Flip the switch above the starting area. Thiis lets you climb the ladder.
On the next floor, use the falling monstar as a springboard, making sure you move above the point at which you flipped the switch.
Jump on the monstar to go over the barrier, and collect the piece.
This puzzle piece will remain collected due to the green sparkle.
From the first puzzle piece, move to the right-hand side of the map.
You need to spring over teh monstar, but position the jump so that he survives and gets the green-sparkled key. Grab the key from the monstar.
Climb the ladder leading to the upper area, and open the door for the piece. Both the door and piece sparkle green, and persist if you move left.