Lof vs AoPS for Geometry

[kohlby]

My 4th grader has been really enjoying AoPS. He did the first 12 chapters of Intro to Algebra for his Alg I and we're now taking a break by going through Intro to Number Theory. Then, he'll resume AoPS Intro to Alg for Alg II. Since AoPS is a wonderful challenge for him, the plan was to continue all the way through. But my daughter got some Life of Fred elementary books and now my oldest is wanting to try Life of Fred. My question is how does Lof Geometry compare to AoPS in rigor and interest? I want something as challenging as AoPS, so if Lof comes anywhere close, then I'm willing to let him give it a shot. He wants to be an engineer when he grows up and is gifted in math/science so I want a challege for him.

[Tattarrattat]

My DS took AoPS geometry class and learned a lot. He also read through LoF geometry book, but only as extra fun reading (He did think through the math part if needed). He thought LoF was much easier than AoPS. We love LoF books here, but I wouldn't think they're nearly as challenging as AoPS.

[regentrude]

Nothing is as challenging as AoPS.

That does not mean that LoF is not a good program for many kids; but as rigorous as AoPS it is not.

[kohlby]

Is Lof MUCH easier than AoPS or just a little easier? I'm willing to let him go a little easier, but I want to make sure he learns all that a high school honors Geometry class would cover as an absolute minimum. As a former math teacher, I can tell you that AoPS goes WAY above honors Alg I and honors Alg II though! (I realize that the contents is on LoF's website, but without seeing the book as a whole and the specific questions, I can't accurately assess the level and completness).

[Luckymama]

First of all, I haven't seen LoF Geometry. Dd is doing AoPS Geometry right now. My older students took geometry in private high school (at the highest level possible for both schools), using different texts. Of the three texts, AoPS is without a doubt more difficult, more advanced, contains more information, and requires more conceptual leaps than either of the other texts.

 

I <3 AoPS.

[mumto2]

Dd did LOF geometry for her stand alone geometry course. If you decide to proceed make sure you do it after algebra 2. She had a tough time because she did geometry first and taught herself the required algebra as she went. She enjoyed the course. Tests well in geometry but has also completed the Singapore NEM sequence.

 

[PinkyandtheBrains.]

In the high school and up courses LoF is a solid stand alone course, and has the advantage of being entertaining as well. We are using LoF Geometry right now, it is not as rigorous as AoPS though....nothing really is.

[kohlby]

I realize it's not as rigorous as AoPS, but how much less? Like would you consider it harder than a regular high school honors class, about the same, or less? I realize AoPS is above. We've only done Alg AoPS, but I'm a former high school math teacher and it was very obvious that AoPS was harder than a high school honors class. Though we're finding Intro to Number Theory much easier than I expected.

 

If you did AoPS with a younger student, was the student's age a disadvantage? The plan is for Geometry the year after next, so after Alg II. But he'll only be 6th grade then. That's part of the reason why I'm considering LoF, since I've heard AoPS Geometry is harder than Intro to Alg and that's the perfect level for My oldest now - but going higher would be too much at this point. I realize I have a full year to figure this out, but figured it might take a while to find parents familiar with both curriculums and/or ones with opinions about using it AoPS geometry with younger students.

[lewelma]

My ds(12) started AoPS algebra at age 9. He finished the book before moving to AoPS geometry at age 11. At age 6, he tested as a 25 year old in spatial reasoning, and he is still finding Geometry challenging. But it is his favorite of the AoPS books, and he would know, just look at my siggy.

 

The geometry book starts off easy but then accelerates rapidly. I would suggest that your ds finish intro algebra and side step to number theory or counting before starting geometry.

 

Given his talent, I would definitely have him do AoPS over LoF.

 

Ruth in NZ

[PinkyandtheBrains.]

I realize it's not as rigorous as AoPS, but how much less? Like would you consider it harder than a regular high school honors class, about the same, or less? I realize AoPS is above. We've only done Alg AoPS, but I'm a former high school math teacher and it was very obvious that AoPS was harder than a high school honors class. Though we're finding Intro to Number Theory much easier than I expected.

 

If you did AoPS with a younger student, was the student's age a disadvantage? The plan is for Geometry the year after next, so after Alg II. But he'll only be 6th grade then. That's part of the reason why I'm considering LoF, since I've heard AoPS Geometry is harder than Intro to Alg and that's the perfect level for My oldest now - but going higher would be too much at this point. I realize I have a full year to figure this out, but figured it might take a while to find parents familiar with both curriculums and/or ones with opinions about using it AoPS geometry with younger students.

 

 

 

I wouldn't consider it harder than the honors geometry my nephew (16) is taking at his school, it might be equal based on what I've discussed with him. I would consider it more challenging than your typical high school geometry course. Does that help?

[lewelma]

Oh, one more thing to consider. Do you plan on graduating him early? If not, do you have the resources for him to take upper level math in high school? Does your state require a certain number of math credits to graduate and do they count classed taken in middle school? Given the answers to these questions, you may want to go deeper rather than speed through the standard math sequence. A lot of people use number theory and counting/probability for this purpose.

