I also did not know where I belonged in college. I applied only to one school, the local college (as a national merit scholar), but my math teacher called Harvard about me and they called and accepted me over the phone in late spring of senior year, based on one local alumni interview, well after admissions had closed but before I even applied. I had seldom left my home state and did not know even what state Harvard was in. Boy, what a fish out of water.

To Slackermom, I think the "imposter" syndrone is one of some importance. When we get to a school like Harvard, where giftedness is average, we suddenly have whatw e always wanted, namely we are average. But we are not used to that. Too bad you did not stay long enough to find out "imposter syndrone" is a universal disease at Harvard, but maybe most of us admitted it only at reunions much later. At epsilon camp, the kids were so used to knowing more math than their teachers it took them a while to get used to otherwise. But they were only 9, getting an early dose of change.

One of my own most negative experiences was in 8th grade with a teacher who always thought i was malingering or cheating. He had a lot of creative ideas for teaching but did not understand kids or giftedness. He asked us to bring in 10 words each week we could not spell, to make our own spelling words that were custom made just for us. Seems great, but what he didn't know was I was then participating on a radio quiz show and regularly winning money for my unusual spelling ability. I literally never encountered a word I could not spell at least not in everyday life, not in the newspaper (I read the NY Times, imperfectly, as a 5 or 6 year old) or in books we were assigned to read. I could spell disestablishmentarianism or phthisic or daguerreotype or what have you. When I showed up without my 10 words he blew his stack , grabbed the dictionary and pulled out 10 of his own. i had never heard of any of them but spelled all correctly except one he mispronounced, amanuensis, which he pronounced as if it began with an "e". When he saw how I did, he looked suspicious, but never praised or apologized to me. I went home terrified and told my mom I had to have 10 words a week. So we spent every Sunday afternoon with her old "blue back" spelling book, but I never found ten I could not spell. (We did not have the Scripps contest level books then.) Eventually I compromised and allowed words I could not define. The only two words I encountered that year I could not spell were hemorrhage and chartreuse. But when the teacher saw my words, he did not know the meanings or spellings of any of them, and he still gave me a hard time about it. He did not know the difference between words that sounded the same but had different meanings and had certainly never heard of asafoetida. If I erased to make my paper neater he would accuse me of cheating, especially since he occasionally graded my paper wrong. I was so stressed out I finally had to go to a psychologist. The benefit for me was the psych made me realize there was nothing wrong with me. But we need to realize gifted kids are more sensitive and more perceptive and we don't always know all the answers, sometimes they are right and we are wrong. The point of this story is to agree on the importance of reassuring the kid there is nothing wrong with him/her. That psychologist gave me the confidence to go back and make it through the year. Unfortunately I had lost all respect for the teacher and for the school, and that may well have been when things started going downhill for me for the rest of high school. With one exception, a great chemistry teacher sophomore year (a junior level class into which I was accelerated), I literally never did another lick of work there, and made all A's. (In the chemistry class my scores were so high the teacher threw them out before setting the curve.) I would suggest if you have a similar experience that you seriously consider changing schools. We considered it but my math teacher begged us not to and we stayed, essentially an academic disaster. I apologize for telling my story, but maybe it can help someone facing the same challenges.

I thought this meant non verbal skills like noticing when someone else is getting upset but not saying anything about it, i.e. basically awareness of surroundings. It took me decades to learn that stuff. If it means spatial perception, then perhaps sports help. Soccer teaches one to gauge how far the goal is and so on.

Some of these issues are still hard, like trying to participate in a group when the others want to do something really mindless, like watch shallow tv shows, or talk about something that seems trivial. Even meditating hard about valuing other people's differing perspective, it is not easy to sit through some things. If some kids want to play with mud and yours wants to build a robot, that is hard to pull off as group play. I guess if they all have their own laptops, one can look at Barney and another can read Shakespeare, but it isn't too interactive or social. Of course I do like watching "Cars" or "The Incredibles" with my grandson. It helps if the activity has something for everyone. It is also possible to "get into" something just because the other person likes it, but it takes flexibility and effort. This is a worthwhile struggle and is not directly related to intelligence. There are many occasions in interactions between professional mathematicians where one thinks something is obvious and the others do not see it at all. Since that can happen to everyone at different times and in different contexts, they learn to be very considerate of each other. The level of politeness and patience on a website for professional math discussions is much higher than on one inhabited mainly by students, in my experience. I guess since we all want to be understood, we can all benefit from trying to understand others.

