@creekland: "I appreciate schools that have different calibers of math classes to allow students to choose how deeply they want to delve. My pre-med or med researcher wannabe needs Calc, but not as a be all, end all. A super tough class could kill his med school dreams if he ended up not being able to handle it or spent too much time on it and hurt his other classes or extra curricular stuff (ALL important for med school admission). I'd never recommend a pre-med teacher take a tough math course unless they really, really, wanted to. The world could lose out on some great future physicians." Actually it is my conjecture that a math course in which creative thinking is encouraged would be better preparation for a medical student who intends to make diagnoses from limited information, than one which is all computation. Again I am not talking about impressing an admissions committee looking for all A's, but impressing a medical professor (think "House") who is looking for depth of understanding and reasoning ability. Hard math courses also help teach students how to learn. (My wife is a math major and a physician - I think her favorite math course was abstract algebra, but she also took logic and proof, complex variables, and advanced calculus. After finding out public school teaching was not her thing, she went to night school (with two children), took the pre med courses and nailed the MCAT's. Her scores were quite high and she was admitted to the only med school she applied to, Emory.)

If anyone wants to teach proof to their child, I strongly recommend the Green Lion edition of Euclid's Elements for under $20. One will also benefit from using it together with Hartshorne's companion book, Geometry: Euclid and beyond (about $50), or the more modest, but free, notes on my webpage at UGA math dept, from epsilon camp.

Here for comparison, are some questions from my second semester honors course. Unlike those on an AP test these are meant to be easy for someone who understands the concepts. There are however concepts that are not usually found in high school courses. E.g. it is common to define integrals and then to state but not prove, that continuous functions have integrals. Thus continuous functions are a subclass of integrable functions. Then the fundamental theorem of calculus is stated for continuous functions. A curious person (e.g. potential mathematician) might ask what the fundamental theorem would say for the more general class of integrable functions. Here the concept of "Lipschitz continuity" occurs naturally. Another type of functions for which integrability is easier to prove, and which suffices for almost all practical cases, is the class of monotone functions. Thus one can easily treat this case in more detail. 2310H Test 2 Fall 2004, Smith NAME: no calculators, good luck! 1. (a) Give the definition of "Lipschitz continuity" for a function f on an interval I. (b) State a criterion for recognizing Lipschitz continuity in the case of a differentiable function f on an interval I. © Determine which of the following functions is or is not Lipschitz continuous, and explain briefly why in each case. (i) The function is f(x) = x^1/3, on the interval (0, infinity ). (ii) The function is G(x) = [t] , on the interval [0,10], (where [t] = "the greatest integer not greater than t", i.e. [t] = 0 for 0≤t<1, [t] = 1 for 1 ≤ t < 2, [t] = 2 for 2 ≤ t < 3, etc....[t] = 9 for 9 ≤ t < 10, [10] = 10.) (iii) The function is h(x) = x + cos(x) on the interval (- infinity, infinity ). 2. (i) State the "fundamental theorem of calculus", i.e. state the key properties of the indefinite integral function G(x) associated to an integrable function f on a closed bounded interval [a,b]. You may assume f is continuous everywhere on [a,b] if you wish. (ii) Explain carefully why the definite integral of a continuous function f on [a,b], equals H(b)-H(a), whenever H is any "antiderivative" of f, i.e. whenever H'(x) = f(x) for all x in [a,b]. Justify the use of any theorems to which you appeal by verifying their hypotheses. (iii) Is there a differentiable function G(x) with G'(x) = cos(x^2)? If so, give one, if not say why not. 3. Let S be the solid obtained by revolving the graph of y = e^x around the x axis between x=0 and x=3. Define the moving volume function V(x) = that part of the volume of S lying between 0 and x. (draw a picture.) (i) What is dV/dx = ? (ii) Write an integral for the volume of S, and compute that volume. 4. Consider a pyramid of height H, with base a square of side B. Define a moving volume function V(x) = that part of the volume of the pyramid lying between the top of the pyramid, and a plane which is parallel to the base and at a distance x from the top. (i) Find the derivative dV/dx. [Hint: By similarity, b/B = x/H.] (ii) Find the volume V(H). (iii) Make a conjecture about the volume of a pyramid of height H with base of any planar shape whatsoever, and base area B. EXTRA: Either: Prove the FTC stated part 2(i); you may draw pictures and assume your f is monotone and continuous if you like. Or: ask and answer your own question.