 

Hopefully, someone with more expertise in this area will comment, but I have seen it discussed before.

 

Ruth in NZ

[PinkyandtheBrains.]

Oh, one more thing to consider. Do you plan on graduating him early? If not, do you have the resources for him to take upper level math in high school? Does your state require a certain number of math credits to graduate and do they count classed taken in middle school? Given the answers to these questions, you may want to go deeper rather than speed through the standard math sequence. A lot of people use number theory and counting/probability for this purpose.

 

Hopefully, someone with more expertise in this area will comment, but I have seen it discussed before.

 

Ruth in NZ

 

 

 

I second the idea of going deeper. We are putting on the breaks once we are done with LoF Geometry and doing "fun" math like Number Theory, Counting and Probability, Cryptology and other projects for a bit.

[Dmmetler]

Why not both? With a 4th grader, you've got a lot of time.

 

My thought, with a young mathy kid, is that I'm willing to let my DD enjoy Fred NOW, when she needs shorter problem sets because her motor skills are age appropriate, and go back and do AOPS in a couple of years, and consider it all good. She's so ahead on math anyway that she might as well get the benefit of both, since they're both good books. I'm not trying to win a race on "See how many math courses you can finish before college". I'm trying to feed her math loving brain what it needs right now and give her a solid background. From what I've seen, AOPS has enough meat to it that even if she's already done Fred, she'll still find it challenging, just as she found parts of BA challenging, and all of it fun, despite being several levels past 3rd grade math.

[keesa]

My DS10 is taking AoPS geometry now. He finished Algebra I and II last year. FWIW, there are lot of confusion even for those mathy kids (most of them 6th or 7th graders). If we can call LoF the "pre-geometry", better start with it first, and check if your kid is geometry ready.

[mathwonk]

For geometry, especially with gifted students, I like AOPS but I suggest Euclid as the best there is, in the beautiful Green Lion edition:

 

http://www.amazon.co...lid, green lion

 

There is a guide written by a famous present day geometer that helps a lot too, and goes much further. The first chapter especially is suitable for high school students, and the later chapters are aimed more at college students.

 

http://www.amazon.co...lid, hartshorne

 

I used these books successfully for my epsilon camp course in Summer 2011, with gifted 8, 9 and 10 year olds.

 

http://www.epsiloncamp.org/

 

One reason to prefer Euclid is that it teaches geometry without presuming knowledge of real numbers. We all know about whole numbers and fractions, but irrational numbers are more mysterious, and the concept of the continuous "real number line" is something that logically precedes and gives rise to the idea of real numbers.

 

Without an understanding of the line first, one has no background to grasp the concept of the continuum of real numbers. This is historically the way the subject progressed, with paradoxes about the real line emerging from the false assumption that all numbers are rational, giving rise to more sophisticated concepts for dealing with all reals. The key that is used in Euclid, is that although not all real numbers are rational still the rationals are "dense" in the reals, so that one can use the tool of infinite sequences of approximations to deal with irrational numbers. This is the underlying idea used later in calculus to define numerical solutions to hard problems as limits. So to my mind Euclid is the proper foundation for the ideas of calculus.

 

Geometrically, real numbers gain meaning as separations of the line, so without the concept of the line, there is no corresponding concept of real numbers. Modern books do things backwards, following Birkhoff, which simplifies the axioms for geometry but at the expense of assuming a more mysterious concept, that of real numbers, with no motivation nor adequate description. In hindsight, one can abstract the geometric properties of real numbers as axioms, but this is so hard that it is not done in many modern books, especially not ones wishing to be considered elementary. As I recall, although AOPS assumes real numbers as its basis for geometry, it does not give a rigorous description of those real numbers, nor state their main properties. Since I no longer have a copy in front of me, my statements are, as always, subject to correction. Please feel free to do so.

 

[For the interested, the key property of the real line distinguishing it from a line with only rational points is: "in any separation of the line into two non empty intervals, exactly one of the intervals must have an endpoint". I.e. removing any point causes a separation of the line into two sides, and every separation is caused by removing a point. The corresponding abstract axiom for real numbers says: "every non empty bounded set of real numbers has a least upper bound." If no version of this axiom is present in your geometry book, then the real numbers have not been fully described there.]

 

I feel AOPS also uses circular reasoning in its approach to area of triangles, assuming similarity to define area as (1/2) base times height (you need similarity to prove you get the same area using a different base), and then using area to prove the principle of similarity later, as Euclid did. So in this sense, some of the apparent rigor in AOPS is slightly illusory. I have not seen LOF. Although to my knowledge the arguments in Euclid are not circular, there are axioms which are used but not clearly stated there. This is clearly addressed in the guide book by Hartshorne.

