Another year and another chance to teach students about Geometric Proofs...however, they are all starting to seem the same to me - how can the students not see another another set of vertical angles - another line that is shared by two triangles - JUST WRITE DOWN REFLEXIVE ALREADY!

I often wonder if the students really understand what we are doing here - or are the students that are successful at proofs the ones that are super good at copying what I am doing? Do students honestly understand that SSA can't be used to prove triangles congruent, or do they think that I just don't want them to use those combination of letters because they spell a bad word if you look at it backwards???

No matter that I have explained until I am blue in the face...

I have honors students, so I think they should be able to understand, but do they...

I think that when I show students later in the year a "proof" of the Pythagorean Theorem - note: no two columns involved...that they appreciate a teenie tiny bit of the beauty of mathematics.

But, back to these congruent triangle proofs...I am open to suggestions on making them more exciting and fun - do you have any suggestions??

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Well, I am a new teacher to Geometry, although I personally loved it as a student. My approach, however inexperienced I might be, seeks to understand whether they comprehend the material by asking specific questions that force them to do the thinking. For example, if I do a proof with them in class about proving the vertical angle theorem, they will have a sheet of reference for the basic theorems and definitions that I taught them thus far in the course. They will not necessarily be only the ones used to come up with reasons, but the answers are on there: they just have to be found. So many times I will either have all/some/none of the statements for the proof filled out, and I will begin by asking them "What format do you need to change the given information into to be able to manipulate it in order to reach your conclusion" or some other essence of that idea. Once I have them establish a general direction in which they need to go to get to their end point, I ask them which statement do they think will be next, using the information in the given, and the definitions and theorems in front of them. From there on, I will do a similar process of getting them to figure out the statement on their own, with some help if necessary, and give the reasons for their statement. With this process it usually becomes pretty clear whether they comprehend the information or not. I will often do an introductory proof for the material of the chapter they are on and then do a more "hands-off" (from my perspective) proof with them in class, and then take it to the level of homework. Usually at then end of this process they will hopefully understand the material. The reason I stress the importance of getting them to think it through during class themselves is that half the battle of doing proofs is that not only do you have to know the rules and definitions to come up with your statements and reasons, but you also have to be able to have a foresight and planning capable of connecting the dots between known definitions and theorems and your eventual conclusion. Hope this helps

ReplyDeleteA way to make it more fun and interesting could be to take a real life example/set up that relates specifically to the problem in question. One example could be to relate triangle congruency in the field of architecture or engineering. You could even have them design a simple structure on paper, verify using proofs to make sure that each triangle on their blueprint is congruent or that the ones that should be are, and have them create their designed product out of something like toothpicks or straws as an engineering competition or sorts.

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