teacher 单数,用第三人称 gives 或 过去时 gave ,你用了 give
write 写字,字 在 纸的表面,要 on.
to write some letters on a piece of paper
把 a piece of paper 提到前面 就是 a piece of paper to write some letters on
省略some letters 就是 a piece of paper to write on
所以是：the teacher gave me a piece of paper to write on.
不是：the teacher gave me a piece of paper to write.
不是：the teacher gave me on a piece of paper to write.