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What a bunch of 3-ring binders have to do with cones

2014/02/18

Many of my ideas on how to teach something with a new twist have come from someone else sharing their successes in the blogosphere.  My experience in geometry today felt worthy of sharing.

Today’s lesson was finding the surface area of a cone.  I had just gotten to this slide:

Image

and was talking about the “wedge” that comprises the lateral surface area.

“It’s like … ” I said, and I got kind of excited.  I walked quickly to the other side of the room.  “It’s like when you stick a bunch of binders together with the spines all facing out.  I frantically grabbed about 10-12 binders from my shelf and arranged them on a desk.”

That was the 2nd period oh-my-goodness-this-might-be-a-really-cool-visual-but-I-didn’t-prepare-for-it version.  When my other geometry class came in, I was prepared.  While they were doing their warm-up problem, I scooped up 8 binders and just plopped them onto an open desk.  “That’s part of our lesson today, huh?” said one student sarcastically.

“Actually, yes.”

When I got to the slide above, I had them write down the images, then put their pencils down and gather around me.  I arranged the binders like so:

Image

I had all 27 students within about 4 feet of me.  They were all starting to get the connection.

“Ah, so the edges of the binders are the blue lines (the slant height),” one said.  The image was being projected right next to me.

As if on queue, another said “and the spines are the green arc.”

“How many of you have tried to do this on a shelf?”  There was a collective appreciation for this dilemma.

“And how do you resolve it?”

A bunch of them said “turn every other one around.”

:”Like this?”

Image

“Yeah,” they all said.  While they made the initial connection, I think many probably thought they were just helping me with an unrelated problem (too many binders on the shelf).

“So what shape is this top surface now?”

Then they saw it.  “A rectangle.”

From this point, connecting the fact that we’d split up the wedge into pieces that we reassembled into a rectangle was obvious.   And this normally pretty complicated conclusion

Imagewas made much less confusing.

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