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triangles and congruence
Posted:
Oct 14, 2009 10:32 PM


I once did a lesson with a class of third graders where I gave them various triangles cut out of cardboard. They were easily able to understand that two triangles are congruent if one can be placed exactly on top of the other. They were also able to understand that two angles are congruent when one can be placed exactly on top of another, even if the two angles belong to triangles which are not congruent. They were also able to see that a 20 degree angle next to a 30 degree angle is congruent to a 50 degree angle, again by juxtaposing the various triangles on top of each other in the right way. (I had labeled some of the angles with their measures, and the students' task was to find the measures of the other angles by experimenting with different ways of putting the triangles next to and on top of each other.)
The sad thing is that many high school students, when they try to deal with triangle ABC and the SAS etc theorems, don't realize that "congruence" is this very simple and natural idea that a third grader can understand, and that what we want to do in high school is make that idea precise so that we can use it to reason and prove. Why do they not see this? Here are a couple of possible reasons:
1. The formal definitions of congruence in many high school texts go like this:
"Two line segments are said to be congruent if they have the same length."
"Two angles are said to be congruent if they have the same measure."
"Two polygons P and Q are said to be congruent if there is a correspondence between the edges of P and the edges of Q, and between the angles of P and the angles of Q, so that corresponding edges are congruent and corresponding angles are congruent."
There are several versions of that last one, depending on the text, and they are often wrong or imprecise. But worse, they completely obscure the clarity and sensibleness of the concept of congruence.
2. Textbooks commonly include "same size and same shape" as an intuitive description of congruence. I think this is misleading, at least if students try to make sense of it in terms of everyday language. Most people would say that a diamond pattern (on wallpaper, for example) is not the "same" as a checkerboard pattern, though the patterns may in fact be congruent. Also, a tall, thin bookshelf is not usually regarded as the "same shape" as a long, low bookshelf, even if the rectangular outlines are congruent.
Ted