I am really enjoying the comments here! To add some data to the philosophy I have espoused, I looked up a BC calculus test online and took part of it tonight. I assumed I would get 100% but did not. To make it "fair" I did it in my head without pencil or paper while watching a couple of Matt Damon/Jason Bourne movies and having dinner with wine, but I missed two of the first 14 computational problems. One was a minus sign and one was a constant I overlooked, but I felt somewhat downcast and defensive. I also cheated and used a pencil to draw the graph of r = 1 + 2cos(t) for 0 ≤ t ≤ 2pi. I did this also partly so those who wish to assess my comments about these tests may have more in formation as to what the comments mean. Indeed if anyone gets all these problems right he/she is a strong calculator who knows a lot of basic formulas. Few of my typical students could have done these at all accurately. I.e. a good performance on this test is something to be proud of. I just like and recommend other types of questions more. These questions are all straightforward, if you know the method, but the computations were still tedious. I usually ask questions that are fairly easy for someone who knows the method. As a mathematician I also found the test uninteresting. To work hard on a problem, it helps if there is some challenge to it or some interest. Thus to me the most interesting ones were the ones that required a calculator, since I did not have one. (By the way, calculators are never allowed in my courses and not in most other college math courses.) E.g. to use calculus to maximize the area of a rectangle based on the x axis, and with upper vertices on the graph of y = cos(x), with -pi/2 ≤ x ≤ pi/2, one needs to solve the equation x.sin(x) = cos(x), not so easy in ones head. But I just tried x = pi/4 and x = pi/3, and found that the maximum must be greater than 1.1 and apparently lies somewhere in between those two points. This ruled out all multiple choice answers except roughly .8 and .9. As to which one, I guessed wrong, but then I looked more closely and by taking the value halfway between those two found the right answer to be indeed more likely. Another easier question not needing a calculator that I also got wrong was to recognize which of 5 or 6 possible sums could be a Riemann sum for the volume of a solid lying over the rectangle between x=0 and x=2 and y=0 and y=1, and with height over (x,y) equals to 1+3x. The first thing that occurs is that only a very compulsive person would use calculus to compute this volume since the shape is so simple it is easily computed as half a certain rectangular parallelepiped. Secondly the question is annoying because there are at least three directions in which to "slice" the volume up into areas, and so one has to try all of them to see which one has been chosen in the problem. What is being tested here, patience? Finally I got it wrong because the answer was written deceptively with a constant factor pulled out that I did not multiply back quite correctly, so i got it wrong for a trivial arithmetic reason. What is being tested here, arithmetic? This is a question that has no use in real life. In real life one might want to write a Riemann sum for such an integral, but not to recognize someone else's. I missed a differential equation question by being lazy and trying all 5 answers but made a mistake with a constant as mentioned above. After seeing my error, I easily solved the equation more easily directly in my head. I eventually lost interest and stopped. There were no definitions to give, no theorems to state, and no proofs. My tests usually involve all 4 aspects, definitions of concepts, statements of theorems (sometimes via true false questions, which is also done on the AP), computations, and proofs. Some picky remarks about the AP: inconsistent use of the terminology for "function", confusing a function (which has a precise domain) with the formula defining it, also deviating from the notation f and f(x) used in the instructions, by using "variable" expressions such as x = t^2 - t +6, instead of x(t) or f(t). Use of terminology not entirely standard in instructions, such as "use Euler's method to approximate a solution to this differential equation" instead of "use differential approximation" or just "approximate and explain your method". Hence to do as well as possible on such a test one should practice beforehand, not just on the math, but on old examples of AP tests, to learn the language and style of questions.

Someone asked about K-12 preparation in my state. I will try to keep this briefer, but I was once charged with evaluating the testing materials that were used to prepare the high school math teachers in my state for certification, so I know something of this topic as well. They were so flawed, (wrong sample answers, wrong explanations, huge gaps in test coverage plus enormously unrealistic syllabus, and so on....), that I eventually conjectured that these materials and tests had been prepared not by experts but by (then) currently certified teachers, in sort of a grandfathering system - i.e. if you are certified, then not only are you qualified but you also determine future certifications. This was indeed confirmed. Everyone in the world knows a more recent news story about K-12 preparation in our state.

