Place the fake fur on a flat surface with the fur side down.
Position the plate on the backing in one corner of the fake fur. Draw around it. Repeat for the other three corners of the fake fur.
Make a mark at the halfway point on each edge of the fake fur. Measure 5" in from each edge at these marks.
Following the curves at the corners to the 5" marks in between, sketch a line around the fake fur backing. This will give the appearance of a bearskin body with the paws at the corners.
Cut the backing along the drawn lines. Cut only the backing, not the fur. You will not need to hem the edges of the fake bear rug, as the fur will camouflage the cut edge and will not fray.
Cut the head off the stuffed bear, 1" below the neck seam. Choose a bear with fluffy stuffing; the pellet stuffing can be quite messy.
Turn the fake bear rug right-side up. Position the head in the center of one short side, facing outward.
Turn the cut edge of the bear's neck under 1/4" and pin in place.
Hand-stitch the head securely to the fake bear rug using the carpet thread. Carefully check that all pins have been removed. Dab a bit of glue over the beginning and ending knots to prevent the thread from coming undone.
Tie the ribbon into a bow around the bear's neck.