Cutting the mattress is pretty much the same as cutting the box springs except there is no wood in the mattress. So cut through the outer covering, cut the springs and sew the ends to form the two mattress sections.
Use short pieces of thread to lessen the tangling and knot often. You don't want the thread to break and a large section of your seam to unravel. It is easier to have a second person help you hold the mattress and the seam as you sew. I also folded the seam part over to have a more finished look.