Ethics of "recycling" assignments
You need to ask your professor. Opinions about this may vary substantially from instructor to instructor. In additional a lot of can depend on the nature of the assignments (are you writing a paper on a topic of your choice, or proving a theorem that has been assigned to you, for example?). 

When writing scholarly papers, you should always cite your own work if it is relevant enough that you would have cited it had it had a different author. Furthermore, if you re-use any figures or substantial amount of text from a previous paper, you need to be careful to indicate this very clearly lest someone complain that you are committing self-plagiarism. (No judgement implied here; my own views on the issue of self-plagiarism are complicated and probably go beyond the scope of the question.)