At the end of the past summer (oh how long ago that was! glorious days of freedom!), I hunkered down at a coffeeshop with @jacehan in preparation for this school year. I was fixing up some of the things we did last year in geometry. One thing I wasn’t pleased with was how we taught the crossed chord theorem…

So I created a totally new approach. Instead of having students discover the theorem, I would work backwards. Here is the TL;DR from my last post — after I had created the activity.

The TL;DR version: students investigate all quadrilaterals where the diagonals satisfy the property that ac=bd. Students are guided to make a conjecture which we as teachers know will be wrong. Then we show a counter-example to blow their conjecture up. And them bam: they have to try again. Using geogebra and some more encouragement, students discover that all cyclic quadrilaterals satisfy ac=bd. And so the circle emerges out of this investigation of quadrilaterals and diagonals. This is, then, the crossed chord theorem. Which students got at by investigating quadrilaterals. Weird. Now they are in a prime place for wondering why the circle shows up. Proof time!

When I shared the activity, I got a couple suggestions from @k8nowak and @bowenkerins and so I modified it with a single tweak which made it oh so much more powerful. In this post, I will talk about my experience implementing the activity, as well as share the modification I made. However I entreat you to read the original post as I’m not going to outline everything again! So go read! Okay? Okay.

(1) First off, the change. I made a change to the very last question on the sheet… Instead of having students look for blermions “in the wild,” I had them fix three points and find a bunch of different locations for the fourth point (so the quadrilateral would be a blermion).

[docx: 01 Crossed Diagonals [new]]

At the end of class, I had students fill out a google form with their possible fourth points location.

(2) In class, when students were filling out the conjecture in #4, I saw a number of interesting conversations happening. Their conjectures were essentially: (a) all blermions need at least one pair of parallel sides, (b) all blermions have supplementary adjacent angles, and (c) a blermion has opposite angles supplementary. Most students didn’t find any kites that were blermions, which is why they came up with conjectures (a) and (b). But when the few students who found blermion kites said this to the class, we realized that (a) and (b) couldn’t hold anymore. But conjecture (c) was a possibility still.

Now to be clear, I was expecting conjectures (a) and (b). I was floored when not one but two groups out of five wanted to persevere and find a good conjecture, and used geogebra to measure angles. It was awesome. And it led to a great discussion later on. More on that later.

(3) I asked kids why I had put question #5 on the sheet… what might have been my motivation? I liked asking that question and having groups discuss, because they all recognized that by only looking at “nice” shapes (which, granted, I asked them to do), they could only make limited conjectures. And as soon as they see the blermion in #5, most conjectures would go out the window. The point? To show students that all quadrilaterals aren’t “nice.”

(4) The moment when students saw all their group’s data together in #6… Well, two of my groups got to this point in class. It was … incredible. Kids had their minds blown. Something totally unexpected happened.

For the other groups, I shared the class data (from the data they entered in the spreadsheet):

Holy cow! It is so beautiful! All possible fourth points of a blermion seem to lie on a circle!

(5) Students then wrote a conjecture, and we said if the conjecture were true, we’d suspect (from the Always Sometimes Never questions in #3) that all squares, rectangles, and isosceles trapezoids could be inscribed in a circle. And we discussed how we were going to prove the opposite: If you have a cyclic quadrilateral (yes, I introduced that term!), that product theorem thingie (a)(c)=(b)(d) holds with the diagonals. Okay, we were a bit more formal, but that was the crux of things.

(6) Before proving that, I wanted to exploit the conjecture (c). I had students prove that all cyclic quadrilaterals had opposite angles that were supplementary. They struggled a bit with this, but once they had their insight, BOOM. (They used the inscribed-central angle theorem thingie — a central angle is half its corresponding inscribed angle).

(7) Then I left students to prove the crossed chord theorem. I gave them this sheet:

[docx: 01 Crossed Diagonals (proof)]

Almost all kids got to the point where they recognized two pairs of similar triangles. And they recognized that if they could prove one pair of similar triangles were congruent, they could set up a proportion and be done! But the problem was proving the triangles similar. Almost all groups got stuck here — and even though I said: you’re almost there! Think about the inscribed-central angle theorem! — they couldn’t progress. I didn’t do a great job of knowing what to say next.

What I did was show them this (which they had created earlier). For some reason, this did not work for them as a hint.

In the future, what I should do is just highlight an arc for them… and say “this arc can guide you!”

Maybe that will work better?

However, eventually all groups got the proof.

(8) At this point, I had students start solving problems. Two with quadrilaterals and two with chords.

 

Again, I asked them why I included had the second two types of questions, and had the discuss in groups. They recognized that the theorem didn’t need to be stated with cyclic quadrilaterals… Instead it held if we are talking about two line segments in a circle (at that point, I introduced and defined the terminology “chord”). Then I had students write the theorem we had proven without reference to the quadrilateral, and we went around and shared and critiqued the wording.

***

I don’t always love the stuff I come up with. Sometimes it flops. Sometimes it’s pretty good. But rarely do I think it’s so awesome that I would give it the stamp of “highly recommended.” This gets that from me. It is interactive, there is a moment when kids’s minds are blown, and it ties together so many interesting ideas.

