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Tuesday, April 24, 2012


I was looking for a way for Geometry students to wrestle with similarity ratios. Daily, I try to escape the "this is how you do this kind of problem" mindset that plagues them, that they prefer, that is a trap we keep falling into. I suspected that Avery's "rank these rectangles according to squareness" problem would be worthwhile. I was excited by what the kids did with it.

Here's the handout I gave out, that I basically copied from Avery:

We had to really unpack 3. "devise a measure of squareness" to get them beyond "it just looks closer to a square." They were able to understand what I meant, but only after I stated it several different ways.

 - Come up with a single number for rectangle E, and a single number for rectangle D, that shows that E is more square than D.
 - Find a way to do a calculation for each rectangle, so that you can compare all the results, to see which is the most square.
 - etc, rephrased a dozen different ways.

Ideally I would like to find a concise way to phrase this for students, that would get them closer to understanding what we are looking for by themselves.

Eventually, lots of groups came up with either or both:
 - shorter/longer, and closer to 1 is more square.
 - longer - shorter, and closer to 0 is more square.

No groups on their own noticed that you could break the subtraction method by choosing uncooperative rectangles (and I don't know how to get them to do this organically.) But after we reported out our methods, and I said "I'll tell you that the subtraction method doesn't always work. See if you can find some examples of rectangles where the subtraction method gives you misleading results." they were interested in looking for ways to break it, and lots of groups found counterexamples.

I asked each group to summarize their method in a google is the raw output.

We had some really interesting methods, too!
 - Draw a diagonal and measure the angles made by the diagonal and the sides. The closer the two angles are, the more square. (Way to draw a connection to the properties of special quadrilaterals! I pounced all over that like...something that pounces enthusiastically on something. So cool.)
 - Attempt to circumscribe a circle around the rectangle. The closer you can get the vertices to all touching the circle, the more square it is. (We did not have time to delve into this today but MAN, THAT IS GOOD. I would really like to find a way to get them to all think deeply about that.)
 - A somewhat crazy method involving comparing the average of the four side lengths to the length and width separately. I encouraged this group to write their method algebraically, but they ended up with an expression that always reduces to zero. However, it doesn't always reduce to zero when they do it with numbers. Would like to spend more time on this, too.

In a nutshell, I had a great time this morning. Philosophical side note: I have been engaging in these thought experiments recently, often, for some reason, about the idea "Could I be happy...?" (Could I be happy if I won the lottery and didn't have to work? Could I be happy working at Walmart? Could I be happy being a housewife? Could I be happy as a math teacher forever?) and what I have come up with so far is that I can be happy as long as I am spending a non-trivial amount of my time learning something fundamentally new - about myself, about other people, about the way things work... ("fundamentally" implying, well, fundamental...specifically not something stupid like how to navigate the menus in a new app or what Loft has on sale this week.) In order to be happy being a math teacher forever, I need more days like today, where I am getting kids excited about learning. As ultimately cheeseball as that sounds.


  1. Thanks for the post, especially the PS. I think everyone needs something for him/herself out of their profession - I think the best schools and workplaces acknowledge that the employees need to learn and be satisfied along with the customers!

    There's a weird guilt thing in teaching (and other professions) where it's supposed to be 100% about the kids. I think that's a nice idea, and I'd love to be totally selfless or whatever... but I don't think I can be! A lesson like this squareness investigation - I did a similar concavity investigation in calculus - is the perfect example of something that's great for the students and great for you! Even if there is some more efficient way to lead students to the same understandings, I think the fact that you discovered and connected with this one is important and can't be discarded.

  2. Well said, Riley. I've never really identified with that martyr mentality either. Of course the goal you keep in mind while working is what is best for the kids' learning, but if the work is not personally satisfying to me (which it isn't, when I'm doing it wrong,) I'm not going to last for very long.

  3. "I can be happy as long as I am spending a non-trivial amount of my time learning something fundamentally new - about myself, about other people, about the way things work..."

