How are we supposed to know which word for teacher is correct?
I do not quite understand what the problem is. If you have to translate "teacher" into Ukrainian and you do not know the word then you can always hover with the mouse pointer over the word to get a tip.
On the mobile app, this was one of those exercises where you pick the words from a list to construct the sentence. That type of exercise doesn't have the ability to see translation tips.
It sounds more like a general Duolingo problem and should be discussed in the general Duoling discussion. I do not think that the authors of the Ukrainian course can do anything about this.
Учитель оr вчитель?
Both, there are some situations where you can switch "в" and "y".
Both variants are accepted, but the best one is "учитель," because the previous word ends with a consonant (й).
What is the difference between вчитель and учитель?
What is a difference between "учитель" and "вчитель"?
What is the difference between учитель and вчитель?