Interesting conjectures

Today was my first day of classes — thirty minutes with each class just as a get to know you. Of course you know me. I dive right in and we did math in all of my classes.

The class I’m most nervous about this year is Geometry. It’s my first time teaching the subject, and it’s my first time teaching 9th graders. I had planned a paper folding activity which would get the students noticing and wondering. And importantly, then conjecturing.

Here are a quick snips from the paper folding activity.

The big idea was to get kids to realize that no matter which two points you chose initial to make that first fold, you’ll always end up with some things that are true in your final figure. Each student at a group folded two different pieces of paper — differently. Each group then looked at all their paper folds and made some great observations.

The couple that stood out to me:

When you make the folds, you will always end up with a pentagon. Some of the kids had a really small side, so they didn’t “see” the pentagon, but with a little prompting they did.

The five sided figure has three right angles. The two at the top (the corners of the original sheet of paper) and the point at the bottom. The point at the bottom is always going to be a right angle. 

As a class, people shared their group’s observations. And then we focused on that third right angle. THAT IS WEIRD.

And so I sent the kids off to try to come up with some reasoning for why that third angle is always a right angle.

And this is where I had my good moments. This is not an easy question for the very first day of class. And with the 8-10 minutes they had, no group completely made a perfectly sound set of reasoning. But so many were making statements that were getting them closer to the answer.

A few groups noticed that the two folded triangles looked like they were always similar. (Indeed, I had totally missed that when I first did this problem…) Some kids were able to come up with the logic that if the triangles were similar, then they could actually prove the bottom point was 90 degrees. Loved it! (As of yet, no kid or group has proved that the triangles are similar.

And a few groups were also noting that when you unfold the paper, there is something really remarkable about the bottom of the page, with the two creases emanating from that point (which is the vertex of that 90 degree angle when folded). They noted that somehow — by folding up the two triangles — they are splitting the bottom of the page (a 180 degree angle) into three angles, where the middle angle is always 90 degrees.So they have a 90 degree angle, and then two smaller angles that add up to 90 degrees. I said: that’s awesome. Now how do you know that middle angle is 90 degrees, no matter where the creases are? (As of yet, no kid or group has explained this halving.)

They are getting there.

Even though I was nervous about this being too hard and even though I wanted to provide more hints and more structure… I let things be. I wanted it to be tough. It wasn’t about the answer, but about the process of talking with each other and thinking and persevering and reasoning.

In fact, I wrote on the board that kids can ask me for one hint to help them if they got stuck. No group asked for a hint!

I am excited to see what happens when I next see them (Monday). I am having them try to figure this out at home, and write up whatever they could discover. If they could figure out the reasoning, great. If not, what did they think about and attempt.

My one good thing from today was watching my kids think aloud and struggle aloud and come up with really interesting ideas. Ideas I thought they might come up with, and ideas that were fresh to my eyes!

Here are all our paper foldings!


[Note: I am posting this both on the one-good-thing blog and my blog, because it both is a good thing and because it deals with teaching my classes!]



    1. YAY! I always love to bounce ideas. This year this blog will probably be pretty geometry centric as this is a totally new adventure to me!

      If you have any thoughts on how you’d finish this off, great! My thought was to have the kids in their groups share what they came in having written, given them a bit more time to see if they have one set of reasonings that really convince them, and then ask for some kids to share their thoughts.

      Some follow up questions I was thinking of:

      1) Do you always need an 8.5×11 sheet of paper for this to work?
      2) How can I make a perfect 45 degree angle out of this 90 degree corner angle?

      Since this is about finding a pattern, making a conjecture, and articulating reasoning, I have a follow up individual activity for kids to do at home which asks them to do all three.

      But I’m open to any suggestions/ideas on different ways to go about “closing” this activity!

      1. Have them fold the 45, then fold in half again and see what happens to the folds. Let them conjecture about what is happening to the angles they are creating. To help them with the 90 degree idea, let the put four angles together and “discover” 360 degrees or the Cartesian grid.
        You can also give them a circle and ask them to fold it into a triangle. There are great geography shapes that come out of that!

  1. Cool!! I love the messages about reasoning and productive math dispositions that you are sending to the kids through this activity. I’m curious to see where things go next!

  2. Awesome.

    In one of your slides it looks like you have a great way for students to know how much time they have to do a task. What is that called? Where can I find it?

    Thanks so much!

    1. I use SmartBoard software, and built in are various thingamabobs, like a timer. I’d bet you can find something alternative that does the same thing, though.

  3. I love this idea for the beginning of the school year! I’m curious about the follow-up homework you referenced. What did that look like? Did you give a second folding exploration or something different?

    1. I don’t totally remember, but it wasn’t a paper-folding thing. I think it was “do this thing and make conjectures about what you notice.” (For example, it might have been something like “draw in the diagonals to the fifteen kites below… list all conjectures you can make…” Just anything that they can come up with lots of observations/conjectures with very low entry points.)

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