Sunday, September 15, 2013

Are these figures the same shape?

Are these lines the same length?

We've got options. We can move one segment until it lines up with the other, and compare them. Or we could get a ruler and compare each length on that. Either way, there's a way to check.

Are these the same shape?

Is there a way that we can check? How about with these two figures? Are they the same shape?


File this under "measure tasks," and see more here, here, here, here, and here. I have nothing much to add except (a) that this is one of them and (b) it went well in class last week. 

There were three major suggestions from the kids:
  1. Check the angles. If they're equal, then the figures have the same shape.
  2. Check the sides. If they're in proportion, then the figures have the same shape.
  3. Take the smaller figure. If a whole number of copies fit into the larger one, then the figures have the same shape.
A big debate broke out between one student who kept on pointing to the picture on the left, and another pointing to the picture on the right:

Meaning, hey, a whole number of triangles fits into the large rectangle on the right. Does that mean that they're not the same shape?

Nice, right?

We settled nothing on Friday, and there's a lot to work with on Monday. We should distinguish between necessary and sufficient conditions. We should create a bunch of examples of pairs of shapes and sort out our thoughts. And, if we think triangles are different than other polygons, then we need to begin to grapple with why that is.

Here's hoping I don't screw it up!

P.S. Why are we doing similarity so early in the year? This is the point in the year when most people would have kids working with congruency, SAS, ASA and SSS and the like. (Or, alternatively, defining things.) So why aren't we doing that? There's a fuller answer to that, I think, but the short version goes like this:

How do you help kids see congruency shortcuts? Do you just list a bunch of combinations of side/angle congruency and then set out to test them? How do you test them? Patterns? Straws? Construction? I don't want to poo-poo all this, but there's a clear path forward for us if we've got similarity in the bag. We can conceive of congruency of two figures as a special case of similarity, with the congruency of one pair of sides fixing the scale factor to 1. And then the question is, what other shortcuts are there out there?

Besides, similarity is really available to us, as I hope this post helped to show.


  1. How does the conversation about the rectangle slide go? I feel like there might be a major clarification there between the same *kind* of shape and the official definition of similar -- how do kids resolve that they are both rectangles but they aren't similar rectangles?

    Also, what happened next?!?

    1. The conversation goes exactly like that. That major clarification is an important one for us to get out there, precisely for the reasons you mention. We need to redeem the language of "same shape" so that we can use that language informally in the long build-up to a more formal definition.

      But, you know, practically how does that go?

      "Are these the same shape?"
      "They're both rectangles."
      "Yeah, but that's not what I mean. What I mean is, do they have the same shape? Like, these [draws weird rectangles] are both rectangles, but they definitely aren't the same shape."

      What happens next? I give them a bunch of scenarios and ask them to decide whether the given information guarantees that the two figures have the same shape. And, at least the way I did it this year, I didn't really formalize similarity at that point. Instead, I transitioned into a conversation about congruence and congruent triangles.

  2. Also, I forgot to sign up for notifications on comments on this post so I am posting again.