Place the slate tile flat on a sanding table with the front side facing up. Draw the design you plan to carve onto the tile with a pencil.
Trace over the pencil line design with a scratch awl to scratch the final design into the slate so that it will be visible while you are carving. Hold the awl at a slight angle to the slate like you would a pencil. Apply moderate pressure as you follow the pencil lines until the design is scratched into the surface.
Turn on the Dremel/rotary tool and lower the tip of the bit at the start of the design lines and guide the tip along each of the lines. Allow the tool to cut into the slate, applying just enough pressure to keep the bit in place against the slate. Continue working along all of the design lines until you have the overall lines of the design carved into the surface of the tile.
Go back over the design and add depth to the carving by moving the bit back over the lines of the design. Add texture to the carving by moving the bit in different strokes across the surface of the tile. Continue refining the design until you are satisfied with the overall appearance of the tile. Turn off the Dremel/rotary tool and set it aside.
Wash off the surface of the slate tile with a cloth rag and warm soapy water to remove all of the dust left from the carving. Allow the tile to air dry before using it in your home project.