None of the students I tutor have ever seemed to have learned how to use a compass and a straightedge to bisect angles and such. In the my high-school days (70’s) I remember doing a little of it. Is it something that school systems have deliberately deemphasized over the years, perhaps because of Autocad and other graphics-building programs?
I work at a tutoring center, and I know it’s part of our lessons (at the pre-algebra level I believe, and then again in geometry), and most of the kids have never seen it before. As a matter of fact, I don’t really remember learning it in school (in the 90s) either.
I learned it, but it was never mentioend again, probably because it’s of dubious utility for actually doing anything in the real world.
No, they don’t, in the mid-90’s when I was in high school.
Actually, I have a story about this. My sixth grade math teacher noticed I was horribly bored in math, so she told me that when I finished my sixth-grade classwork I could start working through this geometry book she had. As the math dork I was, I thought this was awesome! My absolute favorite part of the book was the constructions, and I probably spent most of my time on them.
A couple of years later, I was forced to retake geometry, even though I had already passed the end-of-course test (I forget what the rationale was). Oh well, I thought, at least I get to do those cool constructions again! I was shocked and dismayed to find out that we did not a single construction in the entire class, even though they were in the textbook. (We did do these awful two-column “proof” things, which put me off proofs until I started doing proof-based math contests.)
If you study geometry in an approach based on Euclid’s Elements, it’s not about real world applications, but about logical, deductive reasoning, and the appeal of “beauty bare” and (for centuries, until the development of non-Euclidean geometry) Absolute Truth.
Is geometry still taught this way in high school?
Regardless of whether constructions are actually useful, I think it’s sad the schools appear to no longer teach them. I always thought they were some of the coolest things in high school math.
I did them in high school (graduated in ‘94). It certainly helped that my geometry teacher was ~80 years old. She had taught one of my classmates’ parents. Fantastic teacher, and everyone enjoyed the class (when they weren’t trying to terrorize her…)
I did them in 8th grade geometry in 1988-89. I didn’t like them, because they seemed so pointless. If you need to bisect (or trisect) an angle, that’s what a protractor is for.
Nope. Using a protractor is how to incorrectly bisect an angle.
Same here. We then moved on to painfully slow proofs.
I did them in ninth grade, which would have been about 1984. Loved them. Also loved my geometry teacher, Ms. Flaherty – an old redhead from Ireland with a thick, think brogue. “You must learrrrn your teerrrums!”
As a high school math teacher, who is currently the school’s specializer in Geometry:
I make the students do constructions. They help them learn various of the theorems and corallaries through a hands on approach. Usually, they find themselves frustrated by the fact that they have purchased inadequate compasses (the $1.50 ones made of cheap plastic that break easily and don’t actually hold an angle through the drawing of the curve).
Sadly, modern geometry texts use shortcuts in constructions unacceptable to Euclid’s methods (mostly by having you replicate a distance by lifting the compass and putting it back down as if it doesn’t change its angle). When I was learning constructions in 1974-5, we had to do it the real way: assume that the compass has to be re-set each time you put it back down on paper. If I make the students do that (it’s not that hard, really), they look at me like I’m being even more of a PITA than normal. 
I had no idea what this meant, so I looked it up. Huh. I don’t think I did this in school (or if we did, we didn’t do enough of it to leave an impression). Graduated high school in '01, and had math classes all the way thru AP Calc. (Didn’t take any math in college.)
But WHY is it better to use a compass and straightedge? If you have a decent protractor, you can get as close as you need for any practical purposes. I never understood why using a compass and straightedge should be superior. To me, it seemed like trying to take out a screw without using a screwdriver- deliberately limiting the tools at your disposal for no good reason.
Heh - I recently showed my 10 year old how to do #10 - finding the center of a circle. All I used was a ruler, because I didn’t have a compass. I put the ruler so it formed a chord with 0 at one point on the circle and the other side was a nice even 6 inches for example so I could find the mid-point and mark it before I lifted up the ruler.
Then I used the ruler as right angle with the short side on the chord and the long side passing through the center of the circle. Repeat with another chord and there you go.
Accurate enough for whatever little project we were working on at the time. I need to remember to bring some drafting equipment home sometime to show how it’s really done.
I did in high school geometry in 1997.
Because, as smeone commented earlier, this has little or nothing to do with the real world. You can also come as close as you like to trisecting an angle; we know that. Euclidean geometry is all about the logical constructs and the proofs, not the results.
Quebec abandoned the teaching of Euclidean geometry I don’t know but it wasn’t offered to my kids in the early '80s until '91. Nonetheless my two older ones were able to get a class in it one year by dragooning 14 other kids into taking it with them.
Now, in response to an earlier post, I see no problem in lifting the compass and putting it down elsewhere. In principle, if it can hold its position while drawing a circle, then it can hold it if you pick it up and place it elsewhere. In any case you can do nothing with that procedure you can’t do without it. Geometry is hard enough without adding artificial constraints.
I can think of three ways to answer that, but they all relate to the difference between theory and practice.
(1) The straightedge-and-compass construction gives you the exact answer (assuming an idealized straightedge and compass). The protractor approach only gives you an approximate answer. To a mathematician (at least in certain contexts), there is a fundamental difference between exact and approximate (and in mathematics there are things that famously cannot be done exactly but can be approximated as close as you wish).
(2) With the straightedge-and-compass approach, you can prove that the angle has been bisected (in fact, the construction is the proof).
(3) Doing it with a protractor is just “cheating,” like “solving” a Rubik’s cube by peeling off the stickers and sticking them back on. It’s not playing by the rules of the game.
The protractor approach is more like taking out a screw with a bread knife: it may work, but it’s an inelegant kludge.
From the site I linked:
I didn’t have any geometry until I took it in high school, and then it was all proofs. My older brothers showed me the basic constructions when I was in grade school.
As to the value of the constructions, Bertrand Russell says it much better than I possibly could. From his essay “Study of Mathematics”: