Geometry

[Ricky]

I just thought I'd pick a space to rant a bit:

I think that, as shocking as it may sound, not enough geometry is taught in schools. I know someone from Romania, and he made the point that there, they have at least six or seven years of geometry, whereas in the United States, we're "lucky" if we get two or three. Also, all of this is usually before elementary trigonometry anyway, so it's not only more awkward but less extensive as well.

I think there's a reason for this. I can still remember high school geometry; heck, I'm still in high school. And frankly, I didn't like it that much. Too much of the two-column proof stuff that I hated. The first semester was spent learning "obvious" facts, and we weren't allowed to skip the simplest of steps. It was, to put it bluntly, annoying. And as I think back on it, I didn't really learn too much either, just the bare bones.

If you ask people who have taken geometry in high school what the orthocenter, centroid, and circumcenter of a triangle are, they might still remember, though probably not. They probably will not have ever learned that these three points always lie on a line with the segments between them in ratio 2:1. They'll have never heard of the nine-point circle. They'll have never heard of Ceva's Theorem, Menelaus' Theorem, Pascal's Theorem, or Ptolemy's Theorem. They might know the Pythagorean Theorem but not even one, or if one, not even two, of the hundreds of proofs of it.

Maybe geometry isn't practical enough, despite the fact that it's more concrete than many other fields of mathematics; you can draw a triangle, but you can't really draw a number or a function. Maybe it's that people think that students won't be able to grasp the material, but it's really much easier than calculus and more intuitive to boot. But I think it's just because it isn't taught well enough, so that students can't help but be thrilled when they get to leave that geometry book at school for good.

Well, I'm out of breath. Any thoughts?

[CrystyB]

First of all, you're wrong.

quote:

---

If you ask people who have taken geometry in high school what the orthocenter, centroid, and circumcenter of a triangle are, they might still remember, though probably not. They probably will not have ever learned that these three points always lie on a line with the segments between them in ratio 2:1.

---

I knew 4 interesting points of the triangle, but unless i go learn some "heavy" math, i won't know the line, or the ratio. I found about it only now, in my first year of Math @ University.

quote:

---

They'll have never heard of the nine-point circle.

---

me neither. It's not high-school/elementary-school stuff. It's only for the olympics/olympiads.

[Ricky]

Okay, so I guess my point is not so much that they teach it in other places so they should teach it here, but just that they should teach it here, or at least offer it in more places than the isolated few that they do now.

And now for an actual educator idea:

Suppose I'm teaching geometry, and I want to prove that the altitudes of a triangle concur. Usually, textbooks will supply the following proof (outlined below):

Let ABC be a triangle, and draw lines parallel to the three sides through the opposite vertices. By looking at a bunch of angles, we can see that the big triangle is similar to the little one, and that it is divided into four smaller congruent triangles. The ratio of the small triangles to the big one is 1:2, so the altitudes of ABC are the perpendicular bisectors of the sides of the big triangle, which we know intersect at the big triangle's circumcenter (because it was proved several pages ago.)

Why not present the following, more informative proof:

Let O be the circumcenter of ABC, G the centroid, and M the midpoint of BC. We know G trisects AM so that AG = 2*GM (because we proved it several pages ago.) Draw H such that G lies between O and H on a line and GH = 2*OG. Then by ratios and angles and stuff, AGH is similar to MGO with ratio 2:1. By more angles, AH is parallel to OM, and is thus perpendicular to BC, so AH is an altitude. Repeat the argument for the other two vertices, and get that H lies on all three altitudes, so they concur.

Note that this proof requires a bit more work, but is not much more complex. Moreover, it shows that O, G, and H lie on a line in ratio 1:2. Why not use this proof instead? No matter which proof is used, students will probably not be able to instantly reproduce it, but the second at least gives a better result without too much more effort.

That's all, for now...

[CrystyB]

Actually the second one MIGHT be better because i think it doesn't use Euclid's Axiom of Paralellism.

[mole]

Here in Australia (Queensland especially) we do almost no geometry.

[Ricky]

Without the parallel postulate, any two similar triangles are congruent, so it is necessary for both proofs.

[VinnyQ]

quote:

---

They probably will not have ever learned that these three points always lie on a line with the segments between them in ratio 2:1. They'll have never heard of the nine-point circle. They'll have never heard of Ceva's Theorem, Menelaus' Theorem, Pascal's Theorem, or Ptolemy's Theorem. They might know the Pythagorean Theorem but not even one, or if one, not even two, of the hundreds of proofs of it.

---

I've been jibbed. I want my money back.


I do know the Pascal programming language tho !!!
... and the Pascal Triangle, and the Pascal proof for the existence of god or something ...

[CrystyB]

Ricky, would you please elaborate? i'm confused about that statement...

