C172 Flight Instructor reported inability to take control from student during landing flair resulted in tail strike damage and loss of control.

Narrative

There were two occupants (An instructor-myself and an instrument student) in a Cessna 172 SP model inbound on the visual approach into ZZZ. We entered on a right base and then proceeded to turn final. Student maintained stabilized approach criteria until crossing the threshold. Crossing the the threshold and slightly high; the student had reduced power and allowed the nose to rise in attitude; reducing airspeed to approximately 50-55 knots; yet had not entered ground effect yet. I recognized the need for a gain in airspeed either by lowering the nose and keeping a descent attitude or adding power to settle into ground effect. Just prior to entering ground effect I was able to make pitch inputs with the yoke to lower the nose and continuously tell the student to lower the nose and keep it down. Settling into ground effect the student prematurely pulled power idle and abruptly reacted by adding significant back pressure while simultaneously locking up on the controls. I; the instructor; tried to recover by lowering the nose as we were in a ballooning effect. With minimal altitude between the airplane and the runway and my inability to overpower the student's back pressure; I had little to no time to even try and add power for a go-around. Considering our lack of altitude and high inertia towards the runway we were in a position where a hard landing was inevitable. I tried my best to protect the airplane and continue to keep the airplane lined up on centerline to ensure we would at least not side load. With the student's locked controls; we were set into an aggressive attitude that led us to have a minor tail strike. The C172 tail skid plate was bent and penetrated the bottom skin of the rudder. Upon landing; I notified maintenance personnel; as aircraft rudder was effected.