She waved a piece of paper at Jo.

She waived a piece of paper at Jo. I don't know the context, but this sounds like a scenario where someone is holding a letter, or a bill, or a legal document and they waive it at someone, saying, have you seen this? Or, what is the meaning of this. This sentence sounds like the piece of paper is significant, not merely a device to get Jo's attention. She waived to Jo - this means she waived with her hand, as a form of greeting.