To make the drawer sides, rip 1/2" plywood to the depth of the existing drawer depth. Cut the ½" plywood to length at the full dimensions of the drawer box. You'll want to create two identically sized pieces for the sides, and two to create the front and back.
At the table saw, you'll need to cut a dado groove into the sides of the drawer for the drawer. Place a dado blade into a table saw with the dado thickness set to 1/4". Set the table saw fence so that the dado cut is 1/4" from the bottom edge of the drawer sides. Run the drawer sides through the dado cut on all four sides of the drawer bottom (Images 1 and 2).