[Christ]

If someone told me that the teeth broke off the gear, I'd assume they meant the helical splines that engage from gear to gear, which transfer power between gears.

Probably the furthest thing from my mind would be the dog-teeth, or those things that engage the gear to the input/output shaft (whichever one free spins when gears aren't engaged, usually output, I think) so that it can begin transferring power from the input shaft, to the drive gear, which sends power to the driven gear (via the "teeth" or Helix Splines), which sends power to the output shaft, eventually going to the differential and driving the wheels.

The fact that you're getting an easily reproducible grinding noise suggests that you've destroyed a synchronizer, though.