Update: Here is the most recent post about implementing this activity in class. And here is a post about an extension I did on this — involving merblions. I highly recommend reading those! These were some of my favorite days in my geometry class this year!
Last year in Geometry, the other teacher and I weren’t pleased how we introduced the crossed chord theorem. Basically, we ran out of time to come up with a *great* idea and instead had kids measure some things, take the products, and see that the theorem held true. Not our style. We were doing the heavy lifting and kids were making the connection that was served up on a platter.
When we met last week, we decided to rectify this. We brainstormed some general ideas, and I turned those into this activity.
The setup: Kids are so used to looking at “normal” quadrilaterals in geometry. So we thought we’d exploit that. We don’t mention circles. We don’t mention chords.
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!
Here’s the start: we introduce a new type of quadrilateral called a “blermion.” (.docx here)
We had some debate over whether we were giving too much away with this start , but we decided we weren’t. (We’re going backwards. The students aren’t deriving the formula. They’re using the formula (which we are calling a “property” of quadrilaterals) to come up with the circle part of the theorem.)
So yeah, we gave kids the ac=bd formula, but in relation to the diagonals of quadrilaterals. And we asked: “which quadrilaterals will this property hold for? We’ll call ’em blermions”
So I ask them to look at the standard quadrilaterals they know — investigating this property using a geogebra sheet — and having them making a conjecture about blermions.
The ggb sheet is here.
So students play on geogebra and come up with some understandings (inductively) about which quadrilaterals are blermions. Then they make a conjecture about all blermions.
This conjecture will fail. Because it is based on students only looking at “nice” quadrilaterals. I want the conjecture to fail. I want to emphasize the point that looking at “nice” examples can often lead to blind spots in your logic.
Students will see it fail when they are asked to drag the four points to specific places (see #5 below). The quadrilateral that results is weird looking. There is nothing that seems special about it. But it does have ac=bd. It is a blermion. Their conjecture about blermions was wrong!
Now students are sent on a chase to find more blermions — and they are encouraged to not just look at “nice” quadrilaterals. They record their results. (If they are stuck, a teacher can have the students fix three points and only drag the fourth point; It turns out you will always be able to drag that point to have ac=bd… and that in fact you can find an infinite number of additional points by doing this dragging of that fourth point.)
At this point, once they have found lots of blermions, students are going to try to make another conjecture about all blermions. I wonder if any student is going to get it. It’s okay if they don’t. At this point, I’m going to have every student plot a different blermion (some “nice” quadrilaterals, but mostly not nice ones). Then I’m going to have them pick any three points and change the color of them. Finally, I’m going to have students go to the “draw a circle with three points” tool, and be surprised by the fact that the circle always goes through that fourth uncolored point.
Why is this good? I hope they don’t get it. Because seeing that every blermion works like this (a circle goes through all four vertices of the blermion) is the key wow factor for kids. It’s strange, because even though I will be giving away this key fact, I think all this play will make this key fact interesting and weird.  Once they all see that, they are going to be curious as to how circles even got involved with these quadrilaterals in the first place. And… that is perfect… because then the kids are going to want to know why this happened.
And then we can transition to figuring out how to prove this. Because suddenly the crossed chord theorem is weird and strange and unexpected, and suddenly we kinda want to know why it works!
 We had to decide whether students should discover the property ac=bd for crossed chords. Motivating that from a circle and crossed chords was hard. We needed kids to somehow see similar triangles (which felt like we would be giving away too much) or come up with the multiplication idea of the pieces of chords on their own. We had ways to motivate that multiplication, but they weren’t elegant. So we scrapped that.
 Here’s the thing. Most things in geometry are presented to students in such a way that their wonderment about the geometric thing is killed. In a proof, the statement to be proved is given up front — and suddenly it isn’t interesting. It might be something really cool, but the exercise around doing the proof doesn’t highlight that. Or — as I’ve blogged before — theorems like the ones involving all the triangle centers… we tell kids to plot the perpendicular bisectors of all three sides of a triangle and they meet at a single point. It isn’t strange and wonderful. They don’t see why that’s weird. They just know we told them to plot the perpendicular bisectors, and they know something will happen because why else would we have them do it? We kill the wonderment of geometry in so many ways.
I want the weirdness and unexpected and unintuitiveness to come back to geometry… that’s where the beauty and curiosity are… and only then have my students work on figuring out why the unexpected happens… and get to the point where the weird and unexpected and unintuitive become obvious and natural. Making the unnatural natural. Yup, that’s the goal. But to do that, you have to first get to the unnatural.