C172 flight instructor with student reported the engine began to malfunction right after takeoff prompting the instructor to take over the controls and land in a grass area near a taxiway.

Narrative

I was with my student doing a regular flight we planned a cross country from ZZZ to ZZZ1; runway was XXR in used; we did the run up and the engine was responding within parameters; nothing was different before taking off; we received the takeoff clearance; we did the call outs; engine instruments were in green; airspeed alive; and tachometer was reading normal; the engine was running good without any different sound; right after we rotate; I noticed a decrease in RPMs down to 2000; I said full power and immediately the RPMs drop more; I took controls immediately; in effect my student had full power; I pitch down; meanwhile I hear an increase in RPM and a drop again; I had no runway remaining for landing straight; and all I saw in front was houses and the airport fence; so I turn immediately to the left pitch down to not loose airspeed and I did the flare on the grass and landed there I stop on the taxiway 1 at the beginning of XYR; nobody was injured I did not hit anything on ground. After services came to check on us; we taxi back to the ramp; the engine never stopped completely during priority landing it was only a drop on the RPMs; before flight during the walk around; we drained fuel we check fuel selector was in both position; and we check one more time in run up as specified by checklist; and before taking off as well. It hard to remember what altitude I was I did not take a look to the altimeter; but it was less 200 feet for sure It was after we lift off.