For what it's worth, the term "good state school" is imprecise enough to allow a wide variation in interpretation as well. A search of the top 10 math departments in the US lists, among the 12 schools ranked or tied for "top 10", three (excellent) state schools, Berkeley, UCLA, and Michigan, of which the last apparently accepts over 30% of its applicants. Presumably these are "elite" schools for most purposes, if not ivies.

To me this is an interesting thread, but hard to think of how to add anything to it. I thought of some things. I know a young person who while in high school not only successfully attended and completed college level courses in his field, at a good state flagship school, but also graduate level courses, and he was a star in all of them. He also competed in an elite college level academic competition and performed well, all while technically in high school. When time came to go to college he attended an elite school, where he found a course designed for apparently people like him, miles above the level of anything available at the good state school he aced while in high school. He wrote the notes for that courses and gave me a copy, which I find challenging even though I am a senior professional in the area. I am puzzled as to where they find enough students to handle such a class, even at an elite school. (you can google this class, called something like: the hardest college course in america.) How many high school students have aced graduate level classes before going to college? I really feel sorry for a bright college freshman who signs up for this class with only standard AP/IB preparation from high school, as he/she would most likely be completely overwhelmed. The book used in this course in 1965 is available free here, but I do not recommend it to learn from, just to get an idea of the level of the course 50 years ago: http://www.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf (edit: Notice for instance that differential calculus is defined there on (possibly) infinite dimensional normed spaces (p.140), and that Stokes' theorem is proved for abstract n dimensional manifolds, not just surfaces in 3 space. Hence the presentation of these topics from advanced calculus is at the level of graduate courses in functional analysis and differential geometry.) Anyway, this student I knew could not possibly have been adequately challenged at virtually any strong state school. I.e. the course he took is only available at a top 10 school. But what does this mean to the rest of us? We are not him, and we do not need the course he took. For the vast majority of us, even bright, well prepared and motivated, most good state schools would have more than plenty to meet our needs. I must say in my experience I do believe the famous lecturers at the top elite schools are better, much better, than those at other places, and this benefits even those of us who are not stars. I attended a friend's class once at such a school, featuring a famous lecturer (Walter Jackson Bate), and heard a lecture the like of which I have never heard again anywhere. This was by a man whose biography of Keats, and later of Johnson, was described by critics as virtually the best ever, and as one which it was deemed inconceivable to be improved on. He won Pulitzer prizes for both of them. His lectures were perfect models of prose that he wrote out carefully and delivered impeccably. Apparently he received a standing ovation after every class, certainly the one I visited. There is nothing like this available at most schools to my knowledge, admittedly limited. However if one takes a non honors, routine calculus class, it will be not too different at a state school or an elite school. If one is unlucky, even at an elite school a non honors class may be taught by an inept, inexperienced grad student. At a good state school, one may well be taught by one of the students of the famous people at elite schools, and get a very fine experience. There are good and bad classes at all schools. Elite schools offer much more diverse possibilities, and the courses at the top end are usually not matched at state schools, but routine classes may seem similar, although at an elite school even the routine class may be offered by a star, but one whose abilities are not exercised in the given class. What we want to do is not compare schools in the abstract, but in regard to how the difference matters to us. I attended an elite school as an undergraduate, which was way over my head at the time, and I did not benefit much except by the vision of what heights of achievement were possible, but not within my own reach at the time. 20 years later I returned to the same school and benefited enormously, so for me that elite school was appropriate mainly as a post graduate experience. I reached my potential and became qualified for that experience, only by studying hard at a good state school, where I was an above average student, studying under people who were students of the professors at the top school. So please try to relax, hard as it is. If one works hard wherever he.she finds him/herself, eventually one winds up at an appropriate place, and many times the experience at the preparatory places is highly beneficial. Once while trying to use time well while out of school and working, I signed up for a language course at a state night school in Tennessee. My professor seemed excellent. When I got back to an elite ivy school and told my famous professor the name of my night school teacher he said proudly: "Ah yes, he got his PhD from so and so, and so and so got his from me!" edit: I did a search on the vitae of some of the most brilliant people i know at my uni, and they all got either their BA or PhD at state schools.

i love that teaching/discovery example. The only way i knew to explain this before was the rule 3^(n+m) = 3^n.3^m, so 1 = 3^0 =3^(1-1) = 3^1.3^-1 = 3.3^(-1). and solve by dividing both sides by 3..

my students at epsilon camp enjoyed making platonic solids out of cardboard, but some needed help with the cutting and measuring.

