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Wednesday, April 25, 2012

Squareness, continued

Yesterday there was lots of grappling with what we might mean by the "squareness" of a rectangle, and how one might come up with a way to express squareness with a single number.

Today, I was a little bit grumpy with my first class right off the bat, because first thing this morning a Resource teacher came to tell me a girl complained about her group yesterday not working and talking about weed and making her uncomfortable. Does anyone know how to do group activities and guarantee everyone is on task at all times? Because I don't. (And people who claim they do are lying. Let's be real.) I didn't know how to respond to her. "Oh ok I'll seat them in rows and demand silence every day" is the normal level of sarcasm of my internal monologue, but I'm a professional, so I just said "Okay, I'm sorry, thanks for letting me know."

Anyway. I wanted students to evaluate the validity of some conjectures, so that they will have a vocabulary to use when we start proving conjectures about similar figures. And hopefully learn and/or recall some geometry along the way.

I also didn't want to just drop the idea and move on to something else, even something else related, because I want them to know that what I ask them to do in class has value and meaning, so that they trust that what they are being asked to do is for a reason, even if the reason isn't immediately clear. (I have paid closer attention to this principle this year, and it seems to have paid off...some days more than others. Keep reading.)

I gave them the following on a 1/2 sheet of paper, a piece of graph paper, and access to any geometry tools they wanted to use.

1. Consider the following rectangles with given side lengths:

(a) 5 by 2 
(b) 12 by 9
(c) 15 by 6
(d) 8 by 8
(e) 16 by 4
2. Draw them on graph paper. Come up with what you think the ranking should be, from most square to least square, just by visually looking at them.
3. Here are some methods for ranking squareness that your classmates conjectured yesterday. Some of them are valid, and some of them are not. Use each of them on all the rectangles above. List the ranking from most to least square produced by each method. Which methods are valid and why? Please respond to this in writing, right on the graph paper. Any tools you need are available on the table.

(f) The absolute value in the difference in length and width. closer to zero, the more square it is.

(g) Longer side divided by shorter sides; closest to one is square

(h) Find the areas of the shapes and compare how close it is to a square number.

(i) Create a diagonal and measure the angles on either side of the diagonal. A square's diagonal bisects the 90 degree angle in half, so the measure that are closest to equal would be the square/rectangle that is closest to an actual square

(j) Draw both diagonals. Measure the smaller angle created at the intersection of the diagonals. The closer it is to 90, the more square the shape is.

(k) Attempt to circumscribe a circle around the rectangle. The closer you can get all four corners to touch the same circle, the closer to a square it is.

I found today more frustrating than yesterday. I had to be a bit of an ogre about "I'm going to collect this and grade it" because otherwise, I was sensing kids giving up in response to a little frustration, instead of working through the frustration to understanding. I hate being like that because it sucks all the fun out of the room. But honestly I don't know what else to do. I get that there are more fun things to do with similar figures (go outside and measure shadows and find the height of the flagpole! etc), but honestly I find those kinds of activities a bit juvenile for tenth grade - middle school-ish, if you will. Not that they are bad, but it depends on your goals. We are going for more formality in their Geometric arguments at this level, and I don't know of any of those more-fun activities that stand up to the rigor required to work through what we did today. And honestly, once they engaged and understood the assignment, they were good to go. It just took some prodding to get them to that point.


  1. (k) really bugs me, and it bugged me yesterday (though I didn't mention it then). Unless I'm really misunderstanding something, any rectangle can be circumscribed exactly by a circle.

    Or is this a conjecture the class came up with that will prove less useful as they play with it?

  2. You're not missing anything! That is one of the invalid conjectures, for the reason you said. But we haven't circumscribed anything in a while, so since a kid came up with it I wanted to include it so they would think about it.

  3. After reading your post, I will definitely do this lesson again, this time with my 8th grade geometry kids. I love your idea of taking the conjectures from the day before and asking each student to formally re-think and re-rank squareness.

    I so appreciate your saying that people lie when they claim to get everyone working in group work. Thank you!! I feel the same way about differentiating instruction PDs also, I want to ask the presenter if I may observe him/her in action every day for a week.

  4. Now I feel bad about calling people liars. I was still feeling frustrated when I hit publish. Maybe there are some people somewhere who can pull this off. I just don't understand how it's possible when you're so outnumbered, and it's an activity where you expect a lot of chatter.

  5. How do you form your groups? I read recently that randomly assigned groups are more effective than students self-selecting groups (over time, you will have groups that don't work at all, and you are welcome to disallow certain combinations).

    On the other hand, if you can find an arrangement of groups which seems to work for most people, stick with them.

  6. The groups aren't self-selected but based on the seating chart. The default seating chart is pairs, and they just rearrange in a particular way to form groups. But this is one of those classes where many kids are friends so it can be a challenge to make a chart that minimizes distractions.

  7. I know you didn't mean that people purposely lie about what they strive to do. (I didn't either.) Funny, right after I wrote last comment, I found a brochure for the 2012 Nat Conf on DI in Vegas this July! Maybe that was a sign that I need to shut up and register and learn something.
    But I'm frustrated too when I feel my time has been wasted at some of our PDs because these people from thick-carpet land don't really get it. Thanks!

  8. After reading your first post on the Squareness activity, I shared it on my own blog as a great example of a lesson that includes practice on all of the CCSS Standards for Mathematical Practice. I am so impressed with what you have done so far...the Google Doc template for publishing the groupwork, etc.

    In that Google Doc, you can see a few groups that did not go quite as deeply as the others. It is hard to get 100% buy in from 10th graders. Don't beat yourself up over it. Just dust yourself off and get on with your wonderful work. You should pat yourself on the back for the results you did get. (Boy, do I have a lot of cliches in the previous 3 sentences!...all meant to boost your morale, of course!)

    You are pushing them in ways that they have not been pushed before, and there will always be teenagers being teenagers. Keep on pushing and giving them rigorous tasks. Perhaps add an extension that brings relevance to the task. Why would we care in the real world if a rectangle were closer to a square or not?

    I love your blog! You are secure enough to show your good days...and your bad. All teachers have both!

  9. Hi Elaine - thank you for the kind comment. I tried to leave a comment on the post at your blog, but the comment form won't show up for me (IE8 on Windows 7, if you are troubleshooting.) I am intrigued by your thought of publishing the students' responses for each other to read and giving them a chance to refine them. I will have to think more about how that could be structured in a way that they were motivated to follow through. Thanks for giving me something to think about!

  10. I always appreciate your bracing honesty about the realities of classroom life.

    - Elizabeth (aka @cheesemonkeysf on Twitter)

  11. As far as minimizing excessive chatter and students giving up in frustration during group work, there are two things that I've found helpful: chunking and time limits. (Your mileage may vary.)

    One, break the task into short 3 to 5 minute chunks: it is hard to goof off when your group has 3 minutes to get each step completed; it is incredibly easy to when they have 20 minutes.

    Two, after a couple minutes, have a representative from each group compare notes for one minute: this gives a helping hand to the completely clueless groups. This also gives the rest of the group members one minute to chatter.

    Three, give the groups a couple more minutes to finish and have a group representative turn in the completed work or at least show you that they've completed it correctly.

    Repeat with the remaining chunks.


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