Still struggling for useful suggestions. Recalling my own struggles with division, I recall it is a process of trial and error, and you check your errors by multiplication. So I would recommend reviewing multiplication. E.g. to divide 100 by 12 you guess something like 10, you multiply, and its too big, so you guess maybe 7, then you multiply again, and its too small. so you try 8. But it is really painfully slow unless you know quickly that 10.12 = 120. 7.12 = 84 and 8.12 = 96. So if it were me I would drill multiplying by single digit numbers a lot. E.g. 13 divided into 859. You start with 13 into 85, and you guess 5, but 5.13 = 65, which is 20 too little so you try 6, and then 6.13 = 78, which is off by only 7. So the guess of 6 is a keeper. Then you "bring down" the remaining 9 and you have 13 into 79, which you recall is again 6. So 6.13 = 78, with remainder 1, so i got 66 with remainder 1. Then since I have no faith in this I check it by multiplying out, 66.13 + 1= 78 + 780 + 1= 859, I hope. By the way, when I was 8 years old I was stumped in learning this, because the teacher kept saying to "bring down" the 9. She meant write it further toward my stomach on the page on my desk, but to me "down" was toward my feet or the floor, so I was completely puzzled and they could not figure out my problem for some time. Maybe they finally let me watch them do it on my paper, while they talked and I saw where "down" was to them. Boy were they puzzled for a while, since I was the best math student in the class until then. I also second the previous suggestion. When trouble arises I always make the problem easier and then work back "up".

this may not help, but in high school i felt like an alien, but as a freshman at harvard, i felt that everyone around me understood everything i said and thought. so it helps to go to a "good" school. i have only had that experience since then at professional math conferences. it is very expensive to provide that opportunity but it gives your kid the conviction that they are not that odd.

well i feel like a complete nutcase alien for thinking i could help here. my idea was to teach division geometrically a la euclid propositions I.42-45. i.e. given a rectangle and another side, how to construct another rectangle with the given side and area equal to that of the original rectangle. i can see i am on a different wave length (planet?). the connection though of course, is that given the area, and one side, finding the other side is equivalent to dividing the area by the given side. i tend to assume teaching is about conveying meaning rather than practicing skills. what about a calculator? peace. in a possibly ill conceived attempt to show this is not totally wacko, i link a preprint of one of my research papers, which has been published, but in which no two digit numbers are either multiplied or divided. http://www.math.uga.edu/%7Eroy/sv8poscrkdefs.pdf the point is your child may well be a success in math even if two digit division is a struggle.

super duper! congratulations. i am curious, did he like lord of the rings?

that reference to ratatouille is so wonderful! those are indeed challenges, but as one gets older and learns to enjoy other people for who they are, rather than wanting everyone to reinforce oneself, it gets better. It also helps to be married to a wonderful accepting and outgoing spouse.

I want to suggest another source book that may be good for either parent or child or both. This is the book Mathematics for Elementary teachers, by Professor Sybilla Beckmann of UGA. This is used there for teaching elementary teachers math, and it is highly regarded. Indeed the math ed program at UGA which uses this book, among other outstanding features, was rated as the only exemplary math education program in the US by by the National Council on Teacher Quality not long ago. Her book is also rated the best such book in the country by the same organization. http://columns.uga.edu/news/article/math-profs-textbook-wins-national-honor/ http://www.math.uga.edu/~sybilla/ Like many required books, it has appeared recently in a new edition, and hence different editions vary in price from over $100 to one cent! here is a reference for a one cent version. http://www.amazon.com/gp/offer-listing/0201725878/ref=dp_olp_used?ie=UTF8&condition=used This is a good book in my opinion. I have taught from an earlier version. Sybilla's book really emphasizes thinking, understanding, and problem solving. As I recall, the first lesson was to solve this one: you go to a party where there are 30 guests all together and everyone shakes hands with everyone else. How many handshakes occur all together? When I taught addition by "stack and carry" now called "regrouping", I made up examples like counting with old style English money, e.g. how many pounds and shillings could you get with 800 pence? Any pennies left over? Or what if you have 150 coke bottles, to put into 6 pack cartons and then into 4 carton cases. How many cases would you fill, how many left over cartons and how many left over bottles? of course the idea is to reinforce the fact that stack and carry means regrouping ones into boxes of size 10, 100, 1000, etc.... Notice that from this point of view, adding polynomials in algebra is far easier since you never carry. I.e. you just add the constants separately from the X's separately from the X^2's and there is no amount of X's that makes an X^2! I.e. just stack! It dawned on me that this analogy suggests one should multiply polynomials the way we do numbers, namely by writing one over the other. Then I found an internet version of this "vertical" multiplication: http://www.purplemath.com/modules/polymult.htm

