A niche use case, but a neat feature. If you need to demonstrate something from an iPad, iPhone, or iPod, running the app AirServer on your teaching computer is the way to go. It acts like an AppleTV without the need for a physical AppleTV or switching projector inputs or any of that mess.
But, your school wireless network may not support it, or like me, AirServer misbehaves from time to time and you need a back up.
On the off chance you teach from a Mac, and on the off chance you recently updated to the latest OS X, you have a built-in option now.
Plug your device into the computer using a USB cable.
Open QuickTime Player and select "New Movie Recording"
Previously, this was a way to record using any built-in cameras or USB cameras attached to your computer. Now, it supports iOS devices as external "cameras," designed for recording the screen of the device, but also handy as a document camera if you open the camera app.
Handy for recording a workflow for your students (it will grab audio from your computer microphone). Handy for a wired document camera solution without the need to fiddle with your projector.
You could also connect a random student iPad and show off something if that's your thing. And yes, the window will change its aspect ratio if you rotate your device to landscape.
Requires the device to be running iOS 8, and the Mac to be running OS X 10.10 Yosemite.