Thursday, October 28, 2010

Glide Reflection

That's my attempt at a Glide Reflection Frieze.

This week my K-8 students were working on motions again.  Using the Geogebra activities at Motion Sketches and More Motion Sketches.  For K-8, there's really not a need for glide reflections because they're usually not a part of the curriculum.

This doesn't fit with Euclid's vision of motions, which was strongly tied to congruence.  Any two objects are congruent if and only if there is a motion from one on to the other.  (Aka rigid motion or isometry or Euclidean transformation...)  This requires four motions, not just three.  But a glide reflection is just a slide and a flip, you may say, we don't need it.  Well all the motions can be made from just reflections, but we still teach turns and slides.

But I've been stumped as to a good way to present these glide reflections.  Students can recognize them by a process of elimination and students can make a motion that is a glide reflection.  The next level of knowing a motion is to be able to specify it.  Students are good at finding lines of reflection, and can specify direction and distance for a translation.  It is difficult for many/most to find the center of a rotation, without being told.  They can do it in a dynamic environment (cf. MotionControl, a geogebra webpage) but it is difficult for them to construct.  The first guess seems to be connecting corresponding points and trying where the lines cross.  (Which doesn't work.)  So it's really hard to get students to know how to specify a glide reflection.  Mathematicians usually describe a glide reflection with a vector and a point or position of that vector.  The vector indicates the direction and distance of the slide, and the line containing the vector is the line of reflection.

This sketch is my attempt at a glide reflection sketch - the goal was to create an environment where students might be able to notice things that would lead them to construct the idea.  I would love it, if you try it out, to get feedback about ways to make the sketch more supportive.  Thanks!

As a webpage or geogebra sketch.

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