## [3ACTS] Best Squares

I’ve been working on this series for the last two months. I asked four of the most active contributors on 101questions (Andrew Stadel, Chris Robinson, Timon Piccini, and Nathan Kraft — all dudes, sorry about that) to:

I love these problems, but they scare the hell out of me.

You have the #3act hallmarks: a short visual setup, minimal language demand, and a question that can be approached intuitively at first. Have your students write down a gut-level ranking of each contestant. Who drew the best square?

Now we ask the students what information matters and doesn’t matter and how they’ll use that information to make a rule.

We’ll eventually give them all the information they could want — area, perimeter, angles, side lengths, and coordinates (so they can get whatever we missed). The point is, we could very easily hint our way towards an answer by providing the area and the perimeter in advance, but now the student’s task is much harder and much more interesting.

Also, your task is much harder and much more interesting. You have to take whatever rule your groups of students come up with and parry back with cases — large, small, degenerate, etc. — that heat that rule to the melting point.

If the student says, “Let’s subtract each side from the mean side length. All the sides should be congruent,” you offer her a tilted rhombus, which scores perfectly against that rule but shouldn’t.

If the student says, “Let’s subtract each angle from the mean angle measure. All of them should be 90°,” you offer her a short, wide rectangle, which scores perfectly against that rule but shouldn’t.

These problems terrify me because even as I put an answer in the teacher’s guide [pdf], I’m not convinced it’s the best answer. (Should we give the bigger squares more credit because they’re tougher, for instance?) I only know the process is worth the terror.

BTW. This activity owes a debt to The Eyeballing Game and to Patrick Honner’s question, “Which triangle is more equilateral?” where you can find him parrying superbly in the comments.

Featured Comment

Why should it be “our task” (teachers?) to take the students’ rule and parry back bad cases? This is one of the most interesting roles a student can take in this process. I’d much rather have the students coming up with edge cases against their own, or ideally others’, rules. I’d keep a few in my back pocket just in case, but students can drive that conversation in great ways.

“Draw two points and then the point exactly between them.”

Sorry to be nit-picky, but while the above task certainly meets the requirement for “minimal language demand,” I think ONE MORE WORD is required for the sake of precision: “halfway.” There is not a unique point that is “exactly between” two given points; there are many. But there is one point that is “exactly halfway between them.”

I know that “conciseness” and “precision” sometimes compete with one another, and I confess that I often strike the balance poorly.

2012 Dec 15. Fawn Nguyen gave this a go in her classroom.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

### 21 Comments

1. #### Nik

December 10, 2012 - 12:37 pm -
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Dan,

Would they scare you if you were going to teach it to a class? or excite you?

I think it’s very cool that the answer you give might not be the best. It takes away some of the cloak and dagger idea that the teacher holds all the answers and that the students can only be right if they check against what you have.

Keep making me rethink what I do!

Nik

2. #### Michael P

December 10, 2012 - 1:40 pm -
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How does video help this problem? I’m not sure that I see the advantages of video over just asking 2-4 kids to come up to the board and draw some points.

3. #### l hodge

December 10, 2012 - 2:28 pm -
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Very nice problems. I am with Michael on using kids vs adults on a video. Why did you give an “answer” and call it the “best answer” in the solutions?

The rule you propose punishes convex mistakes more than the equivalent concave mistake. Suppose person #1 has drawn a perfect square except one “side” is actually two segments that meet at a point inside the “square”. Person #2 draws the same thing except has the two segments meet at a point outside the “square”. Hard to argue that one square is better than the other, but square #1 wins by the area to perimeter rule.

Can you design a figure with an arbitrarily large perimeter, an area of 1 sq unit, that “looks” very much like a square or circle?

4. #### Chris Robinson

December 10, 2012 - 2:33 pm -
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Michael P:

How does video help this problem? I’m not sure that I see the advantages of video over just asking 2-4 kids to come up to the board and draw some points.

I’m not speaking for Dan, but in my opinion, video allows for a lesson/task like this to work in an asynchronous environment. Having students draw points on a board just doesn’t work there. In a class with “warm bodies,” yes. But don’t forget about the independent learners who might not have that.

5. #### Dan Meyer

December 10, 2012 - 4:38 pm -
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Michael P:

How does video help this problem? I’m not sure that I see the advantages of video over just asking 2-4 kids to come up to the board and draw some points.

The video doesn’t add much on top of the final frame of the video but it adds plenty on top of your hypothetical 2-4 kids. Namely, it allows me to gather all kinds of data in advance about the four quadrilaterals that can then be used in analysis. What’s the plan with the kids at the board? You can sink a few minutes measuring angles and sides, sure, but area is gonna be really annoying.

Superior to both of those, of course, would be a digital platform that lets everyone create their own quadrilateral and then returns all those data automatically. Wonder if anybody’s working on that.

6. #### Chris Robinson

December 10, 2012 - 4:52 pm -
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Dan, I sense a rhetorical question/statement in that last sentence. Or at least I’m hoping I do.

7. #### l hodge

December 10, 2012 - 5:25 pm -
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If you have a smart board (I don’t) I wouldn’t guess that it is much of an issue getting the area and perimeter for a students drawing.

I would have them draw it on the white board, then project geogbra onto the drawings, trace a polygon over their drawing, and have geobera measure – easy.

8. #### David Taub

December 10, 2012 - 10:30 pm -
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For the triangle version how about simply measuring the distance between the center of the inscribed and circumscribed circles? Or am I not thinking of a case where they coincide that this not an equilateral triangle?

