Attention: I'm not a teacher.
There are a few words concerning the matter in question.
paper (n) [uncountable] material in the form of thin sheets that is used for writing on, wrapping things etc
I bought some paper.
piece/sheet of paper
Write it down on a piece of paper.
When paper refers to the material that you write on, it is an uncountable noun. Do not say that something is written on 'a paper'. Say that it is written on a piece of paper.
paper = newspaper [countable] a newspaper:
Have you seen today's paper?
You'll read about it in tomorrow's papers.
document/letters papers [plural]
a) pieces of paper with writing on them that you use in your work, at meetings etc:
I left some important papers in my briefcase.
b) documents and letters concerning someone's private or public life:
While I was organizing Simon's papers I came across his diaries.
c) official documents such as your passport, identity card etc:
My papers are all in order (=they are legal and correct).
examination [countable] British English a set of printed questions used as an examination in a particular subject, and the answers people write:
an exam paper
I have a stack of papers to mark.
piece of school work [countable] especially American English a piece of writing that is done as part of a course at school or university [= essay]
a paper on the Civil War