 

Still AOPS is a very, very good book, I myself learned some geometry from it and you won't go wrong with it in spite of my highly specialized criticisms. In fact it is thanks to the popularity of book series like AOPS that the quality of elementary education where they are used is again improving. It is only because Euclid has received an undeserved bad rap for so many years, that I thought it might help to give some of my reasons for preferring it.

 

In fact I would date the decline of American public education to around 100 years ago when Euclid was largely removed from the curriculum. American math books from about 100 years ago are much better than those used today, as the public schools began to grapple with the problem of educating more students with less background. You can see this by comparing modern geometry books to Euclid and modern algebra books to A treatise on Algebra by C. Smith, from 1895. I mention this since part of the home schooling movement is presumably a reaction to this decline.

 

(But if you try to buy a copy of Smith's algebra, do not buy the highly flawed, poorly scanned modern copies offered by "Forgotten Books", buy an early edition used instead. I have a nice copy of the 1896 5th edition.)

 

I want to acknowledge that a child may find Euclid harder to read on his own than LOF, which I gather is written to be readable without a teacher, as is perhaps, although maybe less so, AOPS. Still, I think Euclid has the most to offer, just as Shakespeare and Homer have more to offer than say Harry Potter or even better modern books for kids. My free epsilon camp geometry guide to Euclid, although not as good as Hartshorne's, may help a child get into it without a teacher. As always I am willing to answer questions about the material.

 

http://www.math.uga.edu/~roy/camp2011/10.pdf

[SnegurochkaL]

Thanks for your post Mathwonk. Being in Russia I had studied Euclid Geometry at school for 3 years so I can appreciate it:).

Does AOPS Geometry book cover any formal proofs? How deep are they covered? Thanks a lot.

[keesa]

@mathwonk: My DS just finished the chapter of Similar Triangle at AoPS geometry last week, so I pulled out his class note and found that they proved the AA similarity by triangle congruency and parallel lines. It did not seem like circular proof to me.

 

@SneguochkaL: AoPS geometry class covers formal proofs (college style). It is not very deep (long winding), but can be very hard. Here is an example of hard problem:

[regentrude]

Does AOPS Geometry book cover any formal proofs? How deep are they covered? Thanks a lot.

 

 

It does cover proofs extensively. (Pretty much everything is proved in AoPS anyway, not just in the geometry text.)

They do not use the two column format that seems to be common in US public schools; they use a narrative format which is more close to how a mathematician would write a proof.

[mathwonk]

I am going from memory, but as I recall, AOPS geometry gives Euclid's own proof that if a line XY drawn parallel

 

to the base BC of a triangle ABC,

 

cuts one side in a ratio of AX/XB, and the other side in the ratio of AY/YC. then those two ratios are equal, i.e. that AX/XB = AY/YC.

 

The proof consists in comparing areas of certain auxiliary triangles constructed within the given triangle.

 

But the concept of area of a triangle itself has been defined as (1/2)(base)(height) in the beginning of the book.

 

 

 

That "definition" actually gives three potentially different areas for each triangle, depending on which side is chosen as base.

 

In order to prove that all three area calculations give the same number, i.e. that the "area" of a triangle makes sense

 

with this definition, requires the use of the similarity principle.

 

 

So similarity is used to justify area and then area is used to justify similarity. That is what I am concerned about as circular reasoning.

 

I.e. similarity and area are equivalent principles. You have to start by justifying one of them before using it to justify the other.

 

Euclid gives an axiomatic approach to area (missing some axioms), but AOPS just makes an unjustifiable "definition".

 

So when they want use area later, it has not been properly introduced.

 

 

 

I wish I had a copy so I could cite pages, but this is what I recall from perusing it this summer. Does this make any sense?

 

 

Harold Jacobs also gives Euclid's proof of similarity using area, but he has given axioms for area beforehand.

 

Logically, if AOPS wants to use their notion of area they need to say that they are going to assume that the definition they gave always gives the same answer independent of choice of base. Do they say this? I did not notice it.

[mathwonk]

duplicate post

[keesa]

@mathwonk: Here is the solution by my DS:

[mathwonk]

Is it possible to delete a post? I just realized that the "solution" was to the previous "hard problem" rather than to the similarity principle.

 

@SneguochkaL: Did you study from Kiselev's Planimetry? I hear that was popular in Russia. It has recently been translated into English as well.

 

Kiselev gives a "limiting" argument for the similarity principle, approximating the ratios by rational ones.

[mathwonk]

Nice argument by your DS, keesa.

 

Two tiny remarks: i think you meant triangle PXD in the first part in place of PAD. Moreover when invoking SAS similarity, one wants to use the exact correspondence between the angles, in the proper order, so strictly speaking it is best to say that triangle RYX ≈ triangle RCD, rather than triangle RXY ≈ triangle RCD, in order to conclude that angle RYX equals angle RCD.

 

Nice work!

[mathwonk]

By the way, the "circularity" I am discussing in the treatment of similarity in AOPS may well be a conscious choice they made to keep the presentation simpler. I.e. usually savvy math book authors are very aware of what they are doing, even when it seems slightly flawed. It is often just too complicated for the reader if we dot absolutely every i and cross every t, so we sometimes just fudge a little in hopes it won't matter too much. Or I could just be wrong. I started to email Richard Rusczyk and ask but I didn't see any contact email on the AOPS site.

