Sunday, September 29, 2013

Because I can never precisely replicate angles in these investigations anyway

Good Lord I'm bad at titling posts. Moving on...

Every triangle congruence investigation that I've seen starts with some sort of prompted inductive investigation.

You can do this on a computer, as in the Illuminations lesson. You can use straws, as in Jennifer Silverman's version.

You can also use compass and ruler, as in the Discovering Geometry version of this activity.

There's a reason for this sort of convergence: the activity is solid. It helps kids notice important things.

But I've found this activity unsatisfying for a few reasons:
  • I had a hard time motivating the investigation. I always ended up with some version of "Hey kids, what if there are some shortcuts here?" in that tone that clearly implies that yes, kids, there are some shortcuts here.
  • Why does ASA work? Why does SSS work? Why doesn't SSA work? Where in the investigative process are we forced to grapple with the "why"? (I'm open to doing deduction to explain the results of induction, but I find it a bit deflating to post-facto explain empirical results that we've accepted as a class.)
  • The investigation itself requires, rather than makes necessary, the SSS, SAS, ASA, SSA abstractions. 
In general, the entire investigation feels a bit rigged. I wanted something that felt more natural.

I started with similarity, which we handled in the way that I described here. I asked: "How can we check whether two figures have the same shape?" The kids brought up equal angles. It seemed to work for triangles, but not for quadrilaterals. (I showed them rectangles and squares, both of which have all right angles.) We walked out of there with clear criteria for two triangles being the same shape, and a fuzzy understanding of why that shortcut worked. We did some conventional practice problems, practiced proportions, and the kids were doing OK.

Then: "How can you check whether these figures are the same?"

Of course, we ended up stuck in a brief discussion of what it means to be the same, but we end up with two suggestions: (1) you could cover one figure "perfectly" with the other, or (2) same size and also same shape.

They didn't know it yet, but they were all going to flock to that second definition pretty soon. Because we've set them up perfectly for it, haven't we? They have the AAA criteria for similarity, and they're looking for insurance of congruence. Using the same size/same shape definition, they just need AAA plus some way to guarantee that the two triangles are the same size. From their practice work, they'll realize that any corresponding sides that are equal with similar triangles will end up fixing the sizes of both.

At the start of the next day I provoked them with these images:

"Those two triangles in the circle have to be congruent!"

Well, OK. Why?

"Because the arcs are the same."
"Because the angles are the same."
"Because their sides are the same."

So I pushed them, and we ended up with our first draft of a congruence criteria for triangles: "An angle and two sides are enough to guarantee congruence."

That required a test. So I drew an enormous angle on the board, and I used a ruler as our side. I pointed out that if that's enough to guarantee congruence, then there should only be one way to finish this triangle. 

And, at first, it seemed like that was the case. There were some skeptics. The bell was about to ring, and for homework I asked them to prepare an argument to settle this issue.

Here's my highlight reel from that next session:

Defeat of the angle and two sides criterion! We showed, using our sketches and compasses, that more than one triangle could fit in per point. (We actually had a whole disagreement about whether there were infinite, two or three triangles that you could make with this starter set of triangle parts, though eventually all was clarified through scribbling and summarizing.)

Here was our summary of the day:

Most important of all, there was a clear articulation of an argument: angles guarantee the shape, and then a side length fixes the size.

We spent the next day tidying up. All we really had was an argument for ASA, but that opened the door for other shortcuts as well. They came quickly, and we made sure that our list of shortcuts was complete, and that we hadn't missed anything. (One kid made a wonderful flowchart that I don't have a picture of.) Then, and only then, did I pass out a photocopy from a textbook and showed them two things:

1. There are clever names for these shortcuts: ASA, SSS, SAS, SSA, AAA, AA, etc.
2. This textbook calls these postulates, and says that postulates are assumptions. Isn't that interesting? Did we just have to assume these shortcuts, or did we prove them?

And we had a brief discussion about that too. And then they got some practice work. And then that was our triangle congruence week.

Addendum: Then we placed with arithmagons for a week. We grappled with finding a general solution for the triangle case. We gave convincing arguments that some square arithmagons were impossible to solve at all, and that others had infinitely many solutions. And then we confronted the difference between triangles and quadrilaterals, and I drew a connection between our work with similarity, congruence and arithmagons: Why are triangles always the exception? That lead to a good conversation about the many interdependencies in a triangle, compared to all other polygons.

Second Addendum: This was all with the Honors class, btw. In non-Honors we've done the similarity stuff but none of the congruence stuff. I'm trying to give that group a lot of informal work that involves noticing side and angle relationships before diving into this investigation with them.


  1. i always taught the triangle congruence stuff by starting with the observation that there are potentially 6 things that need to be identical in a pair of triangles in order for them to be congruent: 3 sides and 3 angles. i set up the question of shortcuts by pleading laziness -- i don't want to have to measure all that stuff. this was very motivating to kids.

    i love these examples and explorations. great stuff, and it sounds like you got your kids thinking really well.

    1. But why are you pleading laziness with only triangles? What would happen if you pleaded laziness with quadrilaterals?

    2. I never characterized this as laziness- instead it was do-able with triangles because testing all 6 possibilities was a reasonable task while the possibilities for other polygons quickly grew too numerous and trying to enumerate and evaluate all of them was less useful as they could be decomposed into triangles

    3. It's a running joke I had with kids about mathematicians liking elegant solutions because of a weird brand of laziness. I meant it playfully. It's actually kind of true that there is a class of problems that seek out an answer like 'what is the smallest amount of information we need to know about X to classify it?'

  2. I've always started out with all 6 things as well. Then the investigation ensued to find the LEAST amount of information necessary. We start with testing 1 attribute, then 2, then all 6 possibilities for 3 attributes. This also opens up the discussion of AAA creating similar triangles.

  3. I've always been frustrated by the lack of consistency in terminology across curricula. As you point out, IF A THING IS AXIOMATIC, IT OUGHT TO BE @$%^%$#@% AXIOMATIC!!!!!

    "Postulates," "shortcuts," "assumptions," grumble grumble grumble.

    Stupid mathematical babytalk makes me cross.

    - Elizabeth (@cheesemonkeysf)

  4. @cheesemonkeysf: If you're in SF, it's a beautiful day outside according to You should go outside for a while and enjoy yourself.

    1. @john - Not to worry. It's been beautiful here for days now, and dog makes sure I get out and enjoy it every day. :)

      - Elizabeth (@cheesemonkeysf)

  5. You may not have seen the entire lesson on my site - links have been funky. In any case, we move beyond PVC pipes (not straws, if it matters) and show through transformation that these combinations of parts are sufficient to ensure congruence. Not very different from your approach.