Smooth out rug-backing felt onto a work surface and place the bear rug on top. Trace around the outline of the bear rug, allowing a 1/2 inch margin, using tailor's chalk.
Remove the bear rug and cut around the outline. Fold over the raw edges of the felt into a double 1/2-inch seam, pressing it flat with an iron and pinning down the seam.
Sew the seam using a double flat stitch on a sewing machine, removing the pins just before you get to them on the sewing machine.
Press the whole backing with the iron to ensure it is smooth and flat. Turn the bear rug upside down and place the backing over the top. The backing should be approximately 1/2 inch smaller than the size of the rug.
Press the backing onto the rug on the lowest heat setting. Double thread the needle with strong thread and tie a knot in the end.
Tack the felt onto the back of the rug, beginning from the center of the rug, working from side to side. Once the middle of the rug is attached, move about 10 inches down and tack on another line. Continue in this way over the whole backing and sew around the perimeter of the backing as well.