Half-similar papers: a PBL in similar polygons

I got the first inklings of an idea for this lesson months ago when I tried to photocopy a two-up layout (two letter-sized portrait pages onto one letter-sized landscape page) using a 50% reduction. Doh! I wasted a piece of paper, but it got me thinking about scaling. Against all odds, I remembered it this weekend when planning my similarity unit.

We are beginning discussion of similar figures in my honors geometry course. The textbook introduces similarity with transformations, in a way that I feel is quite non-intuitive, even though we’ve been using transformations quite comfortably throughout the course to define and discuss congruence. I think the students understand thoroughly that dilations are a size change. We dilated our favorite cartoon characters, and we got some very interesting results. (More on that in a later post.)

But the book first defines similarity in terms of “a composite of a size change and any number of reflections.” That idea requires so much unpacking and consideration that I feel it belongs later in the chapter, after the students get a feel for what similarity looks like. The section immediately after the definition discusses the relationships between perimeters, areas, and volumes of similar figures. So I glossed over the transformational discussion (to come back to it in a few days), and posed two very simple questions to get them thinking about how similarity works.

The half-similar paper question:  I held up two letter-sized (8.5 by 11 inches) pieces of paper.  I folded one in half horizontally and held it up next to the other piece. (I didn’t have my camera at the time, but this is what it looked like:)

Similarity2

“Are these shapes similar?” I asked. There were a flurry of immediate answers.

“Yes!”

“No, both sides aren’t half!”

“No, two of the sides have the same length!”

I thought about it for a second and asked, “So shapes can never be similar if one pair of sides is the same length, but the other isn’t?” (I wasn’t sure what they were getting at, but I didn’t want to wreck the vehemence with which they were arguing.)  Some of the students looked confused, and some tentatively said, “Yeah….”   So I looked around the room and found a pad of square post-it notes.  I folded one of them in half and held it up next to the whole one:

Similarity1

“Does this look the same as the first case?” I asked.  The students immediately noted that these two shapes looked “even less similar” than the first set.  So I asked, “Do you think it might depend on the original shape of the paper?” They looked thoughtful, and I think some of them saw the point: if this case is worse than the first, then another might be better….

I let them wrestle with the problem in groups. I saw much measuring (yes, if you fold paper in half, it’s half as wide) and several approaches using proportions, and every group managed to come to the conclusion that the two shapes were not similar.   When each group reached that conclusion and explained it to me, I set that group loose on the follow-up problem: what shape of paper WOULD work? I made it simpler (maybe too simple, I don’t know) by saying, “The long side is 10 inches long. How long does the short side have to be if the folded paper is similar to the unfolded paper?”  Eventually (although some groups needed more guidance here) they all came up with sqrt(50).

(By the way, this led to an excellent discussion of simplifying square roots, and another justification for teaching that skill in the first place. Unless they can simplify the square root, it’s not at all obvious what the proportional relationship is, and how to scale to other sizes of paper.)

Overall, I felt like this went really well.  I do wish I had spent more time trying to get them to generalize. I originally thought of this as an area problem ( the area is divided by two, but the side length is divided by sqrt(2)), but we never really got there. Instead we moved on to a volume question (which I meant to post about later, but now I don’t remember what it was–Dang).  But I think this helped the kids to wrap their brains around what it really means to ask, “Are these similar?”

Does anyone have suggestions for making it better?

Postscript: When I got home, I did some research, and I came up with an interesting follow-up question, maybe for a few days later: “Why is A4 a more sensible paper size than US Letter?” Besides the metric system, of course…

Post-postcript: Well, wipe that smug smile off my face.  Thanks to assessment the next day, I found out that most of the students didn’t actually learn much.  Are there any ideas for an end-of-class routine that would help the students to individually consolidate the group learning?

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