 

Euclid's approach also has an omission as pointed out by Hartshorne. Namely he axiomatizes not area as a number, but simply the notion of equal area. He says that two triangles have equal area if they are either congruent, or admit decompositions into corresponding congruent triangles, or are the difference of two figures that admit such decompositions.

 

Now he does not do it, but one can prove this does give an equivalence relation, but what he leaves out is the non obvious fact that if one figure fits properly inside another that they cannot have equal area. I.e. it is not so easy to prove that half a triangle say, does not have equal area to the whole triangle. For that one does need some kind of axiom or else a numerical measure of area, where one knows the property that the product of two non zero numbers is non zero.

 

Hartshorne gives a wonderful treatment of all these matters in his guide book to Euclid, suitable for advanced students and/or professionals. I only mention these things because some of you are already living with very young future professionals.

[SnegurochkaL]

 

 

@SneguochkaL: Did you study from Kiselev's Planimetry? I hear that was popular in Russia. It has recently been translated into English as well.

 

Kiselev gives a "limiting" argument for the similarity principle, approximating the ratios by rational ones.

 

 

I have a Kiselev's copy in Russian but I haven't looked through it yet. I used Atanasyan's textbooks at school which I like much better than Pogorelov's. I think I will end up using a mix of Russian/American textbooks to be sure my kids at least know what I learned at school.

 

Thanks for your comments.

[regentrude]

But the concept of area of a triangle itself has been defined as (1/2)(base)(height) in the beginning of the book.

That "definition" actually gives three potentially different areas for each triangle, depending on which side is chosen as base.

In order to prove that all three area calculations give the same number, i.e. that the "area" of a triangle makes sense

 

with this definition, requires the use of the similarity principle.

...

but AOPS just makes an unjustifiable "definition".

 

So when they want use area later, it has not been properly introduced.

 

 

I do not quite understand, mathwonk. The area of the triangle is not simply defined, but derived to be (1/2)bh through constructing a rectangle by doubling each of the right triangles that are created when the base is constructed; since each right triangle is doubled, the area of the original triangle must be half the area of the constructed rectangle.

 

Would not the simplest way to prove that we can choose ANY base/height pair be an indirect one: since the triangle clearly can only have one unique value of the area, and since by construction this area is 0.5bh for one chosen b and h pair, all three b*h values must be identical?

 

I do not see where similarity would enter the problem.

[keesa]

@mathwonk: Here is another note from my DS for similarity proof:

 

[mathwonk]

First you have to prove that some concept such as "area" exists. if you assume that there is such a thing as area,that it is a number, and that it is additive for disjoint decompositions, then yes you can make some deductions. This is the approach Harold Jacobs uses. But you have to ask yourself, what is area?

 

I.e. this statement is what is not at all clear:

 

"since the triangle 'clearly' can only have one unique value of the area,"

 

(In general a geiger counter for detecting a problem, or a mistake, in a math explanation is wherever the word "clearly" occurs. I.e. if it is so clear, then explain it to me like I'm a 5 year old, as Denzel Washington's character said in Philadelphia.))

 

 

If you define it to be base times height for a rectangle, then try to prove that for a parallelogram it equals base times height.

 

if you assume that a parallelogram HAS a unique area, then it does follow. But if you are skeptical as to whether "area" makes sense for a parallelogram, you have to check that base times height is the same no matter which base you choose.

 

if you look closely, (I just did so), you will se that the assumption that both different choices of base for a parallelogram, give the same answer for the are, is equivalent to assuming similarity for triangles.

 

Choose a triangle and drop two perpendiculars to two different bases for that same triangle, You will note that the two right triangles they form are similar, and that the statement of similarity for those two triangles is equivalent to the statement that the two products of base times height are equal.

 

This is very subtle because we are brainwashed into thinking in advance that :"area" makes sense. But once you make that assumption you have missed much of the difficulty. "Area", as a number, is not used at all in Euclid, until after he has proved similarity, assuming only the abstract concept of "equality [of area]" defined in terms of equidecomposability.

 

If you think that "area": means "how many little squares fit inside the figure", then you have to prove that number of squares is always the same no matter how you try to fit them in there. Also you have to make sense of "how many" when it is an irrational number of squares, for example.

[mathwonk]

Let me put the question in the simplest possible terms. If it is true that a triangle has an "area", then it should be possible to tell me how to determine what that area is.

 

And the method must give a unique answer. It is not adequate to give a potentially non unique answer and just pretend that the answers must be unique. You have to demonstrate that the method you give does give a unique answer.

 

On the other hand, it is acceptable to just assume that area makes sense and has these natural properties, but you have to acknowledge that you are assuming that. Harold Jacobs does this. It is not logically acceptable to pass over it in silence and not admit what properties you are using, as I am afraid AOPS does. Unless of course they just don't choose to be that precise. That is ok in a sense, as an informal tactic, but it is not a mathematically rigorous approach.