I have only read Fitzgerald, the one used in my college course, but it is so poetically beautiful I still recall some passages from the last time I read it to my children over 30 years ago. Powerful and beautiful. It is nice there are so many good choices. (Lattimore was the choice for my college course of the Iliad, but it didn't impress itself on me as much as Fitzgerald's Odyssey did.)

I'm not sure what is wanted but here is a link for the epic geometry thread: http://forums.welltrainedmind.com/topic/514183-epic-thread-of-geometry-programs-or-geometry-thread-of-epic-proportions/

here is what came "next" after post #15: We have seen that the solutions p,q of the equation X^2 -BX + C = 0, satisfy B = p+q, and C = pq, and we want to use this to find p and q, in terms of B and C. The trick discovered by the ancients was that this could be done if only we knew the value of p-q. I.e. then we could add B to that value and get B + (p-q) = (p+q)+(p-q) = 2p, and dividing by 2 solves for p. Similarly subtracting p-q from B would give B - (p-q) = (p+q)-(p-q) = 2q, and we can divide by 2 and also get q. Here is the trick: notice that squaring (p-q) and squaring (p+q) gives almost the same thing, namely (p+q)^2 = p^2 + 2pq + q^2, and (p-q)^2 = p^2 - 2pq + q^2. Thus these two squares differ by 4pq. I.e. (p+q)^2 - 4pq = (p-q)^2. This says that the three quantities, (p+q)^2, (p-q^2), and pq, are related so that if you know any two of them you can find the third. But we know two of them since we know B and C. Thus, (p-q)^2 = (p+q)^2 - 4pq = B^2 - 4C. So taking a square root solves for (p-q) = sqrt(B^2-4C). Thus 2p and 2q are obtained by adding and subtracting this from B, i.e. B ± sqrt(B^2-4C) = 2p, 2q, so p and q = (1/2)(B ± sqrt(B^2-4C)), the famous quadratic formula! E.g. to solve X^2 - 31X + 58 this way for p and q, we note that (p+q)^2 = (31)^2 = 96, and thus (p-q)^2 = 961 - 4(58) = 729. So p-q = ± sqrt(729) = ± 27, and thus p,q = (1/2)(31 ± 27) = (1/2)(58) and (1/2)(4) = 29 and 2. I love this because to me it explains the mysterious B^2-4C under the radical in the formula in a way I can relate to.

here are the first 4 algebra problems posed to the epsilon campers: maybe one of your students would have fun with one or more of them. they are harder than average though. These three algebra problems are from the book Elements of Algebra by Euler, the great 18th century mathematician. 1. Find a number such that if we multiply half of it by a third of it, and to the product add half of it again, the result will be 30. 2. Find two numbers, the one being double the other, and such that, if we add their sum to their product, we obtain 90. 3. A father leaves to his four sons $8,600 and , according to the will, the share of the eldest is to be double that of the second, minus $100; the second is to receive three times as much as the third, minus $200; and the third is to receive four times as much as the fourth, minus $300. What are the respective portions of these four sons? here is a problem from an 1895 algebra book, Treatise on Algebra, by Charles Smith: 4. i) {a^2 (1/b - 1/c) + b^2 (1/c - 1/a) + c^2 (1/a - 1/b)} / {a(1/b - 1/c) + b(1/c - 1/a) + c(1/a - 1/b)}. simplify. Hint: the answer is a+b+c. ii) Related problem: Factor this polynomial completely into linear factors: {a^3(c-b ) + b^3(a-c) + c^3 (b-a)} hint: recall that if X = r is a root of a polynomial in X, then X-r is a factor. What do you think happens if a=b is a "root"? By the way, problem #4 from 1895, which is essentially undoable even by most teachers of today, gives you an idea of the decline in expectations in algebra over 100 years of textbook writing. But this is the accelerated learners board. By the way, if you are curious about the 1895 book Treatise on algebra by Charles Smith, do not buy the cheesy reprint by Forgotten Books, as it is an unreadably bad scanned copy. There is no value in an algebra problem if the numbers are not legible. This is a piece of junk as I learned to my chagrin. I eventually bought an old library copy, but have since given it away.

