I love giving students genuinely different ways to show their understanding. In Geometry this year, I have been having students record screencasts to explain paper folding phenomena. Basically, I walk them through a paper folding exercise (details below on the two I have done so far) that has a surprising or interesting result. Then we talk about as a class why it’s happening – they try to figure it out together, and I help them figure it out through a full class discussion. Then, they go home and record a video of them explaining the idea, showing me physically on the paper what is happening and why. I give them feedback and they record again! I have found it a great way to engage them in geometrical argument without the annoying technicalities of written proofs.

(Here, a student is using the physicality of the paper to show why when you fold a point onto another point, all the points on the fold are equidistant from the two points)

For video collection, I use Flipgrid which makes things SO EASY. They all go in one place and no one has to worry about saving or uploading files. I limit them to 2 or 3 minutes so that they have to be efficient and I can view them easily.

PAPER FOLDING CONJECTURE 1:
(from Sam)

1. Fold up one corner of the paper in any direction so long as the crease goes between two adjacent sides.
2. Then, fold an adjacent corner up so that it meets the side of the fold already there.
Any conjectures? Students will come up with lots of things, but the fun ones to argue are: Why is that bottom angle a right angle? Why are the two triangles that you made from the folds similar?

PAPER FOLDING CONJECTURE 2:
(from an Illustrative Mathematics Task that I CAN’T FIND right now, halp!)

1. Draw two points on a piece of paper. Fold the paper so that
Any conjectures? We had been talking about perpendicular bisectors, so most students immediately saw that this was a perpendicular bisector. Can you argue that all the points on this line are equidistant from the original two points?
2. Now draw a third point.
3. Fold the other two combos of points onto each other (so if the first fold was from A to B, then fold B to C and A to C).
4. Locate the point that they all meet.
Wait why do they all meet at one point?
5. Now draw a circle with the center at that point, and use the radius as one of the original points.
My circle goes through all 3 points! Why did that happen?