[mathwonk]

Let me put the question in the simplest possible terms (for my own benefit). If it is true that a triangle has an "area", then it should be possible to tell me how to determine what that area is.

 

And the method must give a unique answer. It is not adequate to give a potentially non unique answer and just pretend that the answers must be unique. You have to demonstrate that the method you give does give a unique answer.

 

On the other hand, it is acceptable to just assume that area makes sense and has these natural properties, but you have to acknowledge that you are assuming that. Harold Jacobs does this. It is not logically acceptable to pass over it in silence and not admit what properties you are using, as I am afraid AOPS does. Unless of course they just don't choose to be that precise. That is ok in a sense, as an informal tactic, but it is not a mathematically rigorous approach.

 

In Euclid's approach also, it is not obvious that one cannot decompose a square into pieces, and reassemble them to form two squares both congruent to the original square. Of course intuitively it seems nuts to think this could happen, but you have to prove it, or else admit openly that you are assuming it.

 

This hypothesis is used in Euclid's proof of the converse of the Pythagorean theorem in Book I. However Euclid does make this an explicit assumption, because he has a general principle that says "the whole is greater than the part". Thus in this sense, he does state what he is assuming about "equality of area".

[mathwonk]

regentrude, you have put your finger on the problem, and have correctly identified the "easy" direction of the problem, with your use of the word "unique".

 

Problem: Assuming we have assigned a number to each line segment (which is the same for all congruent segments),

tell how to assign a number to each plane polygon ("area"), so that the number assigned

 

1) is additive over disjoint decompositions,

 

2) is the same for any two congruent figures,

 

3) and equals (base)times(height) for any rectangle.

 

 

In this problem there is the question of existence as well as uniqueness. Existence deals with whether there is any way to assign areas, and uniqueness deals with whether there is more than one way.

 

In fact you have proved there cannot be more than one way to assign the area to a right triangle, using axioms 1,2, and 3. I.e. knowing the area of all rectangles, and knowing how to decompose a rectangle into two congruent right triangles, determines the area of those right triangles as 1/2 the area of the rectangle.

 

However, there is actually a theorem that any bounded figure at all in the plane can be decomposed into a finite number of "figures" and rearranged to form any given rectangle. In fact any rectangle can be decomposed and rearranged to form a rectangle twice as large.

 

Also just one of the right triangles formed as half of a rectangle can be decomposed an d rearranged to form the entire rectangle.

 

Thus the area of half a rectangle must also equal the area of the whole rectangle. This is inconsistent with the axioms so in fact although there is indeed at most one way to define the area of a right triangle, in fact that way does not satisfy the axioms, and so in reality there is no way to define area, if we allow the decompositions I claimed exist.

 

Fortunately those decompositions are very exotic (Banach - Tarski paradox, 1924) and employ "figures" much more wild than triangles. But one can still imagine that a very clever fellow might be able to find a decomposition of a rectangle into triangles that when rearranged form a larger rectangle.

 

Of course we do not believe this is possible, but that must be proved.

 

 

So you are right, the uniqueness part of the theorem is relatively easy and clear. If we assume there is at least one numerical area function which does satisfy all the axioms (at least for some nice class of figures, such as polygons), then the only way to assign such a number is to use (1/2) base times height for triangles, and then to decompose other polygons into triangles and add up their areas. So there is at most one such function.

 

 

In order however to prove this answer actually works, i.e. does satisfy all the axioms, we have to prove that for triangles it never gives two (or three) different answers, and that when we decompose a polygon in two different ways into triangles, that adding up their areas always gives the same result.

 

These two theorems can be proved, the first by proving AAA similarity for triangles, and the second by "refining" any two decompositions into a third finer one.

 

This is carried out in Hartshorne's book, following ideas of Hilbert.

 

 

 

I should say these are very subtle matters (to me at least), and I am never completely rigorous about them even when I teach calculus. I.e. I always pretend that "area" and "volume" do make sense, and that our task in calculus is to learn how to calculate them. This was essentially the approach of Archimedes, although Archimedes was more careful than I am in my course.

 

 

I.e. one way to calculate area of a circle is to approximate it from within by triangles forming inscribed polygons. But to be sure we are getting the one unique right answer, we should also approximate from without by circumscribed polygons, and show that we get the same answer.

 

Archimedes did this but I tend not to in my class. (This is what goes wrong in the exotic decompositions of the Banach - Tarski paradox, they do not give the same answer when approximated from both outside and inside.)

 

 

I fact I am a lot more careless than Archimedes. I never noticed until I read Euclid that the argument I always gave my class for why a parallelogram has the same area as a rectangle, assumed that I had a really simple parallelogram, one of whose top vertices lies directly over the base. Then I could just drop a perpendicular from that vertex onto the base, and rearrange to get a rectangle.

