That's true of all kinds of paper that you may want to toss. Here are a few other ideas—
Cut up old maps to draw or paint on.
Put out paper scraps, tape, and markers.
Cut up abandoned paintings - or community paintings -
and use the pieces for projects.
Our paper scraps go into a basket that's always available
(along with glue sticks and scissors).
Sometimes those scraps get used in fun ways at the writing table.
Sometimes those scraps become fun creations.
Our scraps often become a part of other projects.