I would like to encourage everyone to have the courage of his/her convictions, and to have faith in your own common sense at evaluating what is working for you. I myself have strong opinions about what materials I like or dislike but I am not teaching anyone else's child. I also want to admit that my math degrees did not make me any more successful as a part time home schooler than anyone else. I was qualified to teach the math, but not so much as to decide when to back off and let the child make some decisions. I forgot that unless he was having fun, it would not "take" permanently. Even that is subject to differences of opinions. We may be criticized when we require extra homework, and sometimes afterwards if we did not. As a mathematician I disliked Saxon books, but for my creative and forgetful child the repetitive drill, even if dull, did help him remember basic facts. I am not sure the unremitting boredom of it was worth it, but I could have supplemented it if I had been patient. And maybe the fact that this material raises test scores, if nothing else, opened some doors for the child. Any book that gives a struggling child the feeling that "I can do this" has some value. Our private school eventually stopped using them after I had objected for years, because they said they found the kids "did not understand anything". Almost any approach can be successful in my opinion, if properly handled, but the early texts available in our state for the integrated approach were especially unfortunate. They were cynically and hurriedly written to appear to conform to some arbitrary standards and were adopted over the teachers' objections. I think in choosing books it would help to formulate some goals for your project. If you primarily want to cement basic skills securely and memorably, then treatments that may put me off because of their drill oriented approach, may be perfect. If you want to encourage imaginative thinking, I would again suggest looking at creatively written books like those by Harold Jacobs. If you think you have a math loving highly motivated child who would benefit from explanations by the greatest minds in mathematics history, I suggest taking a look at he algebra book by Euler (Elements of Algebra), and the Elements of Euclid. These are free online. Hilbert also has an elementary geometry book, very wide ranging, called Geometry and the Imagination (in English), and originally called something more like "Intuitive Geometry" (in German). Here are some for $15, but you may want to look at a library copy first. This is not just traditional geometry. http://www.abebooks.com/servlet/SearchResults?an=hilbert&kn=imagination&tn=geometry&x=80&y=8 The book by Hartshorne mentioned by Kathy in Richmond, Geometry: Euclid and beyond, has an enormous amount of beautiful and deep material, but starts off excellently in an elementary vein, with a guide to books 1-4 of euclid. It was thanks to Hartshorne that I finally read Euclid myself after age 65!! I was always reluctant to read such an old book as Euclid and had been told it was not logically rigorous and so on. I now regard it as the ideal best introduction to geometry and to many other things. Hartshorne costs $35 and up used, but has enough material for many years of study.