 

But it is a little harder for a parallelogram both of whose top vertices are way over to the right of the entire base segment, but Euclid also does this case.

 

 

Even some very rigorous calculus books, e.g. Tom Apostol's book, once used at MIT and Stanford, make clear that they are not proving existence of area, but merely assuming it exists and satisfies certain axioms. Then they calculate it by an integral.

 

 

We very seldom point out to our students that defining area of a plane region by an integral assumes that the x axis is always considered as the base. What happens if we use the y axis as base? Or rotate the figure in the plane? Do we get the same area integral?

 

That we do, follows from the "change of variables formula" of several variable integration, in a much later course, but I have never had a student even ask me that question in elementary calculus. E.g. no student has ever asked whether (or why) using the method of "cylindrical shells" will always give the same answer for volume as using the method of "volumes by slicing". (Such questions can be approached using the principle of "repeated integration", or change of order of repeated integration, i.e. it does not matter which direction we slice in first.)

 

Of course when I am teaching I do not want to introduce more rigor than my students will appreciate, although I might refer to it in passing. Also I often don't notice these subtleties myself except when I am being pressed by sharp questioners, as here!

[regentrude]

So you are right, the uniqueness part of the theorem is relatively easy and clear. If we assume there is at least one numerical area function which does satisfy all the axioms (at least for some nice class of figures, such as polygons), then the only way to assign such a number is to use (1/2) base times height for triangles, and then to decompose other polygons into triangles and add up their areas. So there is at most one such function.

 

In order however to prove this answer actually works, i.e. does satisfy all the axioms, we have to prove that for triangles it never gives two (or three) different answers, and that when we decompose a polygon in two different ways into triangles, that adding up their areas always gives the same result.

 

These two theorems can be proved, the first by proving AAA similarity for triangles, and the second by "refining" any two decompositions into a third finer one.

 

 

Thanks for your long explanation, mathwonk. I will have to sit down with it and thing about it some more.

 

I should say these are very subtle matters (to me at least), and I am never completely rigorous about them even when I teach calculus. I.e. I always pretend that "area" and "volume" do make sense, and that our task in calculus is to learn how to calculate them.

 

 

See, mathwonk, my problem is that I think like a theoretical physicist. Physicists do all kinds of things with math mathematicians will find horrible, LOL. We use the delta function by intuition without really strictly defining everything. We have an intuitive understanding of vector calculus, without pondering the existence of some quantities first, and there are lots of examples.

My math professor at the university, who was teaching our five semester sequence of math for physicists, often pointed out where the physicist would be entirely satisfied that the function is, for example "sufficiently well behaved" - and the mathematician needed to add several more statements to define precisely what properties the "well behaved" the function had to have.

So I guess that means I gloss over the very subtle points - which are probably the points that get mathematicians excited ;-)

[kohlby]

Do you plan on graduating him early? If not, do you have the resources for him to take upper level math in high school? Does your state require a certain number of math credits to graduate and do they count classed taken in middle school? Given the answers to these questions, you may want to go deeper rather than speed through the standard math sequence. A lot of people use number theory and counting/probability for this purpose.

 

 

 

I was thinking AoPS took care of the depth. It has slowed him down a lot compared to what we were using before. He is doing Intro to Number Theory now - between the first and second half of Intro to Alg and zipping along more quickly than expected. He's enjoying number theory but it's much easier than even the first part of Intro to Alg. (However, he did Pre-Alg through a different curriculum so his transition over may have played a part too). The plan is for him to go to the Governor's School of Math and Science. It's a public boarding magnet school which is only 11th and 12th grade. If he gets in, then he'll have lots of college level courses available, including many for college credit. He won't have to sit through wimpy college math courses like college Algebra, but that school actually offers courses like Differential Equations. I really don't see him running out of maths. There's so many on the AoPS website, especially when you add in the online coures. Plus, there's always college level courses, even if I instruct him on some not for college credit or he does some independantly. (I used to be a high school math teacher). He's a fairly normal almost 10 year old boy other than being gifted, and as such, skipping him ahead in grade number hasn't entered my mind as an option.

 

It sounds like LoF won't be rigorous enough - since I want something at least as rigorous as high school honors. As for doing both, I don't know if he'll want to do that until we're through with one. I can see him more willing to go through LoF as a fun review than going through AoPS just for the fun of it. But it sounds like I should start him out in AoPS Geometry. (After we finish Intro to Alg, of course. That's been the plan all along - to finish the rest of the book as Alg II once Intro to Number Theory is done). We bought Intro to Prob as well but he wasn't all that interested in it so it's on the shelf for when he is.

[mathwonk]

Well, regentrude, we are very envious of the intuition the physicists have, which never seems to lead them astray. I.e. in solving any problem it always helps to already know the answer, and that's where physicists have the big advantage over us. Your knowledge of how the world actually works makes you tremendous guessers, whereas sometimes we are too caught up in theoretical but highly unlikely possibilities, to see the real situation.

 

The greatest mathematicians of the past were always well versed in physics: Archimedes, Gauss, Riemann, Hilbert,... unlike some of us today.

