Find a photo of a bobcat so you know what the fur looks like. You are going to try to imitate it with the felt.
Sketch the shape of your rug onto the scrap paper. A rug made from an animal pelt usually has the four legs splayed out and the head and tail still attached. This rug is flat with no head, but you'll still want to draw the shape where the head would be.
Add the spotted fur design to your sketch. You'll notice that on some bobcats, their spots look more like bumpy stripes rather than completely separated dots. Drawing the fur this way rather than individual dots will make it easier to assemble the rug.
Find the center line on your sketch and draw half of the rug pattern onto the grid of the paper so you will have identical sides for the rug. Add a 2-inch border around the edges of your drawing.
Cut out your pattern. Leave the 2-inch border in place and cut out what will be the brown areas of the fur leaving the dark spotted lines behind.
Fold the gray felt in half lengthwise and lay your pattern on the felt aligning the edge of your pattern with the fold of the fabric. Trace around the shape of the rug onto the felt and cut out.
Repeat the step above on the black felt. You are only cutting out the rug shape at this time, not the design of the fur.
Lay the black felt flat and trace the black spotted lines onto one half of the the felt with the chalk. Flip the pattern over and trace onto the other half. Cut out the areas of the black felt that the brown felt will show through.
Cut the brown felt in half and lay the pieces on top of each other matching the edges. Unfold the gray felt and lay it on top of the brown felt and pin the layers together.
Cut the brown felt to the same size and shape of the gray felt. Glue the three layers of felt together, matching the edges and let dry.
Place the black felt over the brown felt, matching the edges. Arranged the dotted stripes and pin to hold in place.
Glue the outside edges together of the black felt to the brown felt first. And survey your fur design. If you are happy with the look continue gluing until all the pieces are secured. For a more dotted look continue to the next step.
Glue down the dotted stripes working from the outside edges in. Cut apart these bumpy stripes where you'd like to show some separation or a very clear dot on the fur and glue. Continue over the whole rug, cutting apart stripes and gluing down until you are happy with your design.
Let the glue dry thoroughly before walking on.