Yes it is very possible and your approach makes good sense. Edison (the utility company) has contracted to have a plant built using 38 ft. parabolic (satellite style) dishes that will provide enough solar energy to produce 25 kW of electricity each. That is about 22 watts per square foot of dish. Your house probably uses much less than that so you can use a much smaller collector (dish).
I have been wanting to try to make a large collector using a method described in Scientific American (and discussed on some home made telescope forums). This method doesn't make very good telescope mirrors but should work great for a solar collector. Essentially you use the fact that a liquid spinning in a container will take the shape of a parabola (liquid mercury telescope mirrors). In the article a round pie pan is used on a record player turntable to spin some slow drying epoxy. When the epoxy hardens you have a near perfect parabolic surface that only needs to be made into a mirror. This can be done with commercially available kits. Slow drying epoxy is used because it will average out record player vibrations, allow time for air bubbles to rise and pop and prevent warping (caused by uneven heating during the cure because the epoxy is thicker on the outsides than near the center). I want to scale up this approach a lot. I'm thinking a 6' sattelite dish. One of the old fashioned solid (fiberglass) ones that will hold the epoxy. It is important to have a parabolic shape to begin with so you don't have to use massive amounts of epoxy which would be expensive and heavy when finished. First glue on a dam around the outside to contain the epoxy, plug/cover any hole in the middle and seal any seams, this is your form. Mount your dish/mold to your variable speed mold spinning thing (probably easier to say than do, I'm considering a electric potters wheel, mabe you guys can think of some better ideas). Mix-up enough epoxy to put a thin layer on the surface of your dish, pour it in, spin slowly up to the speed that makes an even coating from the bottom to the top and hold that speed until the epoxy hardens. Apply the mirror metalization and you are ready to catch some serious solar energy.
P.S. A nail, string and pensil does not a parabola make. If you swing the pensil all the way around you will make a circle not a parabola.
P.S.S. If you want to make a "broken mirror refelector" I would suggest the following: Make the mirrors the same size as the target area you want to heat, this will minimize the number of mirrors you will have to cut and place and will be as effective and probably more effective than many smaller mirrors. As you get further from the center you can make the mirrors even larger in one dimension because of the angle of incedence will be increasing. I think I would mount the mirrors in the sun. Mount a pole with a plate on the end of it (the plate will be slightly larger then the target you will have for your project) to the center of your refelector long enough to establish what will be your focal point. Point the refelector at the sun (the plate's shadow will be centered on the base of the pole) glue the most inner mirror down first making sure they reflect the sunlight centered on the target. Cover that mirror, check your sun alignment (the sun moves in the sky!) and glue the next one. I suggest you cover each mirror as you go so you can see the refelected light of the mirror you are working on and so the pole/plate doesn't burn up! One feature of the broken "mirror" that I think is really cool is it doesn't have to be dish shaped. Because each mirror is "doing it's own thing" the refelector can be a flat sheet of plywood or any shape you like.
Next time I'd like to talk about a low buck ($100-150), high power Stirling engine idea I have.