[Ricky]

So many of you probably know something about the history of the so-called parallel postulate. If you don't, here's a brief history:

Around 300 BC, Euclid wrote the book, the Elements, that is the foundation of geometry today. In it, he states five axioms as a basis for his geometry, along with a bunch of definitions. The five axioms, which he calls postualtes, are, succinctly:
1) Two points determine a line
2) A line segment can be extended indefinitely
3) A circle can be drawn given center and radius
4) All right angles are equal
5) "If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

Naturally, this was a bit awkward. Many people have tried, since then to prove this fifth postulate based on the other four, but none have succeeded. Euclid, too, did not like the statement of this postulate either, because he delayed using it for as long as possible. Without the parallel postulate, he proved all the triangle congruence theorems, SSS, SAS, ASA, and AAS. It was not until afterwards that he proved such things as the sum of the angles of a triangle is two right angles, that a line falling on parallel lines makes equal alternate interior angles, and the Pythagorean Theorem.

The fifth postulate is equivalent to many other statements, such as:
1) If two lines are parallel, any line that intersects one must intersect the other
2) Parallel lines are everywhere equidistant
3) Through a point not on a line, exactly one line can be drawn parallel to the line.
4) The sum of the angles of a triangle is two right angles.
5) Rectangles exist.

Three mathematicians, Gauss, Bolyai, and Lobachevski, were among those that sought to prove the parallel postulate. They managed to show that given the second postulate, the angles of a triangle had to add up to less than or equal to two right angles, but they could not prove it could not be less than. Indeed, the so-called hyperbolic geometry that erupted from triangles having the sum of their angles less than two right angles turned out, as discovered years later by Beltrami, to be just as consistent as Euclidean geometry.

(As a side note, spherical geometry was discovered later by Riemann, in which the second postulate is relaxed.)

So, that was really long-winded. Anyway, in hyperbolic geometry, we have another congruence theorem besides SSS, SAS, ASA, and AAS, namely AAA. The proof is actually quite simple:

Suppose ABC has angles equal to those of DEF. Let's assume that AB<DE. Then draw G on DE so that DG=DE. Draw GH so that H is on DF and <DGH = <B. Then by ASA, DGH is congruent to ABC and <DHG = <C. There are three cases:

1) H does not lie between D and F. Well, then <GHF is bigger than the exterior angle at F, which is impossible by one of Euclid's early theorems.

2) H is F. This obviously doesn't work either (just draw a picture.)

3) His between D and F. Well then look at quadrilateral GHFE. The sum of its angles is four right angles, which is impossible in hyperbolic geometry. Thus, we have a contradiction.

Thus, AB is not less than DE, and by similar argument, DE is not less than AB, so AB=DE. Thus, ABC and DEF are congruent by ASA.

So, AAA is a congruence theorem in hyperbolic geometry. Yay, sort of.

Oof, my hands are tired.

[tinman]

Geometry great stuff:
One of the most important to study is the Elipse.......

Make shure you search out (use data) for shapes dont laugh. Sacred Geometry you need to no these elements and elipse at this time in history.
Use Golden Ratio to stir interest.
.
tinman

Just idea to help topic...

[Leptonn]

Last year, I took an advanced (?) undergraduate-level course in geometry where the first half was spent on Euclid's elements, the 9-point-triangle, Cevans, and that sort of thing. None of it is covered in high schools here.

One of the important points about learning geometry in P-12 schools, however, is learning to think geometrically. Being able to visualize 3D and 2D structures, for example, is essential in physics and engineering and important in the other physical science.

Another important aspect of mathematics that is taught through geometry is the concept of a proof, which is sorely missed here.

[Courk]

I took one semester of Geometry in high school, followed by a semester of Algebra II. The year before I had to take two semesters of Algebra I, despite having learned all/most of it in 8th grade. I would have liked to have taken Geometry my freshman year and perhaps actually remember the stuff, but, alas, no.

[mudbuck]

I'm in the middle of Geometry right now, though I'm in eight grade... I actually take a bus to the high school every other day for it.

The only thing I'm complaining about are too many conjectures and a really monotonous teacher. Other than that, I'm okay with Honors Geometry. The main reason being is that eigth grade Language Arts and History are thrice as much work. Gah. I hope that the rumors are correct and that Highschool is easier than those two classes.
_________________
1000oclock.com

[tinman]

You no in school I hated math.....but reason was I could see little use for it. Then latter life I had dynamic type jobs.......
Here is summery now if in school Id get the math first and typing...
Reason:
I determend that no matter what job tecnicle, Like I have degree in refrigeration also, but if you came out of school expert on math you could get a degree in it just because you can solve the tec date. Helicopters were same it all went back to math and navigation. I started the Company Oceaneering International and was USN Diver Deep Sea then frogman. The worst part of it all was the math.......OI is a monster Co and we have guys who go into space now. You could do all that on nothing but math and be a wimp. I went macho way and all the SEAl stuff but do it again Ill go wimp with math. And just run whole opperation. In long term opperations undewater we now us Remote Vehicles ...tec for them is good job, until you run the division. In Oil Field diving if your good youll be boss in two years of your specialty. Oils Field Service Companys.....not Shell etc

Tinman

just thoughts