PA-28 Flight Instructor on training flight with student reported transponder was inoperative in flight. ATC requested airplane be grounded until fixed.

Narrative

My student and I scheduled to practice traffic patterns at ZZZ. around 1200 PM my student and I filed an SFRA flight plan for the traffic pattern at ZZZ. We did our start up procedures as normal and contact ZZZ TRACON clearance via Comm 2 and Comm 1. When we could not get ZZZ TRACON through the radio. I connected my phone via bluetooth to my Bose headset and called ZZZ TRACON clearance via telephone. I received my squawk code from the controller and had my student put the code in the transponder and then set the transponder to altitude mode. I reminded my student to press the ALT button on the transponder in order to make sure it is reporting. we proceeded with checklists as normal and referenced the transponder again in the before takeoff checklist and it was still reporting Altitude mode. We continued with our checklist with Comm1 active with XXX.XXX and comm 2 on standby with XXY.X. We proceeded to take off into the pattern with no issues for around 7 passes. On the last pass; ZZZ FBO contacted us over UNICOM stating that ZZZ TRACON asked us to land immediately. We complied and landed immediately and taxied to parking. ZZZs FBO gave us a number that ZZZ TRACON wanted us to call and I called the number. ZZZ TRACON explained that our transponder was not reporting and asked if we turned it on. I verified the transponder was in Altitude mode as we had set it and it was still in ALT mode when we landed and explained to them that it had been on the entire time and in ALT mode and that neither I nor my student heard any communications on guard on comm2 and landed as soon as the FBO contacted us. They asked for my contact information and aircraft information and instructed me to ground the aircraft and get it fixed. after the phone call; I promptly went on to my employers flight schedule account and grounded the aircraft for a squawk on transponder failing to report altitude even in altitude mode. The front office and another instructor informed me that that aircraft had that same issue a couple of weeks before. I made sure the front office grounded the aircraft so that the transponder could be inspected.