One of my favorite first grade cutting & glueing lessons: cutting from the edge of a folded piece of construction paper.
I cut and students cut on the folded edge.
I cut and students cut on the open edge.
After students cut on the open edge, I show them how, if they would like, they can use the half pieces cut from the edge to make a head and tail.
This is a simple lesson that has the potential to produce fascinating results. I love seeing all the variations students come up with! And students love this lesson too--they like the comfort and confidence of being able to master this basic cutting technique and they can't wait to start when I announce "you can cut and combine the pieces any way you like."