ASRS 2275053

On Day 0 at approximately XA:15 local time a training flight was being conducted with a commercial ASEL student. After completing a maneuver lesson an RNAV approach was loaded and joined on an 11-12 mile final. CTAF calls were made at 9 7 and 5 miles. An aircraft approached the airport from the south east and was making radio calls asking if anyone else was at the airport. Another aircraft was reporting downwind, base and final calls for the runway in use. The aircraft approaching from the south again asked if anyone was in the pattern. The aircraft in the pattern did report that there was RNAV traffic behind him when the inbound aircraft asked, confirming that our radio was transmitting properly. This was also confirmed by the FBO staff monitoring CTAF. Our aircraft reported a 5 mile final call and the inbound aircraft was making a base turn for the runway after making an improper pattern entry. I repeated our position on final and the aircraft on base proceeded to make a right turn flying head on at our aircraft. After reviewing the track log the aircraft flying head on was flying around 160 MPH and our aircraft was flying at approach speed around 90 knots. I took control of the aircraft and made a sharp right hand turn to avoid a collision well the other aircraft proceeded to fly straight ahead. This can be avoided in the future by properly entering the pattern, monitoring CTAF and verifying the position of other aircraft when unsure. Reviewing right of way rules in the pattern and proper pattern entry after a descent when switching from an IFR flight at high altitude to a VFR uncontrolled airport.