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.

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

Reflective Properties of Conic Sections

One of the things I find most interesting about conic sections is their reflective properties.  Here are those properties, as I understand them (for the sake of a common description, I'll suppose a "beam" is bouncing off each conic section):

1) Any beam parallel to the axis of a parabola will be reflected to its focus.  Or, any beam leaving the focus of a parabola will be reflected to a path that is parallel to the axis.

2) Any beam that passes through or leaves one focus of an ellipse will be reflected to pass through the other focus.

3) Any beam directed toward one focus of a hyperbola will be reflected toward the other focus.

These are a nice talking point when teaching students about conic sections, since they provide some interesting reasons for locating the foci or finding equations for sections with specific foci or focal distances.  Flashlights and satellite dishes are familiar objects for kids, and it's always nice to have an answer to the question "Who uses this stuff?"

One of these days I'd like to team up with the engineering teacher at my school and make a model that demonstrates these properties.  I imagine setting up pieces of conics with shared foci, as sketched out below.   Our technical education department has a computerized lathe, so I think we could get the appropriate grooves cut into a piece of particle board and then put a thin strip of reflective material in each one.  There would be a track for a laser pointer that runs perpendicular to the axis of the parabola, so the beam could move back and forth and always be parallel to the axis.  If it were set up correctly, the beam would always hit the last focus (the second focus of the hyperbola) even as the laser pointer is moved back and forth.  I'm not sure how tight the precision would have to be, but I guess that last focus could just be as big as necessary.

Maybe students wouldn't get as excited about this as I would, but I think it would be a pretty cool demonstration for class.  Here's my attempt at creating the same in GeoGebra, but I'm not sure how to plot only part of a conic section yet.

Teaching Proofs

As I mentioned in my first blog post, on Van Hiele levels,  I have some misgivings about the way proofs are presented in the curriculum from which my geometry classes are taught.  The work that we have been doing in Modern Geometry this semester has opened my eyes to the value of conjecturing.  Ideally, proof in my students' geometry classes would develop naturally out of discussion and defense of their own conjectures.  Instruction on how to construct proofs more formally could follow.

As it is, my students face formal geometric proof head on.  They start by writing "algebraic proofs," which basically entails setting up and solving equations with justifications (maybe a theorem or two, mostly properties of equality) for each step.  Then they practice proofs that are more based in geometry concepts.  Early on most of their work is filling in blanks in two-column proofs (which is the format we stick to, for the most part), providing missing statements and reasons.  The next chapter in the course includes parallel and perpendicular lines, and proofs are included again.  The big finish for proofs in Geometry A (although there are more in Geometry B) is triangle congruence proofs.

Proofs are always a struggle, for the majority of my students anyway.  To help them along, I try to point out common patterns in the structure of the proofs we write in class, such as "the final statement is whatever you were asked to prove" or "to change a word into an equation or an equation into a word, use its definition."  I try to remind them to keep the end result in mind, and sometimes we even start at the end of a proof and work backwards a few steps.  A lot of these things that I do to help students "get through" proofs don't really get at the meaning involved, and to a large extent I often feel I'm just teaching them the rules of a game.

It was hard not to be somewhat encouraged though, when by the end of our chapter on triangle congruence, many of my students were able to put together a pretty solid triangle congruence proof on their own.  It made me wonder if teaching formal proof head on might have some value.  The big question, of course, was whether or not my students could transfer any of the skills they were demonstrating to more varied proofs.  I got a partial answer to that question in my class' response to a bonus question on one of their unit tests.

The bonus problem asked them to write a proof of the Triangle Sum Theorem (the sum of the angle measures is 180 degrees).  Not every student attempted the proof.  Among those who did, only two gave fairly conceptually solid proofs, and theirs weren't entirely rock solid as far as our formal standards from class went.  Below is a copy of the problem on the test (I think more could have been said of what is "Given:"), and the two best proofs from my class.

For only two students to come this close took some wind out of my sails again.  But at the same time, it is still the slightest bit encouraging that a couple of them were able to go "off pattern" and prove something fairly different from the triangle congruence cases we had been working with (maybe my hopes for students' proof fluency are way too low).

So I still think that a heavily conjecture-based lead in to proof would serve more of my students better.  But having gone through the usual proof sequence with my students once again, I don't believe what they've done will have no value for them as they continue in mathematics.  If they aren't able to transfer their proof writing to more novel proofs now, might some ground work still have been laid for future proof writing?  It seems like writing proofs formally will always be somewhat awkward the first time, even if it is preceded by lots of practice with informal arguments.

I will have many of the same students again next trimester in Geometry B, and the first unit is on quadrilaterals and their properties.  We are planning to give students a few days to hopefully conjecture some of these properties on their own and then try to prove them with what they already know.  This will be another opportunity to see how much they were able to take away from their education in proofs this trimester.