These are wise comments. Some bright high school students to whom I have told my concerns have said that their AP courses really were an improvement in many cases over the ones that were in place before. I myself once taught a vector calculus class called "beyond calculus" to students in a private high school who already had AP math, from a book by Marsden and Tromba that was used at Berkeley, and some of my students protested that they did not expect or need such a tough class. One of them who dropped out, later did "just fine" in a presumably non honors calculus class at a state school and claimed that justified his objections to my hair - shirt standards. Another student however who went to Harvard and took second year non honors calculus there from a world famous mathematician, said it was so advanced he "could not have survived" without our prior introduction to such topics as integration of differential forms. Of the 5 finishing students, one graduated Yale and then got a PhD in math, and another graduated Harvard and then got a PhD in physics. Another high school student who only took Spivak style precalculus from me for one summer month (rigorous development of real numbers from Spivak's appendix) went to U of Chicago and survived the honors intro calc class. One has to be very clear about what ones goals are, whether ones goals are to pass a certain test or a certain course, or whether one wants to understand a subject deeply. E.g. if one wants to be a pre med student and does not need a deep grasp of calculus, but just needs a credit in it, or a good grade, then an AP exemption may be just the ticket. (But hopefully she/he learns biology/chemistry as deeply as possible.) The same holds for people using math courses only to satisfy a formal requirement for further study, even in scientific fields such as chemistry and engineering. But if one wants to be in a stimulating, challenging math course with the best other students, and possibly become a mathematician or theoretical physicist, one can probably not afford to allow an AP score to be the only measure of ones math understanding. AP courses are not inherently bad, but using them for more than they deserve is hazardous to ones education. They may have many uses, but in my own experience the one thing they are not good at is the one thing they claim to do, namely justify advanced college placement. If they were called "standardized short answer tests of minimal computational proficiency", then they might be less misleading. Even this however is changing, as I have said. I.e. AP tests are so ubiquitous that they have indeed become the standard in many places. Thus in some schools and some courses, a certain AP score is indeed the de facto correct criterion for advanced placement, because the quality of the college course has declined to allow this to be true. Even in the same department there may be professors who teach at the level of AP preparation and some who vastly exceed it. Uniformity is hard to enforce. So do some homework. But if a student goes to Stanford and enrolls in the second year honors course from Apostol, or Harvard's math 55, with preparation equivalent to only a 5 on the AP test, after learning from a book like Stewart, he/she better be ready to be shocked by what will be expected. One should always ask the person teaching the course what that is. E.g. when I called the Stanford professor teaching the Apostol class he freely told me that the formal AP prerequisite was not the real prerequisite. Unfortunately I did not call him 2 years in advance. One recent success I know in math 55 prepared by taking essentially a complete math major including graduate courses at UGA before enrolling in college. So there is no simple answer. The point is to take advantage of whatever the AP courses and tests have to offer, but not to assume they are always what they claim to be, namely proof of suitability of advanced placement. Always consult with the professor. A basic rule may be that AP scores are useful for impressing admissions, but professors want real knowledge and understanding. Caveat: I am getting much too preachy. Fortunately you guys are wise enough to separate wheat from chaff! In my defense I have been frustrated for decades because AP courses have made much harder my life's goal of trying to do a good job of high quality college teaching .

@ Creekland: I think one placement problem is that AP tests can cause an honors high school student to place into the second semester of a non honors college course, which is the wrong population for a bright kid. They belong in the honors course, which they would have been in had they not gotten "advanced" placement. At UGA we have three or 4 levels of intro calculus, terminal, 1st semester continuing quality, honors level, and future mathematician level. This is unusual today, since AP courses have reduced the audience for many of these courses, or at least reduced the audience that realizes they should be in one of these. Most entering AP students do not take any of these but enter in, and struggle in, some second semester course.

This is obviously a very sophisticated group, much more than my average entering student population. For what it's worth, I reproduce a letter (edited) that I wrote in frustration to my sons' private school in 1997. My main interest here is to suggest being very careful about using AP tests either to measure competency or to determine college placement. A question about the value of AP courses At a parent- teacher meeting some years ago I argued that our school was not particularly difficult, observing that it had only one AP course (then calculus). The headmaster patiently observed that there were a number of courses which, although not technically AP courses, were quite difficult and advanced. In spite of the simple truth of this argument I did not give in. Over the years I have gotten my wish as AP courses proliferated. When our older son went off to college an odd thing happened: he had difficulty in his advanced mathematics course, into which he placed by virtue of his AP preparation, whereas he "dominated" in his English course, for which he had prepared by taking traditional advanced courses in English literature. Suddenly, I regretted not listening to the headmaster’s argument more closely. I now feel that this whole AP revolution is regrettable, in fact that it is actually hurting the cause not only of good education but of good college preparation. I believe there are two reasons for this: first, AP courses are designed to prepare people to answer multiple choice questions on chosen topics, while the traditional courses, especially the honors seminars, are designed simply to teach people to read closely, analyze deeply, and to discuss and write effectively. These latter skills are much more useful in college and elsewhere, than is familiarity with a particular AP syllabus. Second, students do not realize that the name AP is often a complete misnomer, and that AP courses are not at all equivalent to college courses. Consequently a student coming out of a traditional honors high school course is likely to take a beginning college course in the same subject (possibly an honors section) for which he is well prepared, while the AP student often tries to skip the introductory college course in his subject and enter an intermediate course, for which, in my experience, he is seldom even adequately prepared. There may be some miscommunication between admissions officials and professors, but the professors I know actually prefer to teach beginning calculus to people who are well versed in algebra and geometry, but who have not had calculus. To some extent colleges are accommodating the situation by gradually making college courses easier, in response to the weaker preparation students have today, but this is hard to do perpetually. Our difficulty is that students today have a shallow grasp of more and more advanced subjects, when we would prefer them to have a deeper grasp of basic subjects. In my opinion AP courses are a primary cause of this problem, and I hope something can be done to retard their advance at the expense of outstanding and unique honors courses at this school, before it is too late. [i received the following response from a friend who is the author of a famous calculus text.] It may be heretical, but it is 100% accurate. You have expressed the failure of the AP program and the success of your son’s and similar schools as well as it could be expressed. My simplistic version---students who take AP calculus merely learn a superficial calculus course without deep understanding, and thereby waste a year forgetting algebra---is not as well put as your analysis. Congratulations and send it to every editorial page in the country! That last sentence encourages me to post it here.

