Last week I was being observed by a colleague and my class was doing an exercise in GeoGebra about circles, arcs and inscribed angles. I don’t think I can do the experience I had justice as I try to describe to you what happened in this class, but strangely, I just can’t believe that someone else was there to witness it. Have you ever attempted to scaffold learning in a way such that the questions you asked would move the students forward so that they came to the conclusions themselves? Well, this is what I do everyday in the PBL classroom and sometimes it’s a success, sometimes it’s in between and I do more “telling” than I’d like, but on this day I couldn’t believe what happened.

I had the students construct a circle with a central angle and measure the arc and the angle and see that they had the same angular size. This was no surprise to them. My plan was then for them to extend one side of a radius of the central angle and make an inscribed angle that intercepted the same arc so that they would measure that one and see that it was half the central angle and the arc it intercepted. Often when I do this students don’t understand that the angles intercept the same arc, or something else goes wrong. However, one this day, I wished I had been recording the flow of the conversation that went flawlessly after my simple question, “What do you observe about the two angles?” From one student to the next around the table it went:

“Well, it’s definitely smaller…”
“Mine’s almost ninety and the other one looks like it’s almost 45.”
“Maybe it’s supposed to be half?”
“Yeah because the side is a diameter and the other one’s sides a radius – like it’s in a proportion?”
“No, that can’t happen, you know the angles of a triangle don’t work like that..”
“move it around and see if you can get it to be exactly a half…”

After a while, they all agree that it seems like the inscribed angle is half the intercepted arc. So I prompt them again,”So why do you think it might be exactly half? Is there any relationship between these two angles that might make it that way?”

“Is it like because of the midsegment theorem? The radius is half the diameter so it makes them parallel?”
“Well, it doesn’t seem like the other one is parallel though…”
“It seems like the other central angle next to that one adds up to 180 with the one that intercepts the arc.”
“Hey wasn’t there a theorem about that?”
“Oh my gosh, yes”
“It was like about outside angles or something like that being …like if you add the two inside you get the one outside”
“Oh I see, the triangle on the other side is an Isosceles triangle because it’s a circle..”

At this point, I almost freaked out because I hadn’t said anything in almost 15 minutes or so, they weren’t doing it all themselves and almost every student (except maybe 2 or 3, who were still engaged at least) had contributed something to the conversation. I mean, I was in teacher ecstasy, and to top it all off, I had somewhere there to see it all. I couldn’t believe it. Besides that I had been having a really bad day, and this just turned it all around. I don’t know if this was all a function of the practice of PBL, or a function of the kids in the class, but it was truly amazing. I heard at least three of students leaving class that day say “that was a great class!”, what more could I ask for?