Equilateral Triangles
Thank you to Dan Meyer for this great task idea on equilateral triangles.
Act One is this video which asks for a ranking of how well each teacher had drawn his equilateral triangle.
But as soon as I saw the video, two thoughts came up:
We're in a 0:1 classroom right now, and having the whole class come up to the big screen just isn't efficient.
I want to be in the competition! The kids will all want to do this.
So I knew this would have to be a pencil-to-paper activity in my class.
Eyeballing
My instructions to twenty 8th-grade geometry students:
Here's a blank sheet of paper for you to do this task. Be sure your pencil is sharpened. Put your name in upper-right corner.
Using only your eyeballing skill and your pencil, mark three dots in the shape of an equilateral triangle.
Gabe normally asks amazing questions that make my heart sing. Today he asked, "So you want us to draw the best equilateral triangle?" I replied, "No, Gabe, I want the crappiest one you can draw."
Now, connect the dots with a straightedge.
Pass your papers forward, I'm going to make a photocopy of your drawing so I can have a clean copy of it just in case.
While I'm making the copies, I want you to think about how you are going to decide which triangle is most equilateral.
I dashed quickly to the copy room a few doors down. I wish you wouldn't tell anyone that I'd left the children unattended for 3 minutes.
I'm now going to randomly pass the papers back, meaning you should have someone else's drawing to work with. Write your name on their paper as the "tester."
Okay, I want to know which one of you drew the bestest equilateral triangle. To do that, we need to come up with some kind of criteria, a way to test it, a way to score it. Talk to me.
They said that an equilateral triangle had to have three congruent sides or three 60-degree angles. So, I said:
I guess you'll be measuring the sides and angles. Then you get these six numbers. Do you need all six? What are you going to do with the numbers?
We know what perfection looks like: 3 congruent segments, 3 congruent angles. Let us safely assume that no one drew a perfect triangle. So how far from perfect is it? What score would it get? How fair is your test?
I need you to work quietly by yourself for now. Get your tools: ruler, protractor, compass, calculator, whatever. Then figure out a way to test for equilateralness.
Working alone
They worked diligently and carefully. I appreciated seeing this student use her ruler to extend the side length to help her spot the angle measure more accurately. Julia asked, "Measure in centimeters, right? To the tenth?"
Working in small groups
Now I'm going to randomly put you in groups of three. In your small group, share your scoring strategy. Fight about it. Defend your methods. Eventually I will ask you to choose the best method from your group to present to the class.
By the way, just because someone in your group is in possession of a drawing that you know is far from perfect doesn't mean that his/her method for testing it should be dismissed.
Oh, hey, should a larger triangle deserve more points in your scoring system? I mean, is it harder to throw down three dots that are spaced farther apart?
I moved from group to group, listening to their discussions and observing their calculations.
Watching them, listening to them, asking them questions — I didn't want to be anywhere else.
Presenting to the whole class
One by one, the groups were eager to share. They questioned each other: Why 100? Why 30? Why divide by 3
Summary of what they shared on the board. You can see that 4 of 7 groups used either side lengths or angle measures and not both.
Voting on the best one
The scoring method from Gianna's group got the most votes with 9. Gabe volunteered, "None of these is spot on. But I don't know what the best way is either." I said, "Thanks for saying that, Gabe. Me neither. But I love what you guys are all coming up with!"
Over two days, no one mentioned using perimeter or area. And I vowed not to say anything — I wanted the kids to drive this entire lesson to wherever it needed to go.
Okay, Gianna's famous now, we'll refer to her group's method as "Gianna's formula" from now on. I need everyone to go back to the triangle you have and use this formula to find a score for it.
Testing another triangle
I made another set of copies from the originals (during my prep this time) and randomly passed these out. This very diligent work was still human, and I just felt each triangle deserved another pair of fresh eyes on it.
Now, you're going to apply Gianna's formula to another triangle. This way each person's triangle gets tested by two different classmates. Record your numbers on the board.
About 7 of the 20 sets of numbers had enough variance that I had to ask both scorers of each set to re-do their calculations and/or measurements. I then took the average of the two scores.
The results
I went back to the kids' original drawings and measured all the sides (with a ruler, thanks), applied Dan's formula using this calculator and here are the results. The names highlighted in yellow share the same rankings via Dan's and Gianna's formulas. The greens are off by just one.
We thought this lesson was pretty great. Maia said, "Our way was not too bad at all.”