I am very impressed that you used Spivak at home! My impression is that this level of preparation is unusual also in home schoolers, but I may be wrong. If home school teachers do not have the information to design a course at the level of yours, I am afraid they will assume that the AP is the accepted standard of excellence, as traditional schools often do, rather than a degradation of the previous level of calculus instruction. Do you think it is well known in home schooling circles that typical AP courses do not in fact match up with strong college courses? It has been a painful experience for me for decades to watch students with only an AP high school preparation, struggle even in second semester non honors calculus. The failure or withdrawal rate was perhaps 50%, and that was with generous evaluation. The savings in tuition by skipping first semester calculus became somewhat illusory when ultimately I had to downgrade the content of the course for them, and half of them still failed and were forced to repeat the course. It does seem that home schooling offers the potential to avoid this, and I hope this information helps some to do so. In one regard at least, I would expect home schooled students to excel at a key skill for college success which is independent of curriculum, namely I would expect that home schooled students have learned how to learn. This one thing can atone for a great many omissions.

I don't know how the reading level matches up, but its not too challenging. http://www.amazon.com/Blue-Tiger-Harry-R-Caldwell/dp/1930585381/ref=sr_1_1?s=books&ie=UTF8&qid=1341716527&sr=1-1&keywords=blue+tiger This is the true story of a childhood friend of mine, a missionary and famous big game hunter in china, Harry Caldwell.

There is a book by Michael Serra called Discovering Geometry that I did not much care for, too fuzzy. But here is a book called Experiencing Geometry by David Henderson from Cornell that I like a lot better. It is a high school level geometry book. The third edition has more in it than the second, which has more than the first, but the first has enough, so I would suggest getting the cheapest one, which is the third edition in this link. The discovery aspect means it is presented as problems to solve. So it is a significant amount of work for the learner, who will benefit accordingly. http://www.abebooks.com/servlet/SearchResults?sts=t&tn=experiencing+geometry Here is a link to a journal from Russia where unlike the USA, where specialists choose research or teaching, many mathematicians of the highest research level also contribute to the study of pedagogy. http://www.komunikacija.org.rs/komunikacija/casopisi/teaching/group_search_ctype?ct_id=ct01&from=I_1&stdlang=lt

Thank you! Thank you!

AP credit is a mixed bag. In over 30 years, I have almost never seen a high school student who should have been advised to skip college calculus based on his/her high school class, because those classes, and AP tests, are vastly inferior to what I consider a decent college class. But this raises a financial problem for the parent. If the student goes back and takes calculus over in college to learn it well, the course costs the parents money. Moreover it causes a recruiting problem for the school. If we refuse credit because we know they don't really know calculus, we lose the student to a college that will give credit. By and large colleges that refuse credit at least in math, are doing so in my opinion not to gouge tuition, but because experience shows the inferior quality of these high school AP courses and tests. However this is a lost war, AP courses are too entrenched to eradicate, and only the most elite schools can afford to tell the truth about their less than collegiate quality. Since AP prepared students are usually not really well prepared, colleges also have a placement problem. The result at public schools I know of has been a lowering of the quality of college offerings. Thus we give credit for AP courses, but to prevent the AP students from being swamped by the next class, we have lowered the quality of our college courses down to what AP high school classes are. One compromise we maintain at UGA is our "super honors" calculus course, the one for future mathematicians. If a student has a 5, or at least a 4 on the AP calc test, and permission from the instructor, and wishes to take our "Spivak" style (very high quality) introductory class, we will still give AP credit. Another more serious problem occurs at elite schools like Stanford and Harvard. They have so many AP students that they no longer offer even a Spivak style introductory class. Thus top students with AP credit who want the best honors course are thrown into the second year super honors class, from Apostol volume 2, for which most, no matter how strong, are not prepared. (Even the BC calc test has few proofs, whereas the Stanford honors class test is ALL proofs. As the professor at Stanford told me, "the technical prerequisite is a 5 on the AP, but that's not the real prerequisite. The real prerequisite is to be able to handle proofs, no apology.") These elite courses, like math 55 at Harvard, are now populated mostly by students lucky enough to have taken a Spivak style class while in high school, either from a college like UGA, or at an elite private school like maybe Andover. One of the few top colleges that still offers Spivak intro calc is University of Chicago, long famous for good high level instruction. In my experience, Harvard and Stanford are more in the "sink or swim" category in honors math.

