I had my kids make this project while we were discussing windmills. A windmill is a circle of vanes for catching the wind and making the circle spin. A pinwheel is a toy version of the same thing.
You start with a square of paper. You can measure out the sides with a ruler and cut with a scissors, or use a paper cutter, or you can buy pre-cut squares of paper. Or you can just take a normal piece of typing paper and fold it twice. This also has the benefit of locating the diagonals, which you will need for the next step. After folding along the diagonals, you would of course mark the bottom edge with a straightedge or another fold, and then cut it off.
Once you have your square of paper, you need to mark the diagonals, and then cut the diagonals about two-thirds of the way into the center. (In my experience, kids tend to cut too far towards the center, which makes the pinwheels flimsy and floppy. It might help to use a ruler or protractor or just your eyeball to make marks along the diagonal, roughly one-third of the way from center to corner, and say “Don’t cut inside this line!” Sometimes children also press their vanes flat, instead of leaving them puffy. Sometimes young children also don’t do a very precise job of taping the tips of the paper onto the “center.” You might want to help them as they do this, to make sure the tips get taped reasonably close to the center, and to make sure they leave them puffy instead of pressing them flat.)
With all four diagonals sliced properly, you can begin folding. Face one edge of the square, pick up the lower-left hand corner of the paper, fold it in to the center and tape it there. Repeat for all four sides. (You could make a left-handed pinwheel by folding the other side instead. It doesn’t really matter. The important thing is that all four sides are folded the same way.) This should complete the actual pinwheel part of the pinwheel.
Now, how do you mount it? I had the kids press push-pins through the center of the pinwheel into the erasers of pencils, and it seemed to work ok. I suggest angling the pins at a slight downward angle so that the sharp point stops against the inside wall of the metal ferrule, rather than poking out the far side.
This design usually works pretty well for me, although depending on how the children formed their vanes, they might have a tendency to catch on the pencil. In general, I find it also usually works better if you wave it through the air at arm’s length, rather than blowing on it.