No problem! Here are quite a few wireless solutions for you...
(1) Apple TV - Go ahead, spend the money. In its fourth generation, Apple TV is definitely pretty cool and will hook directly to your projector system, but it comes at a cost of close to $150 (the 3rd generation can be found for almost a third of that price now, and completely serves the purpose you'll need). You'll be able to Air Play any Apple device (even newer MacBooks), link in to your Netflix account (plus lots of other content), and might even be able to stream YouTube without it being blocked by your school's web filter (Bonus!!). It's a great solution, that is, if your projector is new enough to have an HDMI input. If you're old-school VGA like most of us, you're out of luck.
(Image source: https://upload.wikimedia.org/wikipedia/commons/7/72/Apple_TV_2nd_Generation.jpg)
(2) Reflector 2 - This is just one of several apps / website solutions that can be used to screencast your device. Reflector 2 allows you to mirror multiple devices to your computer, which then allows for projecting. At $15 for a single user license (multiple users can get a discount) the cost is much less prohibitive than the Apple TV option. With a little bit of work, you can mirror multiple types of devices (iOS or Android) at the same time within the same system. This offers an advantage over the Apple TV which only works with iOS devices.
(Image source: http://airsquirrels-images.s3-us-west-2.amazonaws.com/press/wp-content/uploads/2015/07/23114506/Reflector-2.png)
(3) Zoom Screen Share - The Zoom app is primarily used for web conferencing, and it's one of the better ones I've used when it comes to maintaining a good connection. Like many videoconferencing apps, it comes with a screen sharing feature that supports iOS devices like your iPad and iPhone. And guess what? It's got a free version! Set up a Zoom meeting on your desktop computer and connect to it with your iPad via the meeting code. Voila! Free screen sharing for up to 40 minutes with just the basic version of the App. One thing to note - you can screen share within an app that you're using on your iPad and it works great. True mirroring (so that students can see your desktop icons, every tap you make, etc.) requires both devices to be on the same (wireless) network.
(Image source: https://zoom.us/)
The Zoom app would provide a great way for you to test out the features of the iPad (or connect via Android devices as well) and see how much you would truly use it during instruction. You might find that it's enough for the purposes you'd use it for, or it would be a good way to make the decision about whether or not to spend the money on an Apple TV.
