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Wednesday, January 27, 2010

Triangle Congruence Theorems

are so boring, and there is no nice way to teach them. A google search turns up a hojillion versions of "state the theorem and show an example." Here's what I did this year. It sucks and I'm looking for better ideas.

What makes triangles congruent? They're the same size and shape. This means all the sides and all the angles are congruent. The kids had to sit through one proof like this. What a pain, having to write 6 statements to show that two triangles are congruent.

In class the next day, I had them draw three segments with a ruler on scrap paper: one the length of their index finger, one the length of their ear, and one the width of their palm. I told them to construct a triangle with sides of these three lengths. I passed out compasses but didn't give them any instruction on how to do it. Some of them figured it out. A few of these guys showed everyone what they did with the giant compass and yardstick on the whiteboard.

We had a "discussion" about how there was only one triangle you could make with the three lengths. And when I say "discussion" I mean I said - did you notice that those three lengths would only make one triangle? That the angles were sort of 'locked in'? You couldn't just draw the other sides at any old angle? Then I did a lame little demo of how with 4 sticks, the angles are all wobbly and you can make a ton of different quadrilaterals, but with three sticks, you can only get them to make one triangle. I said something about bridges. Then I modeled and they practiced a bunch of SSS proofs.

Today was SAS. In previous years I passed out protractors, and asked them to draw a triangle with two specific side lengths and a specific included angle, then look around and notice how everyone ended up with the same triangle. I didn't feel this really sent the message that the angle has to be included. This year, I asked them to solve these two problems:

I don't honestly know how effective this was. Yeah "we" eventually made the point, but it took FOREVER, and I'm sure some kids got the point, but I'm also sure that half the class was just sitting politely waiting for the torture to be over. And some kids, of course, get downright indignant when you ask them to do something that turns out to be impossible. Which, whatever, but it's much harder to teach somebody who has concluded you are a crap teacher.

Then we practiced determining which other pair of corresponding things you would need in order to use SAS, then we did one example proof.

I know this sucks but I don't know anyone doing anything better. So let me hear it, hot shots.


  1. I did a slight variation on SSS this year. I gave them 3 lengths and draw two DIFFERENT triangles and then cut them out with scissors. I told them I had a prize for the student who had the two most different triangles.

    Many students worked hard trying to make the two triangles different. They eventually figured out what was going by one.

    One student said, "I am going to make two different triangles out of these." I told him he would be very famous if he succeeded :).

    I collected each of the students triangles as they finished and stacked them in a pile...of course every one was identical.

    I am pretty sure that SSS stuck with them after the activity.

  2. >And some kids, of course, get downright indignant when you ask them to do something that turns out to be impossible.

    But it is possible. I just got an answer. It's ugly, but it's there. I made an assumption from your picture that geometry doesn't allow. And then I proceeded something like I would in the first problem. But it was way messier. Much more interesting than the first, though.

    I've never taught geometry, so I don't know how the game works, exactly. But I've taught these variations in trig, and the SSA case usually has 2 solutions. (Under certain condition, it will have just one solution.) Your picture told me which solution would fit this triangle.

    I don't suppose this helps with the problem of how to teach it...

  3. SSA does show up in astronomy, and I did this fun thing with students pretending to be planets and rotating around each other, but I am baffled at how to convey what went on with only text. I'll try to get something up on my blog (likely it'll have to wait for next week).

  4. I don't exactly have a tried-and-true technique, but in general I frame the whole unit as, "We'd like to be able to answer two questions about congruent triangles: (1) If two triangles are congruent, what does that tell me about their corresponding sides and angles? (2) What do I need to know about the sides and/or angles of two given triangles in order to be able to conclude that they must be congruent?" (The fact that these questions really are different takes a while to sink in, but it's worth making that distinction very clear.) I tell them right off the bat that (1) is very simple, but (2) is less simple. They get (1) almost immediately, but then we talk about (2) more in-depth, and I encourage them to a) make conjectures, and b) defend them.

    I have yet to get through it to my satisfaction, but I do believe in that framework.

  5. Ms Kocmoud (in a neighboring district) has a nice set of little Sketchpad things to play with, which makes the playing around with triangles go a bit faster. You should be able to set them up with the Smart board so that the kids can do the drag-ing and experimenting too. Check them out at: Follow the links: Geometry and Algebra Information-> Geometry Chapters 0-6 -> And then Java Sketchpad links under the Chapter 4 heading.

    I teach geometry to college students, and what I like to do is point out that SAS has to be a postulate and not a theorem, because it's job is to say that a plane is flat. If you make a 3D graph of something like z=1/(x^2+y^2+2) (window suggestion: -5,5, -5,5, 0,1) You get something that's pretty flat once you are far from the origin, but not flat at the origin. If you made a triangle by constructing SAS near the origin--lying flat on the surface, which is not flat like on a plane, and another with the same SAS far from the origin, the other angles and side would not be the same. So, these triangle congruences tell you something important about the Euclidean plane, that you don't get from the other axioms. (PS, the rest are really theorems--you just need one to be a postulate)

  6. I'm a statistician so I think about this problem in terms of degrees of freedom - what are the minimum number of pieces of information needed to describe a triangle. SSS does, as does SSA and SAA, but not AAA - why?