 

You probably know the recent story of the mathematicians trying to count all the rational curves, of every degree, on a quintic threefold some years back by ad hoc mathematical methods and getting stuck at degree 3. Then the physicists came along and produced a differential equation based on something called "quantum gravity" or "string theory", and a consequent "generating function" that just spat out all the answers for every degree.

 

Worse yet, the answer for degree 3 differed from the mathematicians' answer. This led the mathematicians to doubt the physicists' answer, until they checked their calculations and found the mistake that proved the physicists were actually right. I was myself one of the hapless mathematicians trying unsuccessfully to crank out the answer to this problem for low degree.

 

see the crossed out sentence in the middle of page 17 of this paper:

 

http://arxiv.org/pdf...m/9202004v1.pdf

 

Of course some of the rigor needed to prove the physicists are really right, at least in some suitable interpretation, is still ongoing, and very challenging for the mathematicians.

[Miss Marple]

We used both LOF geometry and AoPS Geometry. FWIW my boys could not figure out how LOF worked. It was an incredibly frustrating program for my most math intuitive son (who is now in college). AoPS was much more their style. I'm sure there are others who have made LOF work, but it just seemed to be disjointed. Whether it is less rigorous is difficult for me to say because, for us, it was impossible to use (perhaps too rigorous?) However, it might be fun for your son and you can always go back and have him do AoPS geometry later.

[mathwonk]

By the way, there are more free geometry notes on my web site, in addition to the epsilon camp notes,

 

i.e. there are class notes for math 5200, FWIW:

 

http://www.math.uga.edu/~roy/

 

You might get something from the discussions of similarity or euclidean content. I seem to have forgotten to write the section on parallel lines.

[mathwonk]

The penalty for your patience is more posts. This is my introduction to a geometry course for teachers, from 20+ years ago. In those days I wanted to share everything I thought about geometry. Later my goals became more modest.

 

MATH 522: Day One. July 16, 1992.

 

What is Geometry?

 

Some people think of geometry as the study of space, the physical universe we live in. The origin of the word is apparently related to this, geo - metry = “measuring the earthâ€. I tend to think of geometry as the study of shapes, or simply as that aspect of an object which relates to shape. I might ask someone, “what is the geometry of that building? i.e. is it round? is it oblong? is it tall and thin?â€

 

In one sense, then geometry is an attribute of the physical world, those aspects of it that are susceptible to the senses of sight, and touch, and thus at least the raw material of geometry is related to things which are physical, or visual. In another sense, the word “geometry†means the study of the spatial aspects of the world. Since the world has many spatial phenomena, this meaning is very broad and allows many very different focuses. An architect may study the geometry of cathedrals, an artist the geometry of maple leaves, or a physicist the geometry of black holes in space.

 

A “classical†algebraic geometer like myself studies the geometry of solution sets for polynomials in several (complex) variables. In a still more restricted sense, the word geometry refers to the traditional high school course dealing with points, lines, and planes which has been a part of western civilization since Euclid wrote his famous books of the Elements.

 

Professional geometers puzzle over such questions as “how do you describe a sphere by a few simple axioms?†or “how many specific measurements do you have to make on a complicated geometric object of a given type before you know completely its shape?†“How can you tell when a geometric object consists of more than one piece?†An airline pilot concerned with navigating the earth might be most interested in spherical trigonometry, and an astronomer or physicist should study three dimensional geometry, or even four dimensional geometry, (to incorporate the combination of time and space used in the theory of relativity).

 

Some students of geometry in high schools may be interested only in learning to solve a few specialized problems on the mathematics portion of the SAT exam. So geometry has many different aspects, and different people have need of different topics within the realm of geometry. I hope as future high school teachers of geometry, or as educators and researchers in other roles, that you will think about the specific needs and interests of your students in choosing what geometry to teach them.

 

What is Euclidean geometry?

 

Historically Euclidean geometry refers to the content of the (first part of ) the books of the Elements. Although there is much more in those books, including some higher dimensional geometry, we shall be interested primarily in the part about “plane†geometry. Today, after thousands of years of careful study of those books, the basic content has been modified only a little, and that primarily in the direction of greater precision in the statement of the axioms, and the proofs of the theorems. There are several different possible sets of axioms available today, which are logically equivalent, and one of the most appealing is the set included in our book, the “Euclidean†axioms due David Hilbert.

 

These axioms are divided into in five groups:

(i) incidence, 3 axioms dealing with the way points and lines intersect. ( for instance, through two distinct points there passes a unique line.)

 

(ii) betweenness, 4 axioms dealing with the sense in which a point separates a line into two parts, and the way a line separates the plane into two parts. (for instance if A,B are on the same side of a line, and B,C are on the same side, then A,C are on the same side of the line.)

 

(iii) congruence, 6 axioms dealing with criteria for when line segments, angles, and triangles have the same size and shape. ( for instance on any ray, you can lay off a segment congruent to a given segment and an angle congruent to a given angle; and “SASâ€: two triangles having two congruent sides and congruent angles between them are congruent).

