C172 Flight Instructor reported engine roughness during cruise with a student. Instructor was unable to stop the roughness and diverted for a normal landing.

Narrative

The Instructor Person A and Student Person B planned a flight to ZZZ2 from ZZZ at 5;500. Pre-flight of the aircraft revealed nothing out of the ordinary. Performed an engine runup where the engine performed normally. Nothing unusual was noted. We departed ZZZ around XA:30 A.M.in [a] Skyhawk. I noticed the engine sputtering. I assumed I had leaned the engine too much and slightly enriched it. The sputtering continued and grew more aggressive. I applied mixture rich and the sputtering persisted. I took controls and turned on Carb Heat and turned back towards ZZZ. I realized that we were unable to maintain altitude and I decided to divert to ZZZ1. At this point I started picking out fields to land in. The RPM gauge fluctuated between 2000 RPM and1400 RPM. Carb heat was kept on for the duration of the flight. We were unable to maintain altitude and ended up descending to 3;200. Reaching 3;200 the engine was able to maintain 3;200 and the sputtering subsided. We were able to climb back up to 3;500. We maintained this altitude until we reached ZZZ1. Upon landing at ZZZ1 I performed another ground engine runup; where nothing out of the ordinary was noted. The magneto check showed nothing unusual. We shut down the engine and I checked the inside of the cowling. Nothing out of the ordinary was noted. We then secured the aircraft.