C172 flight Instructor with student reported departing traffic on runway forced a go around. The departing traffic turned toward the Instructor causing a NMAC and requiring evasive action.

Narrative

Me and my student were doing pattern work at our home airport (ZZZ). While on a left downwind for runway XX; another Ce172 announces on the CTAF that they are on a 3 mile final for the RNAV XX. We decide to extend our downwind slightly to allow for spacing between us and the RNAV traffic; and we announce our intentions on CTAF. Shortly after turning final; and before the RNAV traffic was clear of the runway; a Ce150 [Aircraft Y] announces on CTAF that they are departing runway XX; and planning to depart to the west. Aircraft Y lines up on the runway; but has to wait approximately 15 seconds for the RNAV traffic to clear the runway. This moves the situation from a tight squeeze between departing and arriving traffic; to a runway incursion. I recognize this; and tell my student to conduct the go around; and to side step to the right so that we have lateral separation between us and the departing traffic. While the student is doing this; I announce on the CTAF that we are going around and side stepping to the right. I then take control of the aircraft; and tell my student that we are going to stay below or level; and to the right of the departing aircraft; as to not enter its blind spot. I know that many pilots depart ZZZ to the west using a right crosswind turn; so I ask the departing aircraft; using its callsign; if they are going to do this. I receive no response. I explain to my student that if the departing traffic decides to make a right crosswind departure to the west while we are side stepped to the right; it would likely cause a collision or near miss. As soon as I finish explaining this; the departing traffic begins a right crosswind departure to the west without making a radio call. At this point; I am level with the departing aircraft; so I add full power and climb above; missing by about ~100ft vertically and ~100ft horizontally. Both aircraft proceeded normally after the near miss.I think the lack of communication between our two aircraft was a primary cause of the near miss. We should have been more persistent in our questioning of the other aircraft; and should have announced a second time where we are. The other aircraft could have listened to use when we announced our intentions the first time; and could have answered our question when we asked if they would be turning right. I also believe the decision on the part of the other aircraft to force a go around; and then to make a right crosswind departure from the pattern without making a radio call is another major factor. I don't believe any human factors affected anyone in our plane; I don't know about the other.