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I've just started teaching congruent triangles to a class of 14/15 year olds in the UK. All that they are required to know for the purpose of national exams here is that two triangles are congruent if they have the same size and shape (obviously not a precise definition) and how to write out SSS / SAS / ASA / RHS proofs.

I have read in a few sources that congruence is often badly taught in schools, and that most students do not have a very strong grasp of what congruence is. For example, most students have no idea why SSS / SAS / ASA etc are sufficient for two triangles to be congruent. Maybe they can convince themselves by drawing pictures but that doesn't demonstrate they fully understand. This bothers me - what is the point of learning how to prove two triangles are congruent if one does not understand why the proof works?

Am I doing my students a disservice if I omit any discussion of why the proofs work? That's what most teachers here do. Is there a nice way to explain this without introducing lots of other precise definitions and proving the relevant theorems?

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    $\begingroup$ I actually like the idea of having students draw pictures to convince themselves that SSS/SAS/ASA are sufficient. Like, if you draw a triangle with two fixed sidelengths and a fixed angle between them, then you can't help but notice that the rest of the triangle (side-lengths and angles) are determined. Making that into an engaging lesson/activity for the students that encourages them to get a full understanding will be the real trick, but I can see this being more engaging to students then working through any sort of Elements-like proofs. $\endgroup$ – Mike Pierce Apr 16 '18 at 20:18
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    $\begingroup$ How would just justify SAS? Depending on your formalization, this could be an axiom (it is in my preferred system). What is your definition of congruent? The existence of a plane isometry mapping one to the other, or what? These questions are subtle and difficult. $\endgroup$ – Steven Gubkin Apr 17 '18 at 4:31
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    $\begingroup$ What is the point of learning how to prove two triangles are congruent if one does not understand why the proof works? I know there are lots of things I did by rote when younger and now understand better; if not for the success at the rote phase I never would have gotten anywhere at all. We have to help kids stay successful at mathematics while they are still kids and therefore less capable of abstraction than they might become as they mature. This is so in all fields, I suppose. $\endgroup$ – Chaim Apr 20 '18 at 17:47
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The main points are that knowing a length restricts you to a circle, and knowing an angle restricts you to a ray. So for SSS, you can draw one side, then for each endpoint draw a circle for the adjacent side length. Those two circles should intersect at two points, given two different, but congruent, triangles. For SAS, you can draw an side, then draw a ray from one of the endpoints, and a circle centered around the same endpoint. Obviously the circle and ray will intersect at only one point. For ASA, you can draw a side, and then rays at both endpoints. Obviously those rays intersect at one point. For SSA, you can draw a side, then draw a circle centered at one endpoint, and a ray from another. The circle and ray will intersect at two points, given two non-congruent triangles.

I think this is about as rigorous as you can expect your students to follow, and it's a sufficient level of understanding. Using this, they should be able to re-derive the rules, rather than just rote memorizing which ones work and which don't.

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I teach Euclid to groups of young students...

The 3 tests for congruence of triangles all appear in Elements Book I: SAS (I.4), SSS (I.8) and ASA/AAS (I.26).

SAS is proved by superposition, ie one triangle placed over the other starting at one point and is rather unsatisfying. The proof for SSS is a little more interesting as superposition is only used to align the two bases; the remainder is proved from the fact triangles with two given sides form a unique rigid polygon on one side of the same base. Lastly, ASA is proved using reductio ad absurdum to prove another side is equal and then SAS is employed to complete the proof.

Compared with other propositions in Book I the above do not stand out as being either interesting or satisfying to do. They are certainly far from deductive geometry's finest moment! I only include the proofs as part of the overall work to maintain consistency and because the results are so important.

If I had to teach the three in isolation I would be very tempted to approach them using practical geometry - ie the circle and ray method already alluded to in earlier replies - starting from just one given side or angle and building up from there, part by part, until students discover sufficiency for congruence.

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  1. You might consider defining a pair of triangles to be congruent based on 3 pairs of equal sides. That would take care of SSS.

  2. Make some puzzle worksheets asking students to pick out the congruent triangles based on this definition. Give the length of each line segment. Do not mark the angles.

  3. Explain how SAS works. A "shortcut" -- you only have to know that the length of 2 pairs of sides are equal, but you have will need a third piece of information: The angles between pairs of sides must be equal. (OK, not much of "shortcut," but it sounds good!)

  4. Give students another puzzle sheet for SAS only.

  5. Give students another puzzle sheet mixing up SSS and SAS.

  6. Explain how ASA works.

  7. Give students another puzzle sheet for ASA only. A "shortcut" -- you only need one pair of equal sides. But you need the angles at either end to match.

  8. Give students another puzzle sheet mixing SSS, SAS and ASA.

Forget Euclid's "Elements." Avoid words like "proofs," "theorems" and "postulates." Think of them as "rules" and "shortcuts." You are developing students' spatial sense. Rigor should come later.

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My experience with trying to focus on rigor with slightly older students (18-19 years old) is that they don't get much out of it when 1) they don't see what they get out of rigor that plain old "memorizing what the teacher says didn't get them", 2) they don't have the skills to distinguish between rigorous and non-rigorous argument.

The later, in particular, causes a lot of problems when trying to introduce rigorous content, especially if it's done piecemeal by dropping a few rigorous proofs into a largely formulaic course. Since students can't tell the difference between rigorous reasoning, loosely correct but non-rigorous reasoning, and plainly incorrect reasoning, the whole project becomes a game of "find the thing the teacher agrees is rigorous enough".

I would argue that it's more important, at that stage, to be setting the groundwork for appreciating the value of proofs and informal reasoning. For example, that's a great topic to begin thinking about counterexamples - students are likely to consider the idea of an SSA rule, which can be refuted by an example. That can then raise the question of how one knows that the SAS rule doesn't have a counterexample.

This isn't based on concrete experience teaching geometry, but based on what I've seen with other subjects, I would expect students to get more out of trying and failing to find counterexamples to the rules that work, and perhaps finding informal ways to express why they can't seem to find a counterexample, than they would out of seeing a formal proof.

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