For the record, the FAA registration faq is here
https://www.faa.gov/uas/registration/faqs/
The two entries of note are...
Q8. Will an operator be required to have proof of registration while operating the UAS?
A. Yes. You will be required to have your FAA registration certificate in your possession when operating your unmanned aircraft.
Q28. Is there a minimum age requirement?
A. Yes. You must be 13 years of age or older before you are permitted to register an unmanned aircraft. If the owner is less than 13 years of age, then a person who is at least 13 years of age must register the unmanned aircraft.
Q43. If I let someone borrow my drone do I have to give them the Certificate of Registration?
A. Yes, anyone who operates your drone must have the Certificate of Aircraft Registration in their possession. You can give them a paper copy, email it to them, or they can show it electronically from the registration website.
Note that your usage must be recreational and not commercial. This might be a bit of a grey area for a school. If you fly as part of an after school club which is optional and fee less then clearly you can argue it's recreational but if its part of the school class and you fly outside, technically you may be flying RC Aircraft as part of a business which might be considered commercial use. Currently you are required to have a full scale pilots license to do that and register in a different way and there are lots of other hoops designed for full size aircraft you have to deal with. Clearly you don't want to have to to that.
I should say that I'm in no way an expert on this and you should make your own decision as to what it all means and possibly seek expert counsel. If you are a member of the AMA you might start there since at least they are on your side.