Remove the molding or trim off the upper edges of the existing half-wall with a pry bar.
Measure the half-wall from end to end with your measuring tape to determine the length. Put on goggles to protect your eyes while sawing.
Cut two 2-by-4-inch boards according to your measurements. You will use these boards as the bottom and top plates for your wall frame.
Stack the bottom and top plates one on top of one another and lay them both on top of the half-wall. Use your measuring tape to measure the distance from the top board to the ceiling to determine the length needed for your studs.
Cut the 2-by-4-inch boards according to your measurements, creating enough studs so they can be placed at 16-inch intervals in your frame.
Lay the bottom plate on a flat surface on the floor. Nail in the end studs, then nail in a stud at each 16-inch interval between the end studs. Secure the top plate to the frame by nailing it to the studs.
Enlist help to lift the frame to the top of the existing half-wall. Secure the new half-wall into position by nailing it to the original half-wall, the attached side wall and the ceiling at 6-inch intervals.
Cut 1/2-inch drywall pieces, using a utility knife, according to the measurements of the sides and the exposed end of the new wall portion.
Secure the drywall to the framing with drywall screws. Insert a screw at 16-inch intervals until the head is embedded just below the drywall surface. Provide a smoother finish at the outer edges of the wall by nailing corner beads into place.
Spread joint compound over the joints between drywall sheets, over the corner beads and at the seam between the new and old half-walls. Use your drywall knife to smooth the compound over the surfaces.
Lay drywall tape over the joint compound. Use your drywall knife to push the tape into the compound. Allow the joint compound to air-dry overnight.
Smooth the dried joint compound with fine-grit sandpaper to avoid bumps and wrinkles that can detract from the final appearance of the wall.
Repeat steps 10 through 12 two more times to finish the drywall process. Vacuum the area and the wall surface to remove dust.
Paint primer over your wall extension. If you intend to paint the original half-wall or any other attached walls, paint them with primer as well. Allow the ´primer to air-dry for at least six to eight hours.
Paint your newly extended wall with one coat of paint. Allow the paint to air-dry for six to eight hours before adding a second coat.