Use a sheet of paper and a pen or pencil to trace the hole. Place the paper directly over the hole and trace its outline. Use a pair of scissors or a utility knife to cut out your paper outline. Hold it up to the hole in the drywall to make sure that it matches up.
Cut a square section of drywall a few inches wider (in every direction) than the paper outline. Use a piece of drywall that is as thick as, or thinner than, the depth of your hole.
Score the backside of the drywall patch section. Place the drywall section face down on a flat surface. Place your paper outline in the center of the backside of the patch section and trace around it. Using a sharp utility knife, cut a score line about halfway through the chalk just to the inside of the trace line (you want to make your patch slightly smaller than the hole so that it will fit while covered in joint compound). Then also using the utility knife, cut score lines every inch or so from the trace line to the edge of the patch.
Crack and peel the unwanted drywall off your patch. The end product you are looking for is a square front made of facia paper only, with a back section of drywall and backing paper the same size as your wall hole. To do this, gently apply pressure to each of the score lines until the drywall "pops." Basically the pressure will force the chalk to break along your score lines all the way to the face paper. Once you have popped all of your score lines, slowly and gently peel the chalk sections off the face paper.
Adhere the drywall patch to the wall. Using a putty knife, apply drywall joint compound or patching filler to the back side of the drywall patch. Place only a thin coat, as you will want the face paper to remain even with the rest of the wall. Place your patch into the hole in the wall or ceiling, making sure to press firmly on the face paper to squeeze out excess joint compound. Using your putty knife, cover the front of your patch with joint compound and use drywall tape on any wide gaps if they exist. After several hours of drying, sand and paint.