a straight in. I noticed another aircraft [Z] on our ADS-B display on the opposite downwind Runway XY. We did not hear [Aircraft A] intentions on downwind. At this point we reported turning left base Runway XX. The straight in airplane landed Runway XX.There was another Aircraft A reporting and entering the pattern; left downwind Runway XX behind our airplane. At this time we were on final approach for Runway XX. As we crossed the threshold stripes we heard a radio call from Aircraft Z on the opposite Runway XY asking; 'Aircraft X are you a full stop?' The Aircraft A behind us announced; 'yes'. Even though the radio call was intended for us. At this time the Aircraft Z was on left base Runway XY. My student and I were in a critical phase of flight and did not have a chance to call out on the radio to announce we were a touch and go. We had been making radio calls on each leg announcing position and intentions.As we completed our touch and go we were on the upwind Runway XX and saw Aircraft Z continuing on final approach to land Runway XY. At 200-300 feet above the ground I took controls and banked the airplane right to avoid the Aircraft Z. At no time did I notice the opposing Aircraft Z make any attempts to avoid our airplane. Or to avoid the Aircraft Y landing behind us. They seemed to continue with their touch and go without regard to other traffic.We called up the Aircraft Z by radio and let them know we were on the upwind and asked if they did see us. We continued left closed traffic at ZZZ Runway XX. The Aircraft Z replied; 'we thought you were a full stop.' When in error the Aircraft Z behind us landing Runway XX was the one who thought they were being queried. Aircraft Z continued in left closed traffic Runway XY and while we were in left closed traffic Runway XX. At this point other traffic joined in left traffic Runway XX. We were number two on the left downwind Runway XX behind Aircraft A. As we completed our second touch and go we had to avoid Aircraft Z on the upwind. This time Aircraft Z was on base Runway XY; and I took controls and made an early turn to left crosswind Runway XX turn to maintain distance from Aircraft Z.After a brief conversation with the Aircraft Z by radio...their reason for using Runway XY; while clearly other traffic and us were established in the pattern Runway XX was the wind was favoring Runway XY. Winds did change from calm to 290 at 04 kts. They couldn't understand why we wanted to land with a tailwind. The winds being calm; and other traffic already established in the pattern Runway XX is our reasoning for remaining in this pattern. After the second exchange I elected to depart the pattern and return to ZZZ1. There was a lot of confusion on the radio at this point as one airplane was trying to land Runway XY and others landing Runway XX. The pilots worked to resolve which runway they were going to all use. While we departed the area to the NW.




Synopsis: PA-28 instructor pilot reported an NMAC event during landing pattern training at a non-towered airport. After two incidents with same aircraft which required evasive maneuvers flight Instructor elected to depart the landing pattern.