To whom it may concern:
My name is Chris and I am a second-grade teacher. I would like to know what the difference is between congruent and symmetry, and how do I explain this to my class? I know that congruent means the same, and symmetry is two identical sides. Is there a difference between the two? I know there must be, but I don't know what or how to explain these two terms. Help! (I hope this isn't a dumb question!)

Thanks, Sincerely,

Hi Chris,

Not a dumb question at all. But the fact that you (and many of my in-service high school teachers and honours math students ask it) says something about a gap in the 'geometry' teaching and texts in North America.

To a geometer (and it its root meaning) congruent means two things which can be placed on top of one another without changing any lengths (e.g. by a rigid motion with perhaps a reflection at the end).

So two line segments of the same length ARE congruent. One can be placed exactly on top of the other. - Same lenght actually means we probably took a measuring device, like a ruler and marked it off on one and moved the rigid measuring device onto the other acting out the rigid motion).

Similarly, congruent triangles means we could pick one (or a copy of one) up and place it down on the other.

This extends to any figures in the plane (or in 3-space). Congruent MEANS one could be placed on top of the other.

Of course IF they are congruent THEN anything you would measure in one will have the same measurement in the other. Because they are congruent, everything you look at will look and measure 'the same'. Same lengths of sides, same angles, same area, .... .

For congruence, I said rigid motion and reflection. In physics, or higher math, rigid motions, reflections, and combinations which preserve all distances are called 'isometries' or symmetry operations. However, I suspect that at grade 2, the word symmetry is being used to describe the fact that, if you made TWO copies of object, ONE could be picked up and put back down on the other, in several ways. In a sense, one copy is congruent to the other - while some of the vertices etc. move!

So an isosceles triangle has 'symmetry' because you could pick up a copy, flip it around the central line, and put it back down onto itself. Or - you could fold it and one side would drop down on the other (as you suggested).

However, there are OTHER symmetries than just a single mirror.

Take a paralellogram. You probably can't fold it over like a mirror. However you CAN pick it up, turn it 180 degrees (half turn) and put it back down. That is also a rigid motion and is a symmetry. In a carefull study of patterns, we work with general combinations: translations (if things are seen as infinite - going on forever), rotations, translations and combinations of these.

I would primarily start working on these ideas with manipulatives, including paper folding, objects like blocks of wood, etc. Take to things. See if one can be moved to take the place of the other. Then they are congruent.

Take an object. Imagine moving it and then seeing whether it looks the same before and after. Then it has symmetry. Or have two copies of a plane shape and picking one off of the other and placing it back down after moving it (turning, flipping in 3-space which is the same as reflecting in the plane). Students could play a game. One closes their eyes the other can move it so it 'looks the same afterwards' and the first student has to guess WHETHER it was moved - and how. If you CANNOT tell if it was moved - there was symmetry!

Also, for plane figures, you can try a mirror (or MIRA) to see if it looks the same before and after reflection. Testing for another symmetry.

With an object with no symmetry you could ALWAYS tell if it was moved!

Try a ball. You cannot tell if it is rotated. If you had a mirror it looks the same before or after. Lots of symmetry.

Symmetry, in general is one of the KEY ideas in all of geometry. It is used in chemistry, in physics, in engineering, in looking at images for thinking about algebra. It is, as they say, a BIG idea. Students do have some kinds of experiences. They should recognize and connect those. They should also realize that there are new things they do not know yet. Look at the patterns of tiles on the floor of the room. Do these (perhaps imagined to go on for ever) have symmetries? Look at the patterns bicks make on the wall, or wallpaper makes? Do these have symmetries. There are BOOKS about the possible 'symmetries' of wall paper patterns. Or strips of paper dolls. ... How about the atoms in a crystal? How about the big shape of a crystal?

Here is something they all will have looked at: Dice. If you wipe off the numbers - are dice symmetric? If they were NOT symmetric, would the shape work for dice? Perhaps a student has seen the dice of an older brother or sister which were not cubes (e.g. in Dungeons and Dragons). Are THOSE shapes symmetric? Can any face have the same chance of being up or down BECAUSE of the symmetry?

Well probably more that you wanted to know right now - but enough to convince you it is well worth playing with, asking about, and having your students play with.

A big idea which I also teach to undergraduate math majors, to graduate students, and even use in my own (geometry) research!

Walter Whiteley
York University,
Toronto Ontario.

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