Breaking a ChickenLet's start with terminology so you don't sound like you fell off the turnip truck. What you're about to (try to) do is called "breaking," or "breaking down," or "piecing," or "piecing out" a chicken. "Cutting it up" could mean anything. More generally, it's a part of kitchen prep called "portioning."
Fortunately, there's one right way to start. With a very sharp chef's knife. A Very Sharp Chef's Knife. A VERY SHARP CHEF'S KNIFE. Getting through? If your teacher recommends using a different knife, listen to her. What do I know?
Alternative 1: Breasts boned out.
If you're right handed, turn the chicken so the legs face away and up, the wings are toward you, and the breast is up. Grab the leg to your right in the middle, and gently pull on it so the thigh opens from the body. Run the point and front part of the edge of your chef's knife in the area between thigh and body and gently bend the thigh back so the joint between thigh and body is exposed.
Now use only the edge, not the point, to open the joint as you continue bending the thigh open. If your aim is good your knife sharp, you will wedge the joint open and cut through the meat cleanly. If your aim is off, act like it isn't and cut through the joint cartilage. Set the leg and thigh (leg quarter) aside on your board.
Turn the chicken around so the other leg is on your right, and separate the leg quarter in the same way. Opening and slicing -- not digging around with the point.
Move the chicken to one side, and lay the leg quarter on the board, skin side down. Use one hand on each end to bend the leg and pay attention to the location of the joint. With your off hand on the end of the leg, use the edge of your knife to slice through the meat at the junction of thigh and leg. If the joint isn't exposed, put your knife down and bend the leg again. Slice through the joint as cleanly as possible. Repeat with the reserved leg. Trim the thighs and legs as necessary to make them tidy.
Get the leg and thigh pieces off your board and onto a pan, to reserve. If your board is very sloppy, wipe it.
Move the body back to the center of the board and remove the wings in the same way you removed the leg quarters. Cut through the joint where the upper wing joins the breast using your edge as much as and your point as little as possible. Repeat with the other wing. Reserve the wings onto the pan.
Feel for the keel bone in the center of the breast. Run your knife all the way along the bird on the right side of the breast. Make several gentle cuts until you start to free the breast from the keel bone. Use your off hand to help peel the breast away from the bone. Without cutting down the ribs or removing breast. Repeat and loosen the breast meat from the left side in the same way.
Return to the right breast. Using your off hand, peel the breast meat out and away from the ribs as your run your knife along and parallel to them. Remove the boned breast and set it on the board. Then remove the other breast in the same way. Reserve both breasts on your tray. Reserve the carcass for roast chicken stock.
Alternative 2: Bone-in breasts.
This starts with splitting the chicken. Turn the chicken so the keel bone is down and cut down one side of the back bone, right through the ribs from front to tail. A sharp knife won't require much pressure. If your knife is at all dull, you'll have to put some pressure on top of the tip with your off hand, and use the curve of the blade to rock through the ribs. Repeat on the other side, and discard the back bone.
Open the chicken as best you can, and lay it breast side up on the board. Get your knife edge on one side of the breast bone, but as close as possible to the center. With an extremely sharp knife you can cut through breast bone. With most knives, you'll have to rock through. In any case, split the chicken. You do not have to remove the breast bone, but can leave it with the breast.
Separate the leg quarters by holding the chicken at the thigh and bending, then cut skin-side up, behind the last rib to remove the legs and thighs. If there's any back bone left, you may have to cut through it. Piece the quarters into leg and thigh, as before. If there is any back bone with the thigh, trim it. Reserve the legs and thighs.
You may either remove the wing as before, in order to save the whole breast; or split the breast into two pieces, one with the wing and one without. If the first, check the back of the breast for back bone or any sharp ribs which may be hanging out. Trim as necessary. If the second, try and portion the breast into even weights.
The two I went through are the most common. Others include, leaving the thigh with the leg, but boning it out; taking the breast off the bone, but leaving the wing with it (very French); etc.
Hope this helps,