Here's another fun observation about congruent triangles. I have seen it argued in books that SSS congruence for triangles is illustrated by the fact that a triangle made of three stiff straws with string running through them cannot be wobbled. I.e. there is no way to slightly alter this triangle without changing the length of the sides, even though the angles are not rigidly fixed. The argument is that the lengths of the three sides determines the angles. But is this true? If you think about it all it says is that there is no other triangle "near" this one with the same sides. But so what? Does that mean no other triangle with the same sides could exist which is a discrete distance away from this one? I.e. just because every triangle which is a small variation of this one has different sides, why should EVERY other triangle have different sides? Think of integer points on the number line. If you wiggle an integer a little, it is no longer an integer. E.g. there are no integers other than 1 in the small interval [.8, 1.2], but does that prove 1 is the only integer? Well, no. It doesn't work for triangles either, remember that although SSS and SAS and AAS and ASA all imply congruence, that SSA does not. I.e. for a given pair of sides and an angle not between them, there are usually exactly two non congruent triangles with those same two sides and angle. But they are not too near each other. I.e. imagine a triangle with two given vertical sides, and one given acute base angle. There are usually two triangles with that data depending on how you choose the other base angle, to be acute or obtuse. I.e. you look at the shorter vertical edge and let it flop over to point in the other direction. If it points left and makes say a 50 degree base angle , you flop it over until it points right and now makes a 130 degree base angle. So there are two different triangles with the same SSA data, which cannot be slightly wiggled into each other, since they are a certain finite distance apart. So if not being able to be wiggled, were enough to prove congruence, then SSA would also be a test for congruence, but it isn't. So what goal do people have who give this illustration? Are they just happy to convey part of what is going on, or have they really confused what is called "rigidity" with uniqueness? Gifted kids are much better at spotting these inconsistencies than the rest of us. Here is a related question for you. What if two triangles satisfy side-side-obtuse angle? are they congruent? I.e. if two triangles have two sides equal and one obtuse angle equal, which is not necessarily between the two given sides, are they congruent?

Let me pose a question in this regard about the AOPS treatment of area and similarity. In the early pages as I recall, area of a triangle is defined as (1/2) base times height. Then later the basic theorem on similarity of triangles, Euclid's Prop. VI.2, is proved using area, as Euclid does. What is odd about this? Well if you think about it, a triangle has three potential sides to be used as the base. Why does one get the same area from all three choices? I think I had never thought of this in my entire school career as a child. But in fact this proof requires the theory of similarity! So to use similarity to define area and then to use area to treat similarity is "circular" reasoning. I may have missed something in their explanation, but if not, then I wonder what goal did the AOPS authors have in mind for doing this?

To try again to add something to the similarity remarks above, ones explanation always depends on ones goal. Professor Givental states that in his experience the similarity proof was the first one some of his friends had felt they really understood. So he was willing to use the concept of real numbers, which most people today find familiar, to illuminate the Pythagorean theorem by giving it an explanation. He wanted an argument that explained why the theorem is true rather than just a clever demonstration that still left us amazed at what we had seen demonstrated. he wanted his audience to say "oh THAT'S why that theorem is true." I now appreciate better his motivation. However my goal is motivated by the fact that after teaching calculus for 40+ years to average college students, I have found that most in fact do not know anything useful at all about real numbers. Hence I was motivated instead to use the historical approach to Euclidean geometry to explain real numbers. I wanted my students to say "Oh THAT'S what a real number is, namely a way to compare lengths of line segments." Of course I could achieve both goals by carefully explaining lengths and real numbers and similarity, as I did in my geometry class in college, and THEN doing Pythagoras as Prof Givental does, but that takes more time. Besides, the geometry is by definition more elementary than the similarity, since it takes less preparation. So in my opinion the Proposition VI.19 of Euclid cited by Prof. Givental, which establishes the fact that area scales as the square of length for similar figures, is more basic and hence more important than the relatively easy application made of it to reprove Pythagoras. Moreover this principle is not at all clear to my incoming college students. In deed this is why area formulas tend to have squares in them π r^2, and so on, but kids actually do not know this. So he is taking something basic, perhaps intuitive, but difficult, for granted and using it to derive something else easily. So to me his easy argument is easy precisely because he left out the hard part! Nonetheless he may be pedagogically correct in this, since it seems to appeal to many people. I think I have said enough to make it clear I believe there is no absolute best way to do anything, it depends on your point of view. As long as you have a pedagogical goal, you can feel good about trying to achieve it! But it does help if we know why we are doing something the way we do. Or as my brilliant colleague used to say, if you want to achieve your goal, first you have to have one. This may also explain the endless discussions I have heard in coffee rooms (and online forums) over the "right" way to present something. The speakers simply have different goals in mind. If your goal is to keep your child engaged and thinking, then anything that achieves that should be enough, without worrying too much whether the curriculum they are using is approved by some official body or other. Just my view.

