Using Discord to support online teaching
Answer to "How do I log onto discord? How do I invite my students?"

Head to https://discordapp.com/ and follow the quick directions to make an account. You should then see a screen that looks like this:



Discord is broken up into "servers", which are subdivided into "channels". There is a list of servers you are currently logged into on the left bar. Click the plus icon to make a new channel. You'll see a screen like this:



Click "create a server". You'll see the following screen, where you can name your server:



You'll then see this. If you have interacted with people on Discord before, it will suggest you invite them to your class. (I've blacked out the names of the people I've chatted with.) Instead, select the shown link and e-mail it to your students. The channel will remain in existence and be permanently available.



Answer to "How do they add themselves to voice and text channels? How can I break them into smaller groups for group discussion, and how can I monitor those groups."

Each server is subdivided into channels. Voice channels play in your headphones and listen to your microphone; text channels are chat windows. You can only have one of each active at once, but you can easily switch between them; simply click on the name of a channel to switch to it. The list of your channels looks like this; each voice channel displays under it (scribbled out in the image) who is in that channel.



What I've done is made two channels of each type and asked students to split themselves among them. I click back and forth to eavesdrop on and join conversations.

Note that the past history of the text stream is continually available. So, if students have been typing a lot, it is quick for you to scroll back and see what has been said. I would suggest you encourage students to type, the same way you would encourage them to write things down when doing group work in person.

Answer to "How can I and students type math in discord, if we know LaTeX?"

You can add bots to Discord channels. The way I know to do this is, after you have already created the channel, go to the webpage associated to the bot and click on the invite button; invite it to your channel. For example, go to  https://top.gg/bot/510789298321096704 and click on the Invite button as noted here:



MathBot provides a quick text interface to LaTeX, a basic calculator and WolframAlpha. If you type =tex before something, MathBot will render it in LaTeX mathmode. Beware that this means that, if you want to mix text and equations, you'll need to say something like 

=tex \textrm{Let}\ k \textrm{be a field and let}\ S \textrm{be the polynomial ring}\ k[x_1, \ldots, x_n]

Preceeding a line with =calc will access a simple built in calculator, for example, =calc 9*43 should produce 387. Using =wolf should pass whatever you type along to WolframAlpha.

TeXit allows people to type LaTeX (demarcated with dollar or double dollar signs) and have it turned in to TeX. This is far more user friendly than the TeX feature in Mathbot: Simply typing Let $k$ be a field and let $S$ be the polynomial ring $k[x_1, \ldots, x_n]$ should work with TeXit running. 

For more help regarding TeXit's commands (which also include a calculator and WolframAlpha), type ,help and it will send you a direct message.

Answer to "How can I and students embed photographed images or hand drawn sketches in Discord?"

You can also drag and drop files into a Discord channel. PDFs turn into links; standard image files such as JPG, PNG, GIF are embedded in the stream. See here. This could be useful if your students have tablets, or have paper and cell phone cameras, so they can take pictures of their work and upload it.

I will report, however, that in a recent test run, most students found the "make image, transfer to desktop, drag-and-drop to Discord" workflow to be clunky and awkward. We are considering whether we want to use shared whiteboard software as an alternative. Other suggestions are welcome.

An easier way to share quick sketches is by pasting images directly into discord. You can draw a sketch in MSPaint or some other program, copy it the image itself from within the program (no need to save it as a file), and then paste the image into any text channel. This is also convenient way to share screenshots; Windows can snip portions of the screen with Win+Shift+S, and Mac and Linux have similar features. After taking a screenshot into your clipboard like this, it can be instantly pasted into Discord.

For more complex things, you can share your screen in realtime if you are using the desktop Discord application. Simply click the "Go Live" button in the bottom left:



You can then share either a specific application or your entire screen. Others can watch by clicking your username (it will have a bright red indication that you're live), and then clicking Join Stream.

More information here.

What are some other tips?


Note that you can mute your microphone when you are not talking, to cut down on static. If you frequently forget to do this, Discord's "push to talk" feature makes it so your microphone is only on when you hold down a specific key on your keyboard. See here for more.
A protocol which has worked well in large VOIP calls I have been in in the past is that, if someone else is talking and you want to talk next, type the letter "Q" in the chat window. When they finish, all the people with Q's speak before anyone else starts. That wasn't much of a problem for my class today, but I plan to enforce this protocol if it seems like some people are never getting a word in edgewise.
You can adjust the volumes of other people in the chat individually. See here.
You can edit past messages to remove typos. Click on the ellipses in the upper right of the message, then click on the edit option. Other options here are to pin a message, so it is permenantly at the top, or delete it.