I think the problem is the misplaced emphasis on grades rather than knowledge. As a college professor teaching an honors calculus course I gave test 1, and several people got A's of various sorts. I tend to hope someone with less than 100% will interpret it as "room to improve". When I handed them back, a young woman with a 92 or 93 (an A-) on test 1, one of the top grades, remarked "I don't like the direction this is going", and immediately dropped out of honors math. I thought my job was to teach her as much as possible, and she seemed to think her job was to get as high a grade as possible, while not expanding her horizons any at all. (I seem to have misinterpreted the title of the thread.) On the topic of that article, to me that's just drug abuse. I never needed any drugs to stay awake and work long hours, and I have worked regular 14 hour days, and up to 30 hours in a row on some weekends, before reaching the age of 60 or so. Young people can easily work long hours if they avoid alcohol. I love that quote from Maimonides, Guide for the perplexed?

This came up at epsilon camp when i had to order compasses for the 28 kids ages 8-10. I tried a lot of different ones and bought several including an expensive thumb wheel one (staedtler) that locks firmly in place. The only drawback I found to the "good" one is that although it does stay where you want it, it does not change easily to another setting. You have to slowly screw the thumbwheel out a microscopic amount at a time. that takes forever to go from a small setting to a large one. i found it so tedious to use for more than one radius setting that I stopped using it and never used it again. Maybe Kareni has some tips to share on this. The plastic ones were useless to me because they slipped while I was trying to draw a circle, although my 10 year old granddaughter could manage it oddly enough. i eventually found lots of simple metal ones for one or two dollars. I bought about 40 of the one dollar ones and that worked fine. They were metal but so cheap they did break occasionally, but I had extras. But just one won't suffice, so I would suggest maybe the two dollar ones as more substantial. But I myself would only use the super precise twelve dollar ones or more, if someone is very patient. Oh and the precise one also has a rather sharp point, I stuck myself painfully. I would use the most basic metal one, like I had in high school, but sturdy. Here is a picture of it, but this ridiculous price is about 6 or 8 times too high. http://www.schoodoodle.com/home/sch/page_1679/metal_ball_bearing_compass_12_pack_with_safety_poi.html?rid=thefind&utm_source=thefind&utm_medium=feed&utm_campaign=thefind Maybe Kathy in Virginia remembers where we got those. I spent a long time searching for a decent price. As a disclaimer, I did not achieve terrific accuracy with my basic compasses every time. It was somewhat hit or miss. So maybe those here who advocate staedtler are right, if you want really good results. Here are some of those. Mine looked like the 6'' student one. I got it at an art supply store for about $12. http://www.google.com/#q=staedtler+compass&hl=en&tbm=shop&prmd=imvns&ei=KJr4T6qSJ4ec8gTVgd2VCQ&start=10&sa=N&bav=on.2,or.r_gc.r_pw.r_qf.,cf.osb&fp=ecd0620c0a53f36f&biw=1268&bih=653 which one is being recommended? My staedtler says "Mars 551, and Comfort".

By the way, if anyone has a child who looks at the law of cosines |C|^2 = |A|^2 +|B|^2 – 2|A||B|cos©, and remarks that the right hand side looks kind of like what you get when you expand (A-B)^2, or asks whether C = A-B, then you have a potential mathematician - certainly a child who is thinking like one. Mathematics is about looking for patterns, and analogies. (By the way, I believe the college board recently removed analogies from the verbal SAT, which renders it much less useful as a gauge of reasoning ability. We noticed long ago at UGA that the verbal SAT measured math aptitude, or at least success in our entry level math courses, better than the quantitative part. In fact there were two measures that tracked entry level math success, the verbal SAT and the social security number, which latter seems to reflect what state the child went to school in. The quantitative SAT was unrelated to entry level college math course success. Hence efforts to raise that number are misguided, unless of course it is related to college entrance. Oddly there is apparently a big difference between what admissions officials look for and what professors look for. I may have damaged my child's entrance chances by recommending he skip taking advanced college level courses while in high school and master basic high school level courses thoroughly instead.)

