Tuesday, November 26, 2013

Compass and Straight-Edge Constructions

Compass and straight-edge constructions are a part of my school's curriculum for Geometry A.  The constructions that we teach and assess include:
- copying a segment
- copying an angle
- perpendicular bisector of a segment
- bisecting an angle
- constructing a line perpendicular to a given line through a point not on the line
- constructing a line parallel to a given line through a given point

I personally enjoy doing and teaching these constructions.  It's hands on, and I enjoy seeing how they work out.  I'm intrigued by all that can be done with just a compass and straight-edge, especially when it comes to things like quadrature and more complicated constructions.  I like the history of it, too, and it seems amazing what early geometers and mathematicians were able to figure out with these tools.

I wonder sometimes what value these constructions have for my students, though.  Some of the basic constructions can be used to draw other things that we study, like a triangle with certain lengths or angles, or the incenter of a triangle, although we usually have them explore concepts with difficult constructions like that on GeoGebra.  I'm not sure whether these types of constructions really enhance students' visual-spatial reasoning relative to the things they are constructing either.  I know I could do a better job emphasizing the fixed distance feature of arcs, but even if I did that, I'm afraid most of my students would still learn these as mechanical processes.

Eric Pandisico, in his article Alternative Geometric Constructions: Promoting Mathematical Reasoning suggests that students can take the most away from constructions when they have the chance to try them with a variety of tools.  Rather than stick primarily to compass and straight-edge constructions, Pandisico promotes the use of several tools to do the same constructions, including Miras, a simple right angle source, like a notecard, and even a two-edged straight-edge.  He suggests that because each tool has the potential to emphasize a different feature or property of each construction, doing one construction three different times with different tools can give students a deeper and more full understanding of the geometry underlying each construction.

We only teach these constructions with a compass and straight-edge.  We don't use any other methods for creating these things.  After trying these constructions and others with patty paper in Math 641, I could see the value in using more than one method.  The patty paper made the symmetry of the constructions much easier to see, and even though the results were the same, the thinking I was doing about each construction was much different, and for some of them more intuitive.

Classmate Trevor Kuzee did a nice job of comparing and contrasting patty paper and compass/straight-edge constructions in this blog post.  I'm not necessarily drawn toward one method over another, but I will try to give my students a chance and using more than one method the next time I teach constructions.

To help my students review the compass and straight-edge constructions that they had learned, I made the GeoGebra worksheet pictured below.   This didn't represent much new GeoGebra learning for me.  It was a lot of assigning values to show objects again, and there were lots of hidden circles underneath those arcs.  I did set points to which I fixed the text boxes for the instructions, which allowed me to keep those well aligned even between constructions.

Padisico, Eric A. (2002) Alternative geometric constructions: Promoting mathematical reasoning. The Mathematics Teacher, 95(1), 32-36.

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