If you got new carpeting, you probably got some extra pieces once the job was done. A carpet remnant wrapped around a post makes a nice scratching pole for your cat, especially if you sprinkle some catnip before you attach the carpet. However, you may not have a kitty at your house, but I have other suggestions.
Another idea is to keep a small piece by the front door to wipe feet on before hitting your living area. This will keep that new carpet cleaner as a first line of defense.
Scraps of carpet are also wonderful to line dollhouses. Your daughter will be thrilled showing off her new rug like mommy's to her friends.
Elementary art school teachers often depend on makeshift items such as carpet scraps, wallpaper scraps, cardboard tubes and rolls, and other such items to encourage creativity without expense in their classrooms.
The next visit to your child's school, you could always drop some off to the art department, which would greatly be appreciated.
Still, I have one more interesting idea you should try for a pair of work boots. This may sound odd, but it will keep your feet toasty when outdoors for a prolonged period of time. Trace the outline of your feet on a piece of paper and cut out a pattern. Now cut out the carpet using your pattern to insert in your winter boots. These homemade insoles are thicker than the conventional type and will really help keep you extra warm.