Flight Instructor on training flight reported NMAC with another aircraft.

Narrative

Conducting maneuvers in the southwest practice area of Phoenix metro over the north test track about 6 miles northwest of Ak-Chin airport at 4500 msl with a commercial pilot student. We performed an 180 clearing turn and took note of Aircraft Y that looked to being doing pattern work at Ak-Chin. We made our radio call and began a simulated descent. After starting it we noticed that Aircraft Y was heading our direction and initiated a climbing turn away from the traffic. Over the radio our tail number was called asking what we were doing and we acknowledged stating that we were abandoning our maneuver to avoid the traffic querying our flight path. They proceeded to ask what we thought we were doing not making radio calls and that we got too close. In which we responded 'we did make radio calls and performed clearing turns and noticed that their aircraft appeared to be at Ak-Chin and that we seemed to be in a safe position to perform the maneuver since people usually don't fly 200 ft. below another aircraft in airspace that is heavily used for training flights.' Apologized for being so close but we abandoned the maneuver and didn't expect another aircraft to choose a flight path at an altitude right below us and that we would be proceeding to another area. After this the aircraft appeared to follow our flight path; still rather close for a short time until breaking off the other direction. In the end everyone was safe.