Walt Wright has the checkerboarding, which does not involve re-arranging the brood nest.
Expanding the brood nest has couple of flavors depending on when you do it. If you do it early in the year, when you still have a chance for cold nights, you can put some empty frames right on the edge of the brood nest. The bees will expand into them when they are ready for it.
The most widely used version uses empty frames (no foundation) and works the following. You can start doing it when the bees can fill a space when you take a frame out. Then you take 3 frames of brood out and replace them with empty frames. You put the three frames of brood in the box above. Your bottom box is going to look like that (b is brood frame, e is empty): bbebebebbb. Depending on whether it is still cold, in the second box, you can either put the brood frames all together or space them with couple of empty frames, too. This will entice the queen to lay in both boxes, usually.