 

(iv) continuity, one axiom, Dedekind’s converse of one of the betweenness axioms, (expressing the single deep fact that any “separation†of a line into two parts, each without endpoints, can only be accomplished by removing a point.)

 

(v) parallelism, one axiom, equivalent to Euclid’s famous “fifth postulateâ€, (that through a point off a line there is a unique line parallel to the given line).

 

These axioms, with a couple of others, are summarized on pages 469-471 of Greenberg’s book, and I recommend that you review them. A good way to do this is to copy them out longhand, since it takes long enough that way to help remember them.

 

Why study Euclidean geometry?

One reason is to learn some simple properties of lines, circles, triangles, in the hope that some of them will be found beautiful, interesting, or useful. One use for them is to acquire the fundamental geometric background needed to go on to higher dimensional, more complicated geometric topics. (An understanding of higher dimensional geometry is usually acquired by reasoning back to the familiar ideas of low dimensional, plane geometry.)

 

Another reason to study Euclidean geometry by the traditional method adopted in this book, is that this was historically the first instance of the now famous “axiomatic†approach, and it offers a good opportunity to learn to appreciate this important intellectual milestone in the history of logic and mathematics. The fundamental idea here is that if one is very careful in the proofs of the theorems, not to use any intuitive ideas about points and lines, but to stick precisely to those properties that are stated in the axioms, then one can actually apply the theorems in situations where the words “pointâ€, “lineâ€, etc. refer to something quite different, as long as the new meanings still satisfy the axioms.

 

This way you may get theorems about many different subjects, which are not actually geometry but are merely analogous to geometry, just by being careful in the way you prove your geometric theorems. This allows geometric intuition to be brought to bear on other subjects where you might not think of using it. For example, the most striking research in number theory in recent years is based on number theoretic analogues of the theory of plane curves.

 

Another reason to study Euclid’s geometry, even if you are really interested in more exotic geometries, is that Euclid’s is in some sense the simplest, and therefore a good place to start. Moreover, once you understand Euclidean geometry, it is then possible to describe even non Euclidean geometries in terms of it.

 

As suggested in class yesterday, Euclidean geometry is also a very significant piece of the intellectual heritage of the human race, and has been studied profitably by generations of budding scholars for centuries. Such famous figures in the history of science as Pythagoras and Descartes apparently believed that the study of geometry was a valuable tool for training the mind to think rationally.

[mathwonk]

yes thank you keesa. that is a nice presentation of Euclid's proof of AA similarity assuming the theory of area, i.e.

 

that if two triangles have the same height, then their areas are in the same ratio as their bases. this assumes a theory of area of course.

 

 

As I said, as I recall, AOPS does not present a theory of area, so if I recall correctly,

 

this is a gap in their presentation of similarity.

 

 

\Nonetheless it is a beautiful argument that a good theory of area implies the basic similarity theorem.

 

This is Euclid Prop. VI.2, and I recommend reading it there. It may be a motivation for your young

scholar to read Euclid,

 

when he/she realizes he already understands this proof.

 

 

best regards.

[mathwonk]

I do not expect many to dig this, but here goes anyway. Take as given the interval [0,1] and we want to subdivide

 

it into an infinite sequence of disjoint subsets, such that the whole interval is obtained as the union of these subsets.

 

Moreover we will choose the subsets so that each is a translate of every other, hence all are congruent.

 

It follows that the interval of length 1, is decomposed into an infinite sequence of disjoint congruent subsets.

 

Hence the length of the interval, which is 1, is the infinite sum of the equal lengths of the subsets!

 

So what can that length be? There is no number such that an infinite number of sums of it equals 1!

 

So that subset cannot have any length!!

 

 

Here is the construction: define an equivalence relation on the numbers of the interval [0,1]

 

so that two numbers are equivalent iff they differ by a rational number.

 

then form the set S by choosing one member from each equivalence class.

 

 

Then no two elements of S are equivalent, but every element of [0,1] is equivalent to exactly one element of S.

 

By means of the transformation t--> (cos2Ï€(t),sin(2Ï€t)) taking [0,1] onto the circle, we may assume we are dealing with the unit circle.

 

 

Then the infinite sequence of rational rotations carries S precisely onto the circle.

 

And this does what we wanted. I.e. what is the length of a set which covers the circle precisely in an infinite number of rotations?

 

Any positive size implies the circle has infinite length. Zero size implies the circle has zero length!!!

 

 

There is another similar construction that covers the unit sphere twice by a finite number of rotations of two disjoint subsets of the sphere.

 

This would imply that if those goofy subsets had an area, then the area of the sphere would be twice itself!

 

Or maybe the physicists have intuition for this???

 

if you prefer finite operations, note that for every finite number N, N rotations of our set fail cover the whole circle, so if the measure of our set is e then Ne must always be less than 2Ï€.

 

But this is onkly possible if e = 0.