9. #### Fawn Nguyen

December 10, 2012 - 11:08 pm -
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I love these; I have a “draw a circle” contest during pi week. But my mind immediately goes to paper/pencil instead of the videos or interactive board because with 37 kids and one screen, visual access for the audience is limited. (Our laptops are good dead weights. I share Kate’s What’s Annoying http://function-of-time.blogspot.com/2012/12/whats-annoying.html)

Right, I’d want EACH kid to do this because I’m sure they all want to try this! No fair in calling on just 2-4 kids. Fine tip sharpie and white piece of paper for each kid. They do their thing, then I gather all their papers and make a photocopy for safekeeping (my low-tech memory stick). Randomly pass back the originals, so each kid has someone else’s paper. Tell kids, “What makes a square a square? How square is the “square” you’re looking at? Here are some tools… Do some measuring… Calculate. Be prepared to defend your methods. Go!”

After the individual struggle, put kids into small groups. Have them talk it out and decide who has the “best” squareness measurement among them. They use this best one to re-measure and re-evaluate all the squares within the group. Then present and defend reasoning in front of whole class. Class argues some more about whose way is best, then they vote.

Sorry I’m taking up space here thinking this lesson out loud. But I do love the bare minimum teacher instruction that potentially yields lots of great discussion from kids where they ALL are doing and talking. Kids also want to know whose square wins the big price, the one that got it “right.”

10. #### Bowen Kerins

December 11, 2012 - 12:50 am -
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Why should it be “our task” (teachers?) to take the students’ rule and parry back bad cases? This is one of the most interesting roles a student can take in this process. I’d much rather have the students coming up with edge cases against their own, or ideally others’, rules. I’d keep a few in my back pocket just in case, but students can drive that conversation in great ways.

It’s then like coming up with a good definition for “quadrilateral” or “function” — coming up with something sensible, then refining it or throwing it out entirely depending on what comes along.

The task also reminds me squarely of the “Squareness” task from Balanced Assessment:

http://balancedassessment.concord.org/hl002.html

Of particular note is that whatever metric the students come up with should be direction-invariant; a measure like “H/L – 1” is not.

You might replace “the best solution” from the TE with “a good solution”, since there are multiple good solutions to the problem and I don’t know any that I would call “best”. What came to mind for me was an accuracy score on angles (longest – shortest) plus an accuracy score on lengths (|(longest – shortest) / average – 1|), then some attempt to weight those into a score, closest to 0 wins and only a square can get exactly 0. Angle measures are more easily measured or eyeballed than areas, so I’d prefer not to have a measure based on area.

11. #### Kevin Hall

December 11, 2012 - 6:18 am -
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It would be our task because students may need a teacher to model this kind of thinking for them. I’ve always found that it’s surprisingly difficult for students to search for counterexamples, but when it comes to proof, it’s an early rung in the ladder of abstraction.

12. #### James Key

December 11, 2012 - 8:12 am -
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“Draw two points and then the point exactly between them.”

Sorry to be nit-picky, but while the above task certainly meets the requirement for “minimal language demand,” I think ONE MORE WORD is required for the sake of precision: “halfway.” There is not a unique point that is “exactly between” two given points; there are many. But there is one point that is “exactly halfway between them.”

I know that “conciseness” and “precision” sometimes compete with one another, and I confess that I often strike the balance poorly.

13. #### Dan Karl

December 12, 2012 - 7:16 am -
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Dan,

I love these problems.

How did you find the area and points for your circles? Geogebra?

I really like that definition of a perfect square or circle or equilateral triangle as it not only applies to geometry, but into Algebra 2 and calculus as we discuss maximizing equations (in this case the area.)

@Hodge – What in the world are you talking about? If

one “side” is actually two segments that meet at a point inside the “square”

then that wouldn’t be a quadrilateral anymore. And it wouldn’t happen because they are only drawing four points, not five….

14. #### Dan Meyer

December 12, 2012 - 11:25 am -
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Really useful commentary and criticism from James Key and Bowen Kerins. I tossed them up into the post itself.

@Dan:

How did you find the area and points for your circles? Geogebra?

No, I vectorized the paths in Adobe Illustrator and used a plugin that tells you the perimeter and area of an closed shape.

15. #### l hodge

December 12, 2012 - 2:27 pm -
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@DanKarl, Sorry, I lost track of the fact that they didn’t draw a square. Consider the circles instead. Take a metal circle and bang on the outside with a hammer so it dips inward a bit. Take a copy of the steal circle and bang on the inside with a hammer so that it bubbles outward a bit. Both shapes have the same perimeter, and were arguably distorted by the same amount, but the first one would score higher on the area to perimeter metric. Somewhat similar sorts of issues apply with plotting four points for a square.

The area measures are not very sensitive. Even with a really lousy job of drawing a square, like one side or angle being 20% more than another, you would still be within a couple of percent of the area of a square with the same perimeter.

16. #### Patrick Honner

December 12, 2012 - 4:33 pm -
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I’m proud to be cited as a minor inspiration in this task design, which I think is great.

To Bowen’s point, we definitely want students to develop the tools to analyze, probe, and ultimately refine mathematical ideas. As Kevin responded, one of our roles as teachers is to model this behavior so that students can learn it.

In addition, content knowledge plays a big role here in helping the teacher ask good questions and shape the conversation. This is partly what makes the teacher better suited for this role than students (in some, or most, cases). And this role is one that I don’t see technology readily replacing.

On a final note, did anyone suggest scoring larger squares higher than smaller squares? I bet it’s easier to draw a good small square than a good big square.

17. #### Greg Schwanbeck

December 13, 2012 - 10:23 am -
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@Dan After extensive testing I have concluded that the error with the postscript not coming through results from a problem existing between keyboard and chair. How embarrassing.