I am a math guy, and I have often had the feeling that home schoolers are being exploited by publishers who target them with books that are actually not ones we mathematicians have ever heard of or would ever use. These books seem universally used in the home school community but not among either the math community or the closely related math ed community. When a set of books are targeted in that way, they naturally are priced accordingly to a captive audience. Of course the higher priced books are the ones that have been found to be popular by their audience. So it may be that these books are easier to use, easier to learn from and so on. But I myself would not home school from any of them. In the area of history or social studies I would use traditional sources of history, or maybe 19th century books written for children, I am not experienced here so much. Of course even the math books I like that used to be cheap have gone up. The Harold Jacobs books on elementary algebra and geometry that were under $20 new when I bought them are now over $60 used, but it may be possible to find other copies or to share! Why doesn't this community set up a rotating system of sharing books that have been finished with by the youngest child in the family. That certainly beats used books buy back prices.. It is a lot of work but if we just write in and order whole sets of canned presentations designed for us we are essentially, pardon the expression, letting them treat us as "suckers". I realize my naivete is painfully obvious to those of you teaching multiple subjects to multiple students, as to the need for prepared curricula. here is a used book site where i typed in rainbow resource. is that a christian organization? these books are under $5. http://www.abebooks.com/servlet/SearchResults?bsi=0&kn=rainbow+resource&x=0&y=0&prevpage=3

The thread killer strikes again! To summarize that mess, define a product of two arrows A,B in the plane, beginning at the origin, as: A.B = |A||B|cos©, where | | denotes length, and c is the angle between the two arrows. Also define an addition of arrows, by placing the tail of one arrow at the head of the other and taking the sum to go from the tail of the first to the head of the second arrow. Then the third side of the triangle with arrows A and B as two sides, is parallel to the arrow A-B, and of same length. Then the law of cosines becomes the usual rule for expanding a square: (A-B).(A-B) = A.A - 2A.B + B.B and since the angle between any vector and itself is zero, we get: |A-B||A-B|cos(0) = |A||A|cos(0) - 2 |A||B| cos© + |B||B|cos(0). Then since cos(0)m = 1, we get |A-B|^2 = |A|^2 - 2|A||B| cos© + |B|^2, which is exactly the usual law of cosines. Caveat: I have not proved here that this multiplication is distributive, For that I probably need the law of cosines! (sometime later)... Well, duh, I guess that's the whole point: the law of cosines is equivalent to saying this multiplication is distributive, i.e. is a multiplication! So this does not reprove the law of cosines, it just restates it in a more natural form. Then the three term principle says that since the explicit multiplication rule A.B = a1b1 + a2b2, is also distributive, and agrees with the first one on vectors that are equal, it must agree as well on all vectors. My point here is to try tio show how elementary math is illuminated when viewed from a higher point of view. No advanced math is being done here, but we are seeing elementary math more clearly I hope, by exposing its structure. This is what is not obtained from plug and chug treatments. If anyone has a book on trig, Saxon or otherwise, it might be interesting to compare its treatment with the one given here.

Coordinate geometry and trig There is a nice way to do geometry in the (x,y) plane, using coordinates that allows the algebraic tools of addition and multiplication to enhance the geometry. For each point (a1,a2) in the (x,y) plane, we imagine an arrow going from the origin (0,0) to the point (a1,a2). We call this arrow A. Let B be another arrow fro the origin (0,0) to the point (b1,b2). If we add the coordinates getting another point (a1+b1, a2+b2), we can ask how the arrow A+B from the origin (0,0) to (a1+b1, a2+b2) is related to the first two arrows. It turns out that the arrow A+B forms the diagonal of a parallelogram with sides A and B. The vertices of this parallelogram are the points (0,0), (a1,a2), (a1+b1, a2+b2), and (b1,b2). E.g. if A = (1,0), and B = (0,1), then A+B = (1,1) is the diagonal of the square with vertices (0,0), (1,0), (1,1), and (0,1). Consider the triangle with vertices (0,0), (a1,a2), and (a1+b1, a2+b2). The arrows A and A+B form two sides of this triangle with common vertex (0,0). The third side, which goes from the point (a1,a2) to the point (a1+b1, a2+b2), is parallel to the arrow B, and has the same length. Thus if the arrows A and B are perpendicular, the arrow A+B is the hypotenuse of a right triangle whose sides have the same lengths as the arrows A and B. If we denote length of an arrow by | |, then by Pythagoras we get |A+B|^2 = |A|^2 + |B|^2. By the same reasoning, if we consider the triangle with two sides A and B, its third side is parallel to the arrow A-B, and has the same length. Hence by Pythagoras, if A and B are perpendicular, then |A-B|^2 = |A|^2 + |B|^2. We will define a multiplication of arrows that captures this theorem and also the more general law of cosines. To multiply two arrows A = (a1,a2) and B = (b1,b2), we define their “dot product”as A.B = a1b1 + a2b2, which is a number, rather than an arrow. It is easy to check that this multiplication has some of the properties of usual multiplication, like commutativity and distributivity for addition, and so on, but the product of two non - zero arrows can be zero. E.g. (1,0).(0,1) = 1.0 + 0.1 = 0+0 = 0. Moreover, the product of an arrow with itself is exactly the square of its length, i.e. A.A = (a1)^2 + (a2)^2 = |A|^2, by Pythagoras. In fact the dot product of two arrows is zero exactly when the arrows are perpendicular. I.e. consider A and B as two sides of a triangle. Then the third side is parallel to the arrow A-B, and has the same length. Hence |A-B|^2 = (A-B).(A-B) = A.A – 2A.B + B.B =|A|^2 +|B|^2 – 2A.B. But if A and B are perpendicular, then by Pythagoras we must have |A-B|^2 = |A|^2 + |B|^2, so A.B must be zero. If A and B are sides of any triangle with third side parallel to and of same length as A-B, then again |A-B|^2 = A|^2 +|B|^2 – 2A.B. This looks exactly like the law of cosines except that we have 2A.B in place of 2|A||B|cos© where c is the angle between A and B. Thus in fact A.B must equal |A||B|cos©. If on the other hand we knew that A.B = |A||B|cos©, then we get the law of cosines by expanding the dot product |A-B|^2 = (A-B).(A-B) = A.A – 2A.B + B.B = |A|^2 +|B|^2 – 2A.B =|A|^2 +|B|^2 – 2|A||B|cos©. This allows us to remember the easier formula A.B = |A||B|cos©, and then to recover the more complicated law of cosines. It also gives a way to calculate cosines without a calculator. E.g. the angle between the arrows A and B has cosine equal to (A.B)/|A||B|. E.g. the angle between A = (1,0) and B = (1,sqrt(3)) has cosine equal to 1/2. Remember what angle that is?