for what it's worth here are my notes for epsilon camp introducing quadratic equations. I immodestly suggest that if your student learns even what is presented below they are way ahead of a lot of algebra students. ( I am trying to present it Euler's way.) Of course it may be worthless, and that is ok too. Use whatever works, and keep searching until you find it. This is the beginning of the notes for 2013: Some of the problems for this summer involve solving quadratic, or second degree, equations. If you have forgotten that skill, or are just learning it, I will present here an explanation I like, that makes it more clear, to me at least, than the usual one found in some books. Please help me improve it by asking questions. if you have a different approach that you like, please share it with us as well. part I: How the roots of a quadratic equation are related to the coefficients. Remark: The "coefficients" of the quadratic expression X^2 -BX +C, are the letters (or numbers) -B and C. We say -B is the coefficient of X and C is the constant coefficient. Of course the coefficient of X^2 is 1 here. I may be careless sometimes and call the coefficients B and C, instead of -B and C. So you need to look at the expression to see whether it is X^2+BX+C or X^2-BX+C. I like using the form X^2-BX+C, because then B has a simpler meaning as we will see below, but technically, the word "coefficient of X" refers to whatever is in front of the letter X together with its sign. To solve a quadratic equation means to find numbers X that satisfy an equation of form X^2 -BX +C = 0, where B and C are given numbers. The numbers X which solve it are called "roots" of the equation, and B and C are called the coefficients (or technically -B and C). E.g. we may want to solve for X such that X^2 - 20X +75 = 0. Here the solutions are X =5 and X=15, as you can check by substituting X=5 and X=15 into the equation and simplifying. The thing to notice is that the two coefficients in the equation, i.e. the numbers 20, and 75, are obtained by adding and multiplying the solutions together. I.e. the coefficient 20 = 5+15, and the coefficient 75 = (5)(15). This is a general principle. E.g. to solve the equation X^2 - 19X + 34 = 0, we must find two numbers whose sum is 19 and whose product is 34. Do you see what they should be? [answer below] The reason this principle holds is because every quadratic equation can be factored into a product of two factors using its roots. In our first example, we have roots 5,15 for X^2-20X+75 =0, and the left side of the equation factors as X^2-20X+75 = (X-5)(X-15). (Check it.) In the second example the roots are X=2,17 (did you get these?), and the quadratic expression factors as X^2 -19X+34 = (X-2)(X-17). (Check it.) The "factor theorem" of algebra says that X=r and X=s are the roots of an equation like X^2 - BX+C =0, if and only if the quadratic expression factors as X^2 - BX+C = (X-r)(X-s). It is easy to see this in one direction, namely that whenever the factorization is true, i.e. whenever X^2 - BX+C = (X-r)(X-s), then setting X=r or X=s, does make the expression equal to zero. (Setting X=r or X=s makes the right hand side equal to zero, so it must also make the left hand side equal to zero). Let's assume the other direction as well, that if X=r and X=s are the roots of the equation X^2 - BX+C = 0, then the factorization does hold: X^2 - BX+C = (X-r)(X-s). Then what does this tell us about the relation between the roots, r,s and the coefficients B,C? To see that, all we have to do is multiply out the right side of the equation X^2 - BX+C = (X-r)(X-s). This gives X^2 - BX+C = (X-r)(X-s) =X(X-s)-r(X-s) =X^2 -sX -rX +rs = X^2 -(r+s)X + rs. Now since X^2 - BX+C = X^2 -(r+s)X + rs, all the corresponding coefficients on the right and left must be equal, so B = (r+s) and C = rs. Thus whatever the roots are for X^2 - BX+C = 0, they must add up to give B, and multiply out to give C. I.e. to solve X^2 - BX+C = 0 means finding two numbers r,s whose sum is B and whose product is C. This gives a way to always find such numbers, as we explain next.

