There is a quantity of x iPads in your room. Maybe you have one yourself and have items you want to demo, or videos you want to show, or maybe you teach from it (it can be done). Or, maybe you're lucky enough to have some student ones. How do you show off work? A lot of the traditional methods involve a clunky save, attach to e-mail process. Or maybe you have a Dropbox and you have them save and upload it in there. Either way, a little clunky, and a little time consuming to determine if the students put the work in the right place. A solution for this sort of thing is AirServer. It piggy backs off of Apple's AirPlay technology that debuted with the AirPort Express in...2004 (originally called AirTunes). What, you thought it was the iPad? AppleTV? Nope. You know, before iPhones or iPads were a thing and iPod screens were black and white. It was a dark period.
Nowadays, wireless bandwidth is such that you can send video via AirPlay (officially you are only supposed to transmit to an AppleTV or AirPort Express or licensed AirPlay speaker) and it's not a gimmick or half-baked. It's pretty mind blowing to watch.
Anyway, the question of sharing work that's done on an iPad. The foolproof way is an iPad VGA adapter. That can be a little tricky if the projector in your classroom doesn't have multiple inputs or you aren't comfortable swapping out VGA cables, or aren't willing to figure out to integrate a KVM switch. You're also limited to the length of your cable and the cable on the iPad side of the equation likes to pop out in my experience.
AirServer offers a little more flexibility in this regard. You install the app on your main computer (there's a Mac and Windows version), and it shows up as an AirPlay source to the iPad:
The result is a perfect replica of the iPad screen on your desktop computer. It can be made full screen if you so choose:
If you don't have an iPad and perhaps just have an iPhone or iPod Touch, these devices will still work with AirServer. Regardless of device, the camera is in play and will stream video to the destination screen without issue (provided there's enough breathing room on your network). Earlier in the year, one particular group of students got into a heated battle of Inverse Trig War. I tied my iPhone to the computer and did a "live broadcast" from the table. Not sure how to describe how like "what the heck did I just do?" this experience was.
I haven't had much trouble with this when using it in the classroom. All the student iPads comply pretty well. Though I do have the VGA cable as a backup. My only tip if you're going to have AirServer running is to set a password (can be done within the Preference/Options of the app). Otherwise any iOS device can hop on if the connection is free. Three times I showed something via iPad, disconnected it, forgot I had AirServer running and at random some kid's music collection started to play. It was pretty funny once I figured out what happened (I suspect the culprit was in the room). But for the unitiated, it can be a bit unsettling.