As promised, here is the tutorial on making smaller fans. Short and sweet. I totally got this idea from someone else's blog, but totally couldn't find it again when I went to cite it. So, if this looks a lot like one you posted, just let me know and I will give you credit for the idea.
Start off with a 12 x 12 piece of paper.
Cut the paper in half.
Accordion fold both lengthwise. Put them both together. However, I took this pic from the wrong direction; so just rotate it in your head to the right.
Tie your papers together in the middle.
Next, fan out to form a circle and connect all edges with double-sided tape. There should be four edges to connect.