C172 Flight Instructor reported while doing pattern work encountering another aircraft in the pattern resulting in the other aircraft turning to avoid a collision with their aircraft. The other aircraft then aligned to a street and executed a go-around when they realized their mistake.

Narrative

While doing pattern work at ZZZ my student and I were doing pattern work making left traffic runway XX; when turning cross wind a call was made by Aircraft X the Piper PA-28; reporting 4 miles to the North to enter the left downwind runway XX; the plane calling was significantly closer than 4 miles and was at 128 kts ground speed; compared to our 77 kts ground speed as we were climbing; we end up flying nearly side by side in the downwind as Aircraft X was about to overtake us; Aircraft X slowed and remained clear of us to extended their downwind as we continued a normal pattern; Aircraft X extended into the ZZZ1 Delta just under 1 NM; proceeded to descend and make a left 180 degree turn and was lined up and descending for a residential street; West of ZZZ; about to touch down. Field elevation at ZZZ is XXXX MSL; information shows Aircraft X was about to touch down at XXYZ MSL on the residential street until realizing their mistake and proceeded to climb to re-enter the left traffic pattern for runway XX at ZZZ. We realized what happened and promptly departed from the upwind to exit the pattern and return to ZZZ1.