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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.