Is it incorrect to use a sentence fragment to answer a question?
Solution 1:
There's nothing at all wrong with OP's two sentences as written, but that doesn't necessarily mean the teacher was ignorant. I don't know what other factors might be involved, but in general teachers should be respected (not a morality issue - you just won't learn so well if you don't).
There is no 'rule' saying exactly what constitutes a valid utterance. Much depends on context. In this case the first sentence is more than enough context to make the second one valid.
Personally I think it reads very fluently.
Solution 2:
No.