C172 Flight Instructor reported miscommunication and distraction resulted in a NMAC event.

Narrative

While on departure leg of RWY XX we announced via CTAF that we were transitioning to a downwind for RWY XY for x-wind practice. There was traffic rolling out on RWY XX and back taxied on XY. We announced turning base for XY. The student overshot a 1.5 mile final and I encouraged him to gently correct; which he did. Then he announced we were high; and I coached him to begin a forward slip which he did but in the wrong direction. As I queried him about this a radio call was broadcast. I assumed it was the aircraft calling clear of the RWY. We slipped to land on XY which obscured the visibility of RWY XX. as we descended and approached the rwy interections we came out of the slip and I look down RWY XX to verify that it was clear. I did not see any aircraft but I did observe a converging shadow on the ground just ahead of our position… I took controls and immediately added full power and begin a climbing left turn; as we rolled I looked and saw a company 182 at our altitude and at the 3 position. It passed passed behind and below us as I evaded. I later spoke with the pilot and we discussed or mutual mistakes. The communication I missed earlier was his departure call and he mistook our base call as for RWY XX and not XY.