 

For context, I did this after we did our unit on similarity. We then proved that the base angles of isosceles triangles were congruent. We then used that to prove the inscribed-central angle conjecture (download here: A Conjecture about Inscribed Angles). And then this. It flows so nicely.

***

Yesterday, as we were wrapping this all up, I said to my kids:

“This thing we just proved about circles and chords… this is at the top of a mountain… this theorem is based on lots and lots of other things. If I gave you a bunch of circles with two intersecting chords in it at the beginning of the year, and said, give me some conjectures about this, I doubt you would ever have stumbled upon this… or if you did, it would have taken you a long time and it would have been an accidental discovery. You still wouldn’t have known why it was true. But now you have built so much mathematics throughout the year that this wasn’t an insurmountable feat. What ideas did this theorem lie on?”

And we even had more. We had to know a lot to get there. But wow, it might have seemed impossible at the start of the year, but it was totally doable with all the tools we’ve put in our toolbelts. And how wonderful and inspirational is that?!

Update: Here is a post about an extension I did on this — involving merblions.

Brendan — my geometry coconspirator — and I went to the Museum of Math recently to see Erik Demaine give a talk about math and magic. It was a special lecture for me because I saw Prof. Demaine speak at the very first Museum of Math lecture (before the museum was built), and this was the five year anniversary of that talk.

Prof. Demaine and his father Martin Demaine both are mathematical artists — playfully using mathematics and art in search of higher truths. The most mindblowing thing that he discovered was that by folding paper however you want, and making only one single cut, you can cut out any polygon. Evenmoreso, the theorem goes further: “Thus it is possible to make single polygons (possibly nonconvex), multiple disjoint polygons, nested polygons, adjoining polygons, and even floating line segments and points.” [1]

Whoa, right? So say you want to cut out each letter of the alphabet? Done.

Or you want to cut out a swan or jack-o-latern?

You can do it. It boggles my mind.

When we went to Prof. Demaine’s talk, on each chair was a packet of paper and a pair of scissors. We were challenged to “fold and cut” each of the shapes out. The shapes were scaffolded well, and so I got pretty far along and was figuring things out. At that time, Brendan and I realized that both angle bisectors and perpendicular lines were key for much of what we were doing. We also realized that the puzzle nature of the challenge got us obsessed. We both were stuck on a single page [I’ll write about that in the P.S.] and as I was waiting for the subway home, as I rode the subway home, and all throughout the next morning, I grappled with it. I still have no clue how to solve it.

In any case, we both wanted to expose our geometry students to this puzzle. We figure next year we could turn it into a lesson — having them play and then have them analyze what they figured out. But for this year, we wanted to just see what happened if we gave our kids the puzzles.

I faintly recalled my friend Bowman doing this in his class and blogging about it, so I found that post and used his recommendations about what to have the kids cut out in which order, with the scaffolding that Prof. Demaine used in his packet, with some ideas that Brendan had, to create our own packet of fold and cut puzzles.

Fold and Cut Figures [PDF download]

What happened? Well, we gave kids 25-30 minutes. We had extra copies of pages for if kids messed up and wanted to try again. And we said “go at it.” Of all the kids in my class, only one seemed not to get into it… at the beginning. That student was trying too hard to have a “method” and their intuition wasn’t as strong as the others… but they showed me proudly at the end when their star! All the other students were addicted. Paper flew about. Kids called me over to proudly show me their successes, and wailed in frustration when their cut didn’t work (and then hurriedly asked me for another copy of the page they messed up on ). It was exciting to see kids focused but also having fun playing with math. I would say that 25-30 minutes was the right amount of time, because at that point, I saw kids start to fade. (It could also be that we met at the very end of the day, and this was the last 30 minutes of a 90 minute block…) No kid in the time given was able to get the scalene triangle (many got close) or the last quadrilateral. But almost every kid was able to get all the figures before ’em.

Next steps from here? I want to turn this into something more formal. I like the play. I love the play. But then we need to come up with some general conclusions and talk about why they work. Why are we doing lots of folding along angle bisectors? [Hint: the answer has to do with reflections!] Why are we doing lots of folds perpendicular to the lines of the polygons we’re trying to cut out? [Hint: if we imagine a “vertex” at the place where we have a perpendicular fold, we can consider our fold an angle bisector — bisecting the 180 degree angle of the vertex!] If kids understand those two principles (and the scalene triangle is the most perfect shape to make them both come alive!), I will have a way for kids to tie their puzzling to our geometry curriculum.

What most impressed me was how much intuition kids already had with regards to these. It was amazing to see them take to it as adroitly as they did.

And who knows? Even though I say we should tie this to the curriculum formally next year, maybe I’ll get to it this year after we complete our mountains of salt investigation. Because heck if they aren’t perfectly related to each other!

P.S. So… Here’s where we got stuck. We were given the following paper… no polygon, just a line segment that we had to cut.

You might say: duh, fold it vertically in half and make a half cut. But here’s the thing: you have to make a COMPLETE cut. So once you start cutting, you have to keep cutting until you have completely hit the end of your paper. And BOOM! Suddenly I am perplexed.