If I may, let me recommend Euclid's Elements and signal how its presentation of geometry connects with other important subjects like algebra and arithmetic. In Book I of Eulid there is an introductory treatment of basic geometry of lines and triangles, including area and its basic properties. Then in Book II, there are more propositions strengthening the computational properties of area. This is essentially an introduction to algebra, but from a visual geometric point of view. Quadratic equations are solved there from a geometric point of view, using Pythagoras as a basic tool. In Book III there is a treatment of properties of circles using what is then known about triangles as a tool. There is a statement characterizing tangents to circles in essentially the same way as Newton later did using limits (Prop. III.16?). Then later in Prop. III.35 I believe there is a statement equivalent to the basic principle of similar triangles, stated for triangles formed by secants in a circle. Then in boks V and VI there is an introduction to proportionality for ratios that are either ratios of integers, "rational numbers", or that can be approximated by such, "irrational numbers". Thus the geometry of the Euclidean line now leads to the concept of "real" numbers. In the following chapters integral multiples of segments are used to treat fundamental concepts of number theory by means of the concept of "measuring" one segment by another. The basic number theory technique of greatest common divisors and Euclid's algorithm for computing them is introduced, as well as the famous proof that there are infinitely many prime integers. Finally, Euclid treats areas and volumes by the technique of limits. This what I mean by "integrated" mathematics. I.e. one idea is treated thoroughly then used to motivate and introduce others. The treatments in textbooks used here just jumble it all together with no clear understanding of any of it. I.e. one can properly not understand real numbers until after studying the Euclidean line. And algebra is correctly and historically motivated as a tool for computing areas by decomposition of the edges. (E.g. an (A+B) - square, decomposes into an A - square, a B-square, and two AB rectangles. My epsilon camp notes, #10 on this page: http://www.math.uga.edu/~roy/ give a guide to Euclid's first 4 chapters. But in spite of my views, you are the boss in your house! Do whatever works for you. Best wishes.

As a professional mathematician, I am one of many who are puzzled as to the reasons behind "integrated" curricula. The ones I have seen here in Georgia do a poor job of teaching anything. I even believe it is inherently impossible to do more than one thing well at a time, unless you are just memorizing procedures. So unless you have a need for following an integrated program that is unrelated to instructional value, such as proceeding in parallel to public schools in Canada, I would support the traditional approach as much more effective for mastery. traditional books that work well for many home schoolers here in the US include the books by Harold Jacobs on algebra and geometry. There are also vastly better books for profoundly gifted kids by euler (Elements of algebra) and euclid (the Elements), I did try to think of an answer to your question as asked, but I honestly do not know of any high quality books based on the integrated premise. As far as I know, none of the great math books in history have ever been written that way. Presumably there is a reason for that.

here is another link on the great abebooks used book site for an elementary number theory book by Underwood Dudley, for $1. http://www.abebooks.com/servlet/SearchResults?an=underwood+dudley&sts=t&x=56&y=10 This is the book my son used at TIP summer camp at Duke years ago. Compare that to the AOPS number theory book for $70 new, $50 used. (Or the online? course for $262.) Those specially targeted books may be great, but I tend to feel gifted parents are getting charged an awful lot for material that is available for a lot less. Here is a book by Einstein on relativity, the special and general theory, written for anyone, and also $1. http://www.abebooks.com/servlet/SearchResults?an=einstein&sts=t&tn=relativity&x=57&y=15 look a little ways down the page - (the first item, is a dover reprint of his original highly technical papers, which however I have in fact also owned since I was a young man, but not read.)

