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?

In my precal (pre-cal, pre-calc, precalculus, Precalc, p-Rec-aLk) class, I have multi-day homework assignments that I collect infrequently, a structure that works great for older kids who can plan their own time out well. But I was struggling figuring out how to deal with homework in my geometry class, as I think freshman needed the daily *umph* to keep them going. I wanted a structure allows for:

• Accountability to work hard on it for both completion and understanding
• Feedback on their work
• A workload that I can handle

So I adapted a mode from colleagues at my last school who would roll a dice to see what happens. The system incorporates a little bit of randomness and has been kind of fun. Students, as a class, pick a number from 1-6 and behind that black box is the option for what is going to happen for that day:

If 3 is picked day 1, then that is used up, and we pick the other numbers on successive days until we get through the cycle. Then I shuffle the options behind them and we start the cycle anew. My options right now are:

• Homework Quiz (no notes)
• Homework Quiz (notes)
• Sight Check (x2)
• Collect
• No Check – everyone gets full credit

I give them 10 minutes in the beginning of class to check homework, and the homework quizzes are literally just a problem directly from the homework, so the idea is, if they worked hard on the homework and fixed any small issues they had with them in the first 10 minutes, they should have no problem on the quizzes (spoiler alert: the kids who don’t do homework well have a big problem here, but at least they are realizing it?). The sight checks are just for completion, and the collected homework is graded on completion PLUS the corrections that they did in the first 10 minutes of class – I’m trying to encourage them to use that time really well and then give them feedback on how to do that…

Things that I have liked about this:

• It has been fun! The reveal every day is actually hilarious, though I have gotten in trouble with the teachers next door for noise a few times :).
• It has reduced my workload without really reducing what they get out of homework. Sometimes, I tip the scale by hiding what I want to happen behind all the boxes (i.e. if it’s a good day for me to collect).
• The homework quizzes are good for both me and them for REAL feedback opportunities (no one looks at what you write on their homework…) and as a low-stakes way for them to assess their own knowledge
• I also thought the homework quizzes would take forever – they take about 7 minutes or so, but it’s 7 minutes where they are rehashing their thinking about an important problem, so I’ve found it useful and a good tradeoff for instructional time.
• Going through the cycle as opposed to rolling evens out the workload a lot and ensures that there are quizzes at regular intervals.