    Yes, yes and YES!
    I'm right there with you with respect to the whole "can I be happy" thing. The moments like you describe are what make walking out of the house and leaving a freaking amazing wife and five beautiful boys behind for 1/3 of my day kinda tolerable. I think it has something to do with the conversation piece. I am so edified by real conversations whether it's pedagogy, politics, religion or even sports. Good, thoughtful conversations help me make connections. These same conversations are what our students desire. And they know the real thing when they see it.

    Thanks for sharing this.

  4. For dissuading them from the subtraction method, what about including a bunch of rectangles that are obviously similar, which may get them to think along the lines of scale-invariance. The idea is that a lon, thin rectangle is just as long/thin whether it's really big or really small, but the subtraction method doesn't capture this intuition.

  5. :>

    Wondering if I can take a day with my pre-calc students on this problem...

  6. I really like your PS as well.

    As for the activity itself, you may find some misconceptions by offering more examples of rectangles, particularly ones which are in different orientations to the given ones.

    The activity is really good though, as hopefully you can get students thinking about a number which highlights the squareness of a rectangle. It may not be the ratio of the sides, but I think a ratio makes the most sense. Are there other ways to define the squareness of a rectangle sensibly that do not involve a ratio?

    This would probably not be obvious to a student, but the angle between the diagonals (and how close it is to 90 degrees) may be a reasonable measure of squareness.

  7. @John I was thinking about that, maybe putting a much smaller and much larger one on the page that were similar. As it is now, I have two sets of similar rectangles with opposite orientations, and maybe don't need both?

    @David the idea of using the angle between the diagonals is similar to the one student's idea about whether a diagonal bisects an interior angle. I agree it's not obvious but it is definitely something a student might come up with. Maybe the measure could be "the smaller angle made by the diagonals" and the closer to 90, the more square.

  8. I feel kinda shallow leaving just a math comment, but I wanted to point out that all rectangles can be circumscribed by a circle, so... is that what they were really doing, or did they not do very many examples with that idea or...?

  9. It was one kid who came out with it at the very end of class, so no one got to spend any time with it. Sorry if that wasn't clear...I get just as excited by ideas that aren't going to work out. :)

  10. One rephrasing I like (maybe it was one of the many you used?):

    "Find a formula [or write a computer program] that predicts which rectangle your friend will say is 'more square'."

    I work in predictive analytics, so I like framing things as prediction problems.

    It should be easy to automatically generate many, many examples and show anything they come up with in action.

  11. I wrote about this problem a little while ago and Christopher Danielson left a comment with an interesting followup of cubeyness

  12. Thanks, Kate!

    If anyone wanted evidence that following great teachers on Twitter is great PD, they should read posts like this one.

    You've included some great ideas for a simply-expressed, mathematically elegant lesson that will get kids thinking about math even if they don't intend to :) .

    Love it.

  13. Hi Kate,
    It's interesting to read how your geometry students worked on this problem as I had done the same activity with my 6th graders and it went so well (meaning they didn't know the answers, they struggled, argued among themselves, and left me alone to lavishly rub lotions on my hands) that I went ahead and did it with my algebra 8th graders too. I was surprised that my 8th graders' rankings were less homogenous than the younger kids'. But your kids used strategies that mine never thought of (circumscribe, diagonals). I wrote a short post here but credited Justin Lanier instead for the lesson.

  14. (a) I love this, but you know that.

    (b) I was going to suggest, for planting a seed to disrupt the subtraction idea, including a big rectangle that is close to square and a very small rectangle that is very not square and such that the length - width is larger for the big one.

    (c) I LOVE the circumscribed circle idea. I am having a wonderful day dream RIGHT NOW about kids carrying it far enough for it to begin to dawn on them that every rectangle can be circumscribed with a circle.

  15. In my humble opinion, there is nothing remotely "cheeseball" about it.


    Elizabeth (aka @cheesmonkeysf on Twitter)

  16. I like this. I think I may steal this idea from you and give it a try. I will provide my reflection for you.


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