thank you. I want to acknowledge after reading the berkeley link containing the remarks about similarity and pythagoras, that their author, Alexander Givental, is a MUCH better mathematician than I am. Hence everything he said is certainly literally correct. i still do not back down from anything I said however. I conclude that he was interested in utilizing the power of similarity to make pythagoras seem more understandable, and was willing to overlook the task of justifying the concept of similarity rigorously. Sometimes ones goal is to unveil the insight behind something, even if it means assuming tacitly something which is intuitive, even if still hard to justify by logic. I myself have also oscillated between using similarity to make area theorems easier and doing the opposite, using simple area concepts to define sophisticated similarity theorems. there is another reason which I have suppressed, for doing area first. Namely there is a hypothesis called archimedes axiom, that says any finite segment can be repeated until it covers any other finite segment. this is needed to use the similarity approach of euclid, but not to use the area approach. Thus the area first approach is more general and works even in non archimedean worlds, unlike the euclidean similarity approach. Professor Givental knows this of course, but his article was apparently serving a different educational purpose. I recall a similar explanation of a problem solution by the great V. Arnol'd where he reduced a certain problem to the "obvious" fact that a continuous curve joining two opposite corners of a rectangle must meet a continuous curve joining the other two opposite corners. This makes the problem certainly believable. the problem is that obvious fact is rather hard to actually prove. It is very helpful to the average person, or anyone, to see something clearly and simply from the right point of view. A few picky individuals still want to see the full proof with all the details. (spoken by a 70 year old larvae.) Ok, sorry, I realize this is mostly math jargon.

Older kids make the same mistake. I assigned homework in an advanced math class and when i handed it back corrected, one kid complained that he couldn't have been wrong because he had copied his answer off the internet! sure enough someone had published a book of answers to the problems in our book, he just hadn't got them all right! maybe your child would benefit from that story.

i don't know the term geogebra but it seems coined to connote what is in euclid book II. there euclid proceeds as if the product of two segments means the rectangle they form sides of. Then the problem of when two rectangles have equal area is the same as when the product of two segments (think of their lengths) are equal. He proves geometrically certain basic algebra results like A(B+C) = AB + AC. This just means visually that a rectangle with height A and base B+C has the same area as two rectangles both of height A, and one of base B and the other of base C. To prove it you just stick the last two rectangles together, or else you subdivide the base of the first rectangle. He also proves stuff like (A+B)^2 = A^2 + 2AB + B^2, by looking at the four regions you naturally see in a square whose sides are both (A+B). I.e. there is an A square and a B square and two AB rectangles. I first saw this demonstration as a senior math major at Harvard from the famous psychologist of learning Jerome Bruner in 1965, and wondered why no one ever showed me before how easy that is. These are propositions II.1 and II.4 in this version of euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookII/bookII.html Unfortunately!! there are no pictures in this version. That pretty much ruins its usefulness as a visual learning tool, i,e, as geogebra, unless you draw them yourself. let me hunt up a better version. (But he does give the equivalent symbolic algebra.) Ohboy, here we go! it has pictures and even titles Book II as "geometric algebra"! are we getting there? http://farside.ph.utexas.edu/euclid/Elements.pdf On p.104 the circle has two secants forming two triangles (not fully drawn) with vertices ABE and DCE, and they have equal vertical angles, and also equal base angles (because those base angles cut off the same arcs of the circle). So the triangles have equal angles and we want to prove their sides are in the same ratios. But this means that AE/BE = DE/CE or equivalently , without defining what that means in terms of numbers, that rectangles (AE)(CE) = (BE)(DE), exactly the statement of Euclid's Prop. III.35. Compare with Prop. VI.2 page 157, where the fact that the bases are parallel is equivalent to saying the base angles are equal.

from what i read on wikipedia, the hilbert transform is much more sophisticated, a tool in fourier series, and i don't know anything about it. Actually the statements you linked above are arguably correct, I just point out that they can use some context. When I taught Euclid the first time I had trouble remembering all the tricky proofs in Book III that used area to derive facts about circles and triangles. When I saw how easy they were using similarity, I recommended my class to change the order of presentation and introduce similarity first, then use the easy derivations of the results like Euclid Prop III.35. We did the theory of similarity based on our prior knowledge of real numbers from calculus. Last summer when I only had two weeks to teach this to kids who did not know calculus and real numbers, I was challenged as to how to get as far as similarity in only 12 lectures. I had been charged with presenting Books !-!V, and I knew similarity was in Book VI, but I learned that it was clearly a key property that was desired in the course. Stubbornly I declined to present it without theoretical justification. Then I noticed that Prop. III.35 is also a statement of the basic triangle similarity theorem but in area form. I.e. the two secants form a pair of similar triangles in a a circle, and instead of saying their sides are in the same ratios A/B = C/D. it says equivalently that the rectangles they form have the same area, i.e. that AD = BC! Here multiplication of two line segments simply means the area of the rectangle they form. Thus I was able to derive the basic principle of similarity, Prop. VI.2, in area formulation, from a much earlier one, Prop. III.35, using the Props in Book II to justify the algebraic properties of area. I was really proud of this, although I learned afterwards that of this had been noticed many decades before. The bonus was that we were able to simplify all the propositions of Books III-IV following this one, since we now had similarity of triangles available. (It is possible to see that every pair of similar triangles can be embedded in a circle as the ones are in Prop. III.35.) This simplified the proof of Euclid's, Prop. IV.10, for constructing a pentagon, by rendering the brilliant and tricky Prop. III.37 unnecessary. This is all in my free epsilon camp notes on my webpage at UGA math dept, near the bottom of the page, #10. http://www.math.uga.edu/~roy/ I think I have already done so, but I specifically recommend these great books which are free: Euclid's Elements; http://aleph0.clarku.edu/~djoyce/java/elements/elements.html (there is also a lovely paperbound edition from Green Lion for reasonable purchase.) and Euler's Elements of Algebra. http://www26.us.archive.org/details/elementsalgebra00lagrgoog

