My teacher keeps handing out these papers with information, calling them booklets, except it's not a booklet as it doesn't have a cover, just a single staple in the corner of a bunch of printed papers. I know there's a word for it other than booklet, but I can't place it. Any ideas?
Booklet can work and depending on what information is being conveyed it may be the best choice. If the pages expand on an subject the teacher is presenting, booklet is probably best.
Handout, as suggested, is also correct but handout is extremely generic. If I give you one sheet of paper in class, that is also a handout.
Pamphlet is a good word for stapled together sheets but that word also implies that the papers provide information on a single subject. If, for example, the handout is actually your class syllabus then pamphlet doesn't really work.
The simplest solution is to refer to stapled together pages more specifically. If it's the class syllabus call it that. If it's a brochure promoting condos in Hawaii call it a brochure.