Obviously the first time the hands form a right angle is exactly at 9:00 AM
and the last time is exactly at 3:00 PM. 

The minute hand makes 12 revolutions in the same amount of time that it
takes the hour hand to make only 1 revolution. So the angle between the
hands is always either increasing or decreasing at 12-1 = 11 revolutions per
12 hours. That means that the minute hand goes around the hour hand 11 times
in 12 hours.  12 hours divided by 11 is 1.0909... hours.  So the hands are
together every 12/11=1.0909... hours.  That means that they are opposite
each other when half that time has elapsed.  So we divide that by 2 and get
that the hands form right angles every 6/11=0.5454... of an hour. From 8:30
to 3:30 is 7 hours. 7 hours divided by 6/11=0.5454... is 12.8334...  That
means they formed a right angle 12 times during the school day.

The 12 times the hands are at right angles (to the nearest minute) are 
given below.  It's fun to take an old clock or watch and set the hands to
those times and observe that the hands are at right angles at each of those
12 times.

 9:00  <--first time they are at right angles.  Time before is before 8:30.
 9:33
10:05
10:38
11:11
11:44
12:16
12:49
 1:22
 1:55
 2:27
 3:00  <--last time they are at right angles.  Next time is after 3:30

Edwin