Thank you Kathy! Before i look at those I will explain my take on the law of sines, which seems rather easy. (mathematicians have a habit of calling "easy" anything they can even come close to understanding, probably to put subtle pressure on anyone who may think it is harder. This is a sort of childish one upsmanship. So to a mathematician there are two categories, easy, and I don't get it. Technically, in this case, easy means the explanation is short and uses no tricky business.) Take a triangle with base AB and vertex C, and side a opposite angle A, side b opposite angle B, and so on. Assume angles A and B are both acute to make it easy. Now drop a perpendicular from vertex C to side AB hitting it at point Q between A and B. Let the length of that perpendicular CQ be x. Now by the very definition of sin and cos, we have that sin(A) = x/b, and sin(B) = x/a. Hence x = bsin(A) = asin(B), so sin(A)/a = sin(B)/b. voila! I don't use this much. It is also true that sin(A)/a = sin©/c.

Ok here's the law of cosines, from euclid. Prop. II.12. Hmmm... the data limit here is exceeded even by a one page pdf file. Here is the word file but without the pictures. the original pdf file is week 2 day three in this link (epsilon camp notes): http://www.math.uga.edu/~roy/camp2011/10.pdf teaser: there is a point of view in which the law of cosines becomes just the familiar rule (a-b)^2 = a^2 - 2ab + b^2. (hooo hahahhaaaa). Law of Cosines Pythagoras says the square on the side opposite a right angle “equals” the two squares on the sides containing the angle. If the angle is acute, the square on the side opposite it is smaller than the two squares on the sides containing it, and if obtuse the square it is greater. The law of cosines tells exactly how much less or how much greater; in particular it says the discrepancy is twice the area of a certain rectangle. These are propositions 12-13, Book II of Euclid. Prop. II.12 (Law of cosines, obtuse case): Let ABC be a triangle on the base BC, with obtuse angle at C, and vertex at A. Drop a perpendicular from A to the line extending base BC, meeting it at X, outside segment BC. Then (AB)^2 = (AC)^2 + (BC)^2 + 2 (BC)(CX). Proof: By Pythagoras applied to right triangle AXB, we have (AB)^2 = (AX)^2 + (BX)^2. From IV.4, this equals (AX)^2 + (CX)^2 + (BC)^2 + 2 (BC)(CX). By Pythagoras applied to triangle AXC, this equals (AC)^2 + (BC)^2 + 2 (BC)(CX). QED. Exercise: Prove: Prop. II.13: (Law of cosines, acute case): Let triangle ABC on base BC have an acute angle at C, and vertex A. Drop a perpendicular from A to base BC, and assume it meets the base at X, between B and C. Then (AB)^2 = (AC)^2 + (BC)^2 - 2 (BC)(CX). Remarks: What does this theorem have to do with cosines? If you recall the definition of the cosine of angle <C in the picture for the acute case above, cos(<C) = |XC|/|AC|, the ratio of the numerical lengths of the two sides. Hence cos(<C).(AC) = XC, an equality of segments. Substituting this into Euclid’s formula above gives us (AB)^2 = (AC)^2 + (BC)^2 - 2 (AC)(BC).cos(<C), and this is the usual law of cosines in trigonometry. It also works for the obtuse case, since the cosine of an obtuse angle is negative, so the minus signs cancel and give us the formula in II.12 above.