you are so welcome. I may not hold out, or be here all the time, but in principle I am open to any kind of questions, especially math ones, that kids encounter, or parents, in navigating books, tests, etc... I am a professional algebraic geometer, so I know some geometry and algebra as well as other topics taught in college. Still I may not know as much as some people about elementary geometry or other elementary topics, as I have noticed specialists really know a lot! I think it could be feasible for us to start up a reading thread where interested kids read a great book like Euclid, or Euler, or whatever.... Actually kids seem to prefer solving problems to reading, so we could have a problem thread. ????

I like the beautiful similarity derivation of a very general Pythagorean theorem. I want to point out however a certain circularity in reasoning involved here. I.e. in Book VI Prop. 2 of Euclid, the fundamental similarity result is proved using the theory of area (prop. I.39), the same theory of area that immediately implies the original version of Pythagoras (prop. I.47). One can also derive the theory of area from the principle of similarity, as is done in the AOPS geometry book. Thus the theories of area and of similarity are essentially equivalent, but you have to start somewhere. If you do as Euclid did and start with area, then you can derive the original version of Pythagoras and then similarity, and then re-derive a stronger form of Pythagoras. But it is somewhat circular to claim that Pythagoras follows just from similarity if similarity is developed using area, as in AOPS. Thus the AOPS derivation is not logically correct. Hilbert did show how to derive similarity without area, and thus gave a coherent and non circular development of area and similarity. This theory is very well described in Hartshorne's beautiful book: Geometry, Euclid and beyond. I apologize for the pickiness of this comment, but this is the kind of slip up that a professional mathematician notices and feels he must point out. Indeed gifted kids also appreciate this kind of thing, even if their teachers do not. So yes, Pythagoras can be derived easily from a version of the similarity principle, but to be honest one should admit that similarity is not so easy to develop. One must either actually assume a theory of area that would already be sufficient to prove Pythagoras to even start the theory of similarity, or else work much harder to derive similarity without area. So in my opinion the quote above is misleading. Just one man's opinion. In fact I have made the same comment on a site where professional mathematicians have made the same claim about pythagoras and similarity. The point is that Euclid's proof of Pythagoras did not even assume a concept of number, much less a principle of scaling. Moreover it works in a much more general world where in finitely small lengths exist, unlike the similarity approach. http://mathoverflow.net/questions/40337/ingenuity-in-mathematics about 10-12 answers down the page. For what it's worth, a competition approach to math wants to use every possible trick available to solve the problem, but a mathematician's approach wants to understand the logic behind every assumption. By the way I also disagree in a sense with the comment that only Euclidean geometry admits non trivial scalings ("conformal isometries"). I.e. the difference is that in spherical geometry for example there are many spheres, one for each radius. thus when you try to scale a spherical triangle, it is possible, but the scaled triangle lives on a sphere of different radius. I.e. you have to scale the whole sphere! In Euclidean geometry the scaled figure lives on the same plane. This is a property of the curvature, which in the Euclidean case is zero.

got it thanks. you are welcome to this. hope it helps. Here is another free website for questions from "people studying math at any level." http://math.stackexchange.com/faq#questions