"We are going home with the teacher."
Translation:Öğretmenle eve gidiyoruz.
There is a difference, and I did a bad job of explaining. (Both of those endings translate to "with" in English, so describing it as a "with" ending wasn't helpful. =) )
Anyway, "with milk" is different from "with the teacher" in Turkish. The "with" that's the opposite of "without" does, indeed, use 4-way harmony. So you have sütlü/sütsüz, peynirli/peynirsiz, and so on.
The other "with" is the instrumental case, and that one's used if you're doing something with somebody (like going with the teacher) or by means of something (if you go somewhere by plane, the "by plane" is "uçakla"). That ending is (y)LA: it uses 2-way harmony.
I hope that makes sense, but if it doesn't, here's the link to the "with" tips and notes (which I'm pretty sure explain it better than I just did): https://www.duolingo.com/skill/tr/With-or-without?
***EDIT: For some reason, I can't get the lesson to load from the link above. (The link is right, but it's not working.) If it doesn't work for you, either, you might have to get to them by going directly to the "With" skill and reading the tips and notes.