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Round Snowflake
Fold a piece of paper in half. For a standard snowflake, a normal printer paper(8.5 x 11) is just fine. To get fancy, decorate your paper beforehand in colored pencil, crayon, or marker. Or use colored paper!
Find the center of the paper on the fold. Then grab the bottom corner and bring it to the center, forming a triangle. Then, fold it again, turning over the folded corner. The paper should end up being folded into thirds, slightly resembling a cone shape. If you found that confusing, fold one side over about 1/3 of the way and then fold the other side over to cover the first side. Any better? When you're handling your soon-to-be snowflake, always keep the point down. This is the center of your snowflake.
Fold in half. You have a mini-cone in front of you, right? Just like this?
Cut off the top in a very slight arch. You are cutting through all the layers just where the top layer stops. It should be pretty clear to see. You're now ready to create your snowflake!
Start cutting. You may want to begin with a very simple pattern and then move on to more complicated patterns. Or dive in. The smaller the cuts (and the more in number), the more detailed your snowflake will be.
Unfold the snowflake. Unfolding takes some patience (make sure it doesn't tear!), but you'll have your first six-sided snowflake in seconds. Tada! Onto the next!
Angular Snowflake
Get a sheet of 8.5 x 11, or A4 paper. Create a perfect square by taking the bottom corner and folding it over to the top, at a diagonal. A 3" (7.5 cm) or so piece will be sticking out. Cut off the excess so you're working with just the square. At all steps in this method make sure your creases are firm and straight. If they're not clean and symmetrical, you may end up with a wonky snowflake.
Fold the paper in half, in a triangle shape. This is the same fold you made in the first step and is only mentioned in case you unfolded it to cut. Once it's folded once, fold it again, forming an even smaller triangle. At this point, you may fold it again to create a different, much smaller, base for your snowflake. Experiment! However, for kids, folding it again renders it much more difficult to work with.
Start making cuts. This is where the snowflake gets really good. If you get creative with it, you'll end up with a pattern that's intricate, delicate, and detailed. Or you could end up with a couple of slits in a piece of paper. Cut in curves, angles, and swirls to get the most out of the paper. You may find it useful to hold on to the center of the snowflake -- the very tip. Though you can cut it off and the snowflake will hold together! The more paper you cut off, the flimsier it will be -- not necessarily a bad thing.
Carefully unfold your snowflake. If you've made a lot of cuts, the snowflake may rip, so be careful. And if your cuts were very small, sometimes the layers of paper stick together (think election day 2000). If you're not happy with your pattern, fold it back up and take a few more whacks at the paper. Problem solved.
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