heres a $49 copy of jacobs: http://www.amazon.com/gp/offer-listing/0716710471/ref=sr_1_1_twi_1_olp?s=books&ie=UTF8&qid=1415753216&sr=1-1&keywords=harold+jacobs+algebra

wow, when i moved this spring i gave away literally thousands of dollars worth of books, at least if you have to buy them. but you cannot sell them for anything. i looked on powells books site once where they offered me about $5 for a great calculus book that they were themselves listing for sale at several hundred dollars. i decided to just sell all my $50 and up books to local math grad students and profs for $5 each. Dozens of them sold at that price in about two days. The copies of Jacobs I just put out in the hall in front of my office for free. Others i donated to the departmental math major library. I of course miss many of them now, but you can't bring everything when you move. I don't even have copies of many of my own research papers, and articles in books that i wrote myself. i assumed they would be available online but i haven't found them all. I am down from about 300 math books and hundreds of research papers, to about 100 math books and a handful of papers. But I still have Euler's Elements of Algebra, Euclid's Elements (of geometry), and the works of Archimedes, and works of Gauss on number theory and differential geometry, Hilbert on foundations of geometry and a popular work "geometry and the imagination", and Riemann's collected works. I recommend Euler as probably the best algebra book out there for a motivated person wanting to learn from an immortal genius who actually wrote his book for his butler. Back when people learned algebra from Euler, they learned not just the quadratic formula, but also the cubic formula, which I myself only learned in graduate school. Using Euler's clear explanation, I taught this material to 10 year olds at epsilon camp. Almost no one learns this stuff in high school today.

what about the next few listed books on abebooks, called the "Teacher's guides" for $25? are they the actual book plus something? no they are only about 250 pages against over 800 for the actual book.

Thumbnail sketch of elementary algebra: In algebra we try to practice rules for addition, multiplication, subtraction and division that will hold true for all numbers usually used. E.g. not only is (2)(3) = (3)(2), but also (17)(12) = (12)(17), and (103)(76) = (76)(103),...and so on. To state this once for all, we say that XY = YX, for all familiar numbers X and Y. A typical fact true for all familiar numbers X is the equation X^3-1 = (X-1)(X^2+X+1). Thus (2^3-1) = (2-1)(2^2+2+1), and (5^3-1) = (5-1)(5^2+5+1), (19^3-1) = (19-1)(19^2+19 +1),..., etc. To get a better feel for this, work out all these examples, including the one with X’s to see if they are really true. To appreciate the value of algebra, note how much simpler it is to check the version with X’s compared to working out say (273^3 - 1)(273^2 + 273 + 1), in numerical terms. This begs us to make more clear what are the “familiar†numbers, so we do so as follows: there are the integers, or whole numbers, positive and negative and zero: ....,-4,-3,-2,-1,0,1,2,3,....... Then there are the rational numbers, of form n/m where n and m are integers and m ≠0, and then the “real†numbers, which fill in all the holes on the number line. Real numbers can be expressed as infinite decimals like 328.12149836551100911......, where it is possible for all decimal entries to be zero after some point. The rational numbers are a subset of the real numbers and consist of those numbers whose decimal expansion eventually repeats, possibly with all zeroes, such as 1.234565656....., or 317.998200000...... Some examples of real numbers that can be shown not to be rational are sqrt(2), sqrt(3), sqrt(3), cubert(5), ....,Ï€,...... Since non rational real numbers are so hard to write down, impossible really, since it takes infinitely many decimals, we try to do problems that involve mostly rational numbers, or else irrational numbers that have simple names like sqrt(3). Algebra is used to find answers to certain questions whose answer is known to be a number, but we don’t know just which number it is. Since we don’t know the answer in advance, we are required to reason on the unknown number using properties that are true of all numbers. E.g. suppose we want to find a number X such that X^2 - 5X + 6 = 0. We may reason as follows: for all numbers X it is true that X^2 - 5X + 6 = (X-2)(X-3), so we are trying to find a number X such that (X-2)(X-3) = 0. But the product of any two numbers is zero only if at least one of those factors is zero, so we are seeking a number X such that either X-2 = 0 or X-3 = 0. Since adding the same number to two equal numbers again gives equal numbers, we seek X such that either X = X-2+2 = 0+2 = 2, or X-3+3 = 0+3 = 3, i.e.our answer could be either X = 2 or X=3. So a basic algebra skill to learn next is how we knew that X^2 - 5X + 6 = (X-2)(X-3). This is called “factoringâ€, and is worth some practice. A few basic examples are worth learning by heart: (X-A)(X- B ) = X^2 - (A+B)X + AB, X^2 - A^2 = (X-A)(X- B ), X^3-A^3 = (X-A)(X^2+AX+A^2), X^4-A^4 = (X-A)(X^3 + X^2A + XA^2 + A^3), ..... (X^3+A^3) = (X+A)(X^2-AX+A^2), X^5 + A^5 = (X+A)(X^4 + X^3A + X^2A^2 + XA^3 + A^4), .... Actually this is more than I knew when I won the 1959 Tenn. mid-state algebra competition. BASIC CONCEPTS and RESULTS: Dividing polynomials (expressions in X). Degree of a polynomial. Root/factor theorem: If a polynomial f(X) = 0 when X = A, then f(X) can be factored as f(X) = (X-A)g(X), where g(X) is a polynomial of degree one less than f(X), (and vice versa). solving equations by the quadratic formula; graphing linear and quadratic equations; solving linear equations involving one, two, and three unknowns. Exponential notation: i.e. X^3.X^5 = X^8, and in general X^n.X^m = X^(n+m). More advanced: rational root theorem: If f(X) is a polynomial with integer coefficients, and lead term X^n (i.e. with coefficient 1), then the only possible rational roots of the equation f(X) = 0, are integers, and indeed are integer factors of the constant term. Consequence: Since no integer factor of -2 is a root (solution) of X^2 -2 = 0, there is no rational root, so sqrt(2) is not a rational number. If your child knows all this, then he/she surpasses my average entering college calculus student.

