Another paper that would work better than construction paper or gift wrap would be black butcher paper, which many schools have on hand. One would, of course need to cut it to size beforehand, and that might take a little extra preparation time.

I loved this for my second grade class of students! (24 little ones) I just cut 6 inches off 12 in x 18 in black construction paper to make 12 in x 12 squares of black paper. Even though construction paper is a bit thick and therefore more difficult to crease (especially when creasing several layers), the project is large enough that it was easier for them to manage.
Minimal preparation, minimal cutting, and minimal gluing. It can’t get much better than that for school projects.
Since we had studied bats for a week and written about them, this was a great culminating activity.

I ‘m wondering if one might be able to find solid black gift wrap–I wouldn’t think a white backing would create any problems–if construction paper was not working well.

It is just black paper!