We are conducting a high school cross country meet and plan to distribute tongue depressors with place values we wrote on them (1, 2, 3, etc.) as the runners come in, so that they can report to the scorer with their name and school. Could this use of the tongue depressors be in any way construed as a derivative work that would infringe on some copyright?
Hiya, Teach, at first I thought your post was a joke. But then I looked at your other posts and see that you may be a wee bit sensitive when it comes to imperiling yourself of innocent, innocuous and infinitesimal IP infringements. (Got my alliteration gold star for the day!).
Sorry - I'll stop poking fun and get to the point. Eventually.

Have you ever seen any tongue depressors with place values written on them? One fundamental requirement of actual copyright infringement is that you
actually copy. You can not by definition "copy" that which you have never seen. Creating a new thing on your own - even if by happenstance it looks like something another person previously created - is not copyright infringement. (Assumes that in the end, the other guy doesn't dig up proof that you're a scurrilous lying dog and were already aware of his stuff.)
Next. Another fundamental of copyright in loose everyday language is that only original creative expressions are worthy of copyright. Let's presume no one else had numbers written on wooden sticks, then you came along and did it. Then that idiot down the hall teaching one of those "soft" sciences copies your sticks! Do you think you could assert copyright infringement? Are your sticks an original creative expression? I just don't think numbered sticks cross the threshold. And if you wouldn't have a case, then someone else who (may have) made numbered sticks likely wouldn't either.
Now, where I would be worried from a legal standpoint is when the first place finisher sticks his stick in mouth and waves that big ol' Numero Uno at and for purposes of mockery of the 2nd/3rd place finishers, and one of them jams the stick back into the mocker's throat.