## Equilateral Triangles

Thank you to Dan Meyer for this great task idea and to Patrick Honner for the fun tweets 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:
1. We're in a 0:1 classroom right now, and having the whole class come up to the big screen just isn't efficient.
2. 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."

Nachos!

I made the mistake of getting Taco Bell for the kids after we did Taco Cart, now they ask if I could bring in those equilateral chips smothered in cheese for them. Lucky me.

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• December 14, 2012 4:37 PM Shaun wrote:
I'd want nachos too after all that work! j/k

That was a great exercise to get everyone thinking about triangles and the concept of equilateral, but I actually liked more the process of coming up with a scoring method. Learning to think like that and having to consider all the information available to you is just as important as learning about math. I'd be curious to see how the students would factor in perimeter or area if you were to suggest it.

1. December 17, 2012 11:11 AM fawnnguyen wrote:
I'm just very luck to have this group of kids who are good about collaborating and engaging in struggle. Right, in hindsight maybe I could have suggested using perimeter and area and see if they could figure out what to do with those parameters for scoring. Thanks, Shaun.
• December 14, 2012 4:44 PM Jim P wrote:
Love it! I'd prefer to have the kids do it than playing the videos too! I really enjoyed the way you wrote everything that you guys did out -- I definitely have not had the discipline to do that for my posts!
1. December 17, 2012 11:59 AM fawnnguyen wrote:
Writing down everything, selfishly, helps meeeeee for the next time that I do the lesson. You'd think that I could remember, but I don't. Thanks, Jim.
• December 14, 2012 7:50 PM Sue VanHattum wrote:
Wow! Fawn, I wish I could go back in time and be in your class!
1. December 17, 2012 12:03 PM fawnnguyen wrote:
You didn't say that because of the nachos, right, Sue?!
1. December 17, 2012 1:04 PM Sue VanHattum wrote:
What nachos? I guess I missed that part. I'll get nachos with you some other time. ;^) (Speaking of which, I look forward to meeting in person. I'll be in San Diego for the Joint Mathematics Meeting in January. You're not particularly close, are you? All I remember is southern CA.)
1. December 17, 2012 3:31 PM fawnnguyen wrote:
I'm about 3 hours north of San Diego. I'm assuming you're flying into SD, so not driving through my town. Yes, would love to meet you, Sue -- lunch or dinner with a side order of nachos, extra jalapenos.
• December 17, 2012 11:31 AM David Patterson wrote:
I have been following Mr. Honner's equilateral triangle postings. I was happy when I stumbled on your posting here. I have been tinkering with the idea and I wrote a computer program to test out various ideas. I coded up your students ideas and ran my program on them. I was impressed...they were generally not too bad! I dumped all the data to a spreadsheet that lists the triangle side lengths, Mr. Honner's equation ranking the triangles from least to most equilateral (which I think should be the same as Dan's), along with the ranking your students' equations gave.

http://www.filedropper.com/triangles

One interesting thing about the data (there are probably many) is that you can look at the data for the triangles with sides 1,9,9 and 2,18,18. They are similar triangles, so the equilateral ranking should be the same (in theory). For Mia, Julia, LaurenP and Dillon, the rankings are equal! So their teams should get kudos for that. What is more interesting is that while Mia, Julia, and Dillon all just used angles in their formulas (that is the reason why similar triangles will have the same ranking) LaurenP only used side lengths and the two triangles still have the same rank. Extra kudos to her group.

1. December 17, 2012 3:24 PM fawnnguyen wrote:
Holy cow, David! This is fantastic! Thank you so much for crunching these numbers. I will share your spreadsheet and comment with the class. Kudos to you! (And I do hope you'll continue to blog and let us know when you start teaching!)
• December 25, 2012 11:36 PM Kim wrote:
Hi Fawn!
Wishing you the season's best...

Kim

1. December 26, 2012 8:34 AM fawnnguyen wrote:
Oh, of course, Kim! It was lovely of you to write on Christmas Day.
• February 13, 2013 9:29 AM Shane wrote:
What a great post. I've been enthused by Meyer's 3 Acts approach but have been a little lost as to how to work with it in class. Your post does a great job of answering some of my questions. I've subscribed to your blog and will be watching for more ideas.

A question: do you feel the two days devoted to this fantastic project set you back in terms of the "required material" you need to cover in class? I only ask because just when I get on the verge of doing something like this the little "curriculum Nazi" in the back of my head screams "You won't have time to cover everything else you need to cover! You better not get too crazy!"

1. February 13, 2013 8:58 PM fawnnguyen wrote:
Shane, you must kick the "curriculum Nazi" in the groin and do this lesson! Maybe it did set me back in the pacing guide, but for a few years now the pacing guide takes a backseat to what what I believe kids need to do more in math: ask questions, collaborate, struggle. Strangely enough, if kids are exposed to enough rich tasks, their test scores go up anyway! (I'm serious.)

Do go crazy -- crazy in love with your kids and how you teach them. This is hard stuff we're doing, but we're trying and we'll get better. Thank you for your kind words on this space, Shane.

2. February 13, 2013 8:59 PM fawnnguyen wrote:
Shane, you must kick the "curriculum Nazi" in the groin and do this lesson! Maybe it did set me back in the pacing guide, but for a few years now the pacing guide takes a backseat to what what I believe kids need to do more in math: ask questions, collaborate, struggle. Strangely enough, if kids are exposed to enough rich tasks, their test scores go up anyway! (I'm serious.)

Do go crazy -- crazy in love with your kids and how you teach them. This is hard stuff we're doing, but we're trying and we'll get better. Thank you for your kind words on this space, Shane.