Any mic in most situations is only as good as how close you can get it. A shotgun is nothing like a zoom lens, especially for interiors. A camera-mounted shotgun at the back of the class will be getting lots of indirect sound, that is, the reverberation in the room, not to mention the noises students make. This means echo and reverb, sometimes difficult to hear the speaker, not clean sound at all.
A wireless lav like a Sennheiser EW100G2 is a reasonable choice, and $350 is a reasonable price, if the system is one that uses legal frequencies. A few years ago the FCC in their wisdom reallocated a lot of the frequencies that were popular for wireless, so you don't want to buy a unit on the "old" freqs.
Info here:
http://www.fcc.gov …/wireless-microphone-faqs
You don't want anything between 698 and 806MHz, it's illegal to use in the US. Still legal in Canada, I think? The Sennheisers have the freq range printed on the back - very small print.
There are a few potential points of failure with wireless - make sure you have it working at home before going out on a job, monitor with headphones if possible, and listen to some playback. There are some setup guides on youtube...
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College and workshop instructor in video and audio.
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