Cessna 172 Flight Instructor reported upon refilling the fuel; there was a large imbalance between the left and right tanks. Instructor stated that although the selector was showing both; the selector was actually out of the detente and fuel was used from only one tank.

Narrative

I was flight instructing and PIC on a training flight with a relatively new; VFR student. The Cessna 172M that we were using for training had a fuel selector to control fuel flow from two wing tanks that cross-fed when the selector was on both. I had the student do the preflight and asked him how much fuel and oil we had. For fuel; he said there was 12 gallons in the left tank and 14 gallons in the right. For our planned approx. 1.5hr flight; in total; that was enough fuel. When we got into the plane; the fuel selector was on both. The 'Engine Start' checklist has a line for setting the fuel selector to both. My student wanted to cycle the fuel selector between tanks; so I let him. He turned it 90 degrees counter-clockwise until it indicated 'Left' (9 o'clock); then 180 degrees clockwise until it indicated 'Right' (3 o'clock); and then 90 degrees counter-clockwise until it indicated 'Both' (12 o'clock). I visually confirmed this as he did it; but didn't touch the selector. We took off shortly after this (from 213' MSL) and climbed to 7500' MSL for smooth air. When we got to 7500'; we leaned the mixture appropriately. After talking through slow flight for a few minutes; we then slightly enriched the mixture (but still kept it relatively lean) and began working on slow flight. Other than the first round of slow flight and right before landing; my student flew the whole time. The hobbs time in the end was 1.7 hrs. After the lesson; we topped off the fuel tanks. We first went to fill the left tank; and only put 4 gallons in until it was full. The right tank took 17 gallons. I was surprised by this and immediately checked the fuel selector. Although it appeared to be on the 'Both' indication; when I turned it slightly counter-clockwise; I felt the selector detent. During the flight; the selector was just slightly out of the detent and we were only pulling fuel from the right tank. With 19 gallons usable; we were fortunate to have not starved the engine of fuel. If this was a shorter flight; I may not have noticed and the next user could have run into problems if they only looked at the fuel selector. After reflecting on this; my takeaways are:Most importantly; I didn't make it clear to my student that the fuel selector needed to be in the detents. He used the fuel selector only by how it looked and not how it felt. I need to make sure students understand the importance of using the fuel selector properly (with detents) and reporting it when something doesn't feel right.I have a habit to not trust the Cessna 172 fuel gauges and mostly ignore them during flight; but that was complacency and I'm going to change that habit. Especially for a new student; 1.7 hrs is a relatively long flight and fatigue can set in easily. I should have taken the controls periodically to give him a break and then I might have felt the weight imbalance due to the uneven fuel draw. The emergency checklist in the aircraft has 'Fuel Selector - Both' as the first line item in the Engine Troubleshoot section. I think I would have moved it to ensure it was actually on both; but if our engine had quit; there's a chance that I would have only visually checked or wouldn't have had time at all. In the future; I will make it a point to physically check the fuel selector for this checklist item (and teach it as well).