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