Ah yes, Jann in TX reminds us there is also the law of cosines, which actually occurs in Book II of Euclid as Prop. II.12 and Prop.II.13 I think, (and law of sines). So one already knows basic trig if one has had a classical course in Euclidean geometry, I will discuss this after dinner. The law of cosines is the pythagorean theorem for non right triangles, and follows from the right triangle case.

Wow! You guys are a great class! The cute trick is the fact that if you use complex numbers, then sine and cosine are a special case of the exponential function! And the laws of exponents are simpler than those for sine and cosine. Lets just treat this as a magic trick that works. If you have an angle of t radians, i.e. that an angle with vertex at the center of a unit radius circle, that cuts off an arc of length t on the circle, then cos(t) + i.sin(t) = e^(it), where e is a certain beautiful positive real number between 2 and 3, and i is a new "complex" number with i^2 = -1. Think of the numbers on the x axis as the usual "real" numbers, whose squares are all positive, and then i is a number on the y axis, at unit distance from the origin. Real numbers make angle zero (with the x axis) and i makes angle pi/2 (I started to say 90 degrees, as old habits die hard). Every point in the x,y plane represents a complex number, and to multiply them you multiply their lengths and add their angles. So i^2 has length 1^2 = 1, and angle pi/2 + pi/2 = pi = 180 degrees. So i^2 is at -1 on the x axis! Just as you can take powers and exponents of real numbers you can also take complex exponents of real numbers like e, and then for any real number t, the complex number e^(it) is the same as the complex number cos(t) + i.sin(t). I.e. it has x coordinate cos(t) and y coordinate (or "i coordinate") sin(t). Now here is the cool part, just as e^(a+b) = e^a.e^b, when a,b are real, also e^i(s+t) = e^is.e^it. that gives us relations for cos(s+t) and sin(s+t) and the same functions of s and t. E.g. suppose we want to know what cos(2t) is, Well, e^i(2t) = e^i(t+t) = e^it.e^it. Now substitute the cos and sin formula in and multiply out. I.e. e^i(2t) = cos(2t) + i.sin(2t), and e^it = cos(t)+i.sin(t), so from e^i(2t) = e^it.e^it, we get (purely mechanically, as in Saxon!), that cos(2t) + i.sin(2t) = (cos(t)+i.sin(t)).(cos(t)+isin(t)) = cos^2(t) + i^2 sin^2(t) + 2i.cos(t)sin(t), I hope. Since i^2 = -1, that gives cos(2t) + i.sin(2t) = [cos^2(t)-sin^2(t)] + i.[2cos(t)sin(t)]. equating real and imaginary (i.e. x and y) parts, gives both formulas at once! cos(2t) = cos^2(t) - sin^2(t), and sin(2t) = 2cos(t)sin(t), the so - called "double angle formulas for cos and sin. Homework: check similarly the "addition laws" for cos and sin: cos(s+t) = cos(s)cos(t) - sin(s)sin(t), and sin(s+t) = cos(s)sin(t) + sin(s)cos(t). OK this was tedious, but it all follows from just remembering two formulas: e^it = cos(t) + i.sin(t), and e^(a+b) = e^a.e^b, (the addition law for exponentials) and most people find those easier to remember than the addition formulas for sin and cos. Now you know why I didn't give this before! I wanted it to look easy!

Harold Jacobs Algebra uses equations with empty boxes instead of X's. Then the idea is to find the number which you can write in the box and make the statement true. This approach is inherently less abstract hence more understandable than having 'X' mean a variable number. Different letters were represented by different shaped boxes. This method has worked well in virtually every class I have ever used it in. This is variation on the nice suggestion by Crimson Wife. Older editions are the same as newer ones I believe just cheaper. http://www.abebooks.com/servlet/SearchResults?an=harold+jacobs&sts=t&tn=elementary+algebra Oh my! what used to be $17 for my child, new, is now over $60, used! That may be testimony to how popular it has become.

I agree 100%. I don't think it is necessarily good to have a higher level grasp of math or necessary or even useful. I do believe that for those children who would like that, they will not get it from a strict diet of Saxon. I will try not to choose anyone's goals in math for them, but after they have chosen them, I can suggest how to achieve them. Well to be really honest I also believe that a bit of understanding (more than I think Saxon offers) is very useful in trying to apply concepts. Mere computation is often inadequate in practice - it helps also to understand what should be computed. Even driving a car, it helps to understand a little physics when cornering too fast. Enjoyment also helps maintain focus while learning the mechanics. If a child is bored to tears she will not want to spend the time needed to master even the technical side of math. I have fought the battle to have my students at least try to understand the ideas behind the math for years, believing they can use it better if they know how it works. I also admit I wish I had taken typing in school!