Re: when a plane flies, does it continue to rotate with the earth?

If I understand your questio correctly, in this case I guess you are right and 
your teacher might be wrong (or we misunderstood).

How to prove:

The earth has a diameter of about 12600 kilometers (7900 miles).
From this you can calculate the circumpherence at the euaqtor (diameter*pi) to be 
near 40000 kilometers. You also know, that it takes 24 hours for the earth to 
turn around once.
Thus the speed of the earth at the equator is about 1600 kilometers per hour.

If the atmosphere would not turn with the earth the normal condition would be a 
storm of that magnitude. We obviously do not see that. Therefore we have to 
conclude, under normal conditions the atmosphere turns with the earth rotation. 

The same is true for any object embedded in the atmosphere. If it is not moving 
relative to the surrounding air molecules and you assume no wind, it has "a speed 
of more then 1600 kilometers per hour". 

(The same is true for our travel around the sun. This adds little more than 
another 100000 kilometers per hour to our speed.)

But there are a number of noticable effects that work together on the earth and 
on the atmosphere. The most noticable might the the coriolis effect. May be you 
can check out  this 
website  (together with your teacher) for this and other effects.

Best wishes 
J Ziesmann