How do you sharpen a plastic spoon I don't even
the same way you sharpen anything into an improvised weapon
find something with a sandpaper-like texture (concrete, for example) and rub the object against it so that it becomes sharp.
(I used this to smooth rocks at my grandmother's house, there were nice white rocks by the fish pond kept in place by some concrete bricks and I made the rocks smooth. I took hours doing this and never really finished smoothing one. It would take less time with a spoon though because spoons are plastic.)
