Why is the regulator very hot? We have an LED strip connected to output 6 and is powered by the Uno board itself.  Shouldn't be drawing too much current, only have 10 elements on the LED strip.  While this LED strip is connected to the Uno, I noticed that the regulator is getting very hot.

One thing we haven't done yet is to power the LED strips from an external source, which will be necessary since there will be many strips.  At the moment, the students are simply trying to debug their program with the small strip.
If you power the Arduino with an external power supply the on board regulator has to dissipate the voltage difference between input and 5VDC at the total current you draw from the board. 

As Klaus Warzecha wrote in an answer to another question 


  The Arduino is just the brain, not the power station.


If the output voltage of your power supply is 12V, then the regulator will be dissipating 12-5V = 7 V at the total current drawn. Total dissipated power for eg. 150mA (just an estimate) would be P = V × I = 7 × 0.150 = 1W
The regulator will be pretty hot from dissipating 1W. 

Mind you that the current is not limited to 150mA, enough LEDs can easily increase current and power. If the regulator is stressed (often) enough, sooner or later it will fail. It is better practice to power any load directly from the external power supply, taking the higher voltage into account.