    If it takes you 10 seconds to measure an angle and 10 minutes to measure a length (think something really long), which would you prefer?

  7. I send my students off on an hour long quest (we have 84 minute blocks) of cutting out some pre-determined triangles. A few are 3 side lengths. Others are two side lengths and an angle, one side and two angles, etc. Basically modeling the triangle congruence theorems and situations that are a free-for-all (i.e. AAA) and then students try to match up their triangles with their friends' triangles.

    The good: it's a great reference point for later discussion and some students start to "get" it.
    The bad: plenty of cutting and measuring errors create hurdles some students just can't seem to get over.

    It's not really a keeper, but lots better than "just accept my word for it and add it to your notes."

  8. I wish I could take credit for this lesson, but the genius is all Jackie's.

    Note the idea of extending to non SSS theorems via the use of pipe cleaners later on.

  9. @Dyer I'm intrigued but can't envision it. :) Looking forward to your post.

    @Sue - Well...there are really two solutions, of course, not sure what you get when you ignore what geometry allows. It's just that for your average 10th grader who doesn't know the law of sines or cosines, there's no way to do it since you can't get a side and an angle in one right triangle.

    @Everyone else - for now, a quick thanks for taking the time to share all the good ideas. Can't wait to dig into them but for now grades are due aaaaaaahhh.

  10. > for your average 10th grader who doesn't know the law of sines or cosines, there's no way to do it since you can't get a side and an angle in one right triangle.

    I didn't touch trig. But I did have 4 equations in 4 variables at first. I called the perpendicular h, the bottom a + b, and the left side c. Our right hand triangle is 30-60-90, and c =2h. Now we can find b in terms of h, and then a, and then we can finally use the 10 to solve.

  11. I tried it using spaghetti and pre-cut angles (cut the open end w/ jagged scissors). This was really helpful in getting the SSA across because if you plan it right, they can swing one leg to the other side of the height and make a second triangle using the same pieces.

    I'm thinking of using pipe cleaners next so I can have color coded congruences and pre-cut lengths so this would work on their own desks, not just on my overhead.

    Not quite sure if you were looking for a more technical "proof"-type way of teaching this, but visually I'm pretty happy w/ this.

  12. Over the course of a four weeks or so before I jump into proofs involving congruent triangles, I spend time drawing and cutting out triangles according to different criterions. For example, use the side-side-side criterion to draw all the different triangles with sides 3cm, 4cm, and 5cm. It does not take them long to figure out that there is only one you can draw. Furthermore, when you ask them to draw a triangle with sides 3cm, 1cm, and 5cm, it sort of makes the triangle inequality theorem real.

    After measuring and drawing triangles using different criterions, kids really know the ones that produce unique triangles (SSS, SAS, and ASA), some that do when order is specified (AAS) and some that just can't be trusted (SSA or AAA).

    What is nice is comparing drawings using different criterions. Draw a triangle with a 3cm side, a 6cm side, and a 60-degree angle using the SAS Criterion. Then draw a triangle with a 6cm side, a 30-degree angle, and a 60-degree angle using the ASA Criterion.

  13. I would second the spaghetti lesson! I marked linguini with tickmarks every inch and students each got a 3-inch, 5-inch, and 6-inch piece (or something like that) and a pre-cut paper angle. Students were in pairs and the challenge was to try to make a different triangle than one's partner. They set up the pieces and traced them (and completed the triangle in the case of SAS). SSS was great and SAS was good. I didn't try ASA with the spaghetti.

    Students knew from the day before that a triangle is determined by its 3 angles and 3 sides, so the key was to figure out how little information is enough. As a class we eliminated one piece (i.e. a single angle or a single side) and two pieces (AA, SA, SS) as determining congruency before going on to list and investigate the possible combinations of 3 pieces of information.

    (The kits were in baggies to make distribution easy. There were a few casualties, but I have most intact for next year...)

  14. The one I always remember (because it was funny) is that there is no ASS in Geometry...

  15. Last year I taught an off-track geometry (started in February).

    Here's where I introduced triangle congruence

    The big deal was that we had done lots of construction before, (plus, it was an honors group).

    Even still, I was quite happy with "copy a triangle, let's talk about the different ways people came up with"


  16. Does anybody have any fun games to teach Triangle Congruence Theorems?

    Games would help out a lot for teachers!

    Thanks and thank you for this post on a hard subject!

    math games for the classroom

  17. It's really late, I know, but I second the building a triangle by hand idea. Our textbook has an activity where kids build triangles using straws and paperclips. (Paperclips can be twisted to form a rough angle size.)

    I also agree with Jonathan's way of starting with constructions. I've already done triangle tessellations with kids this year, and I'm sure earlier when they did it they had have realized that they didn't need to construct all 3 angles and all 3 sides, but just some combo of the two.

  18. This year I used a lesson from an old issue of MT with pre-cut sides made out of fettucini and precut angles made out of card stock. It worked ok but I didn't post about it because that's all copyrighted and whatnot.

  19. Hey all....I know this is REALLY late to be posting on this but I am about to do triangle inequality and was looking around and then I landed on this chain and thought I would share. I am not up to triangle congruence yet but I will be in about a unit or so. Last year I got so tired of using straws or pasta so I got my department to order me these

    I think they will be a huge help for the quad and triangle units.



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