Flight Instructor on training flight in PA-28 aircraft reported NMAC with another aircraft on final approach.

Narrative

Hi; I am a CFI at a local Flight School. I was training a client in a PA-28-161 and on Day 0; we were doing laps in the pattern working on his performance takeoff and landings. On the 2nd lap in the pattern (using Runway XX); we cleared base; made a call; then turned. We didn't see any traffic at this point. As we are seconds away from turning final; we turned to clear extended final approach when a Cessna (I know the tail # if required) came across our flight path narrowly missing us. Personally I think he was maybe 150-200 ft. in-front of us and he was slightly higher (maybe 100 not too positive). He made a straight in approach; no radio calls; and just about caused a midair collision if we turned final 4/5 seconds earlier. Luckily; since we were slightly below him; we flew underneath his flight path (he was ahead of us at this point); continued on base; turned on the depart leg then started a climb to get re-established in the pattern. He made a touch and go and since were at pattern altitude at midfield on the departure leg (much higher than he was) and there was no one on upwind; crosswind; or downwind; we just tuned a modified crosswind and downwind to get ahead of him. After a couple of more laps in the pattern with him making no radios; he started making radio calls; did maybe 1 or 2 more laps; then departed the pattern. I wanted to give him the benefit of the doubt and thought he didn't have radios or didn't have the correct frequency in; but after he made a radio call; I knew these were both false and he just neglected to make calls. When I looked up the tail number; he is based out of a local towered airport; which means he has operable comms. I truly think there was nothing more I could have done but I wanted it to be reported to bring awareness to midair collisions at uncontrolled airports since the accident that happened. While radios aren't necessary; I think use of them should be required and will make every airspace much much safer.