Here is my favorite used books site, abebooks.com, and a link to several copies of Jacobs' Algebra for under $20: http://www.abebooks.com/servlet/SearchResults?an=harold+jacobs&sts=t&tn=elementary+algebra Correction: see next few posts.

Another odd thing about a "pre-X" course, from the teacher's viewpoint, is that it becomes judged by the results of the next class, i.e. the X class, rather than the present class. Thus the pre-X teacher will be criticized even if he/she covers well every topic on the syllabus, if someone teaching the next class complains that the entering students do not know everything they need. Of course I was also prone to complain that my entering students had been ill served by their previous classes, all the way back to grade school, if they failed to know what I wanted to assume. Eventually I began to make every class somewhat self contained, or at least include a quick review of previous material I wanted to use. I always tried to learn from Jesus' parable of the sower, in regard to prerequisites, i.e. sometimes even the message from the best teachers fall on infertile soil, but one keeps trying.

In that sense, I guess ideally a pre algebra course would be prerequisites for algebra, i.e. if you know this material then you are ready for algebra. That would explain why it might contain both arithmetic and some elementary algebra. Precalculus could ideally be taken in the same vein. The realist in me immediately recalls however that in actuality our precalculus students at university seem to have had a somewhat mediocre success rate in calculus afterwards. Historically, it does seem that such courses are created in response to an observed need, i.e. our students are not doing well in X, so we create a pre-X course to try to help out. it does not mean this succeeds, but that is the goal. In college we had such a course to prepare students for proof oriented classes like abstract algebra and analysis. So when I taught it, it covered logic and proof and some of the early parts of the abstract algebra course. Then I realized that my students were not learning any analysis concepts, such as rigorous limits, so I considered I should have covered some of the easier parts of analysis as well, but there was never enough time for everything. The most difficult challenge was always to get students to realize that each statement is only true under certain conditions. For instance, if a product XY of integers is divisible by an integer n, then is it true that n also divides one of the factors X or Y? This is true if n is prime, but not usually otherwise, and I never figured out how to make this sink in, even though I called it the "prime divisibility property".

Please forgive any perceived derision. I think this is a result of the conditions under which some of us encountered various topics, and has nothing to do with the value of the material itself. I am so old that in my day there was nothing but arithmetic, algebra, geometry (plane and solid), trig in high school, then calculus in college. So to me precalculus seemed like a made up subject introduced for students who had not mastered algebra, (coordinate) geometry, and trig. Then I saw some books called "college algebra", and I thought those were a fake too, another course made up to remediate college students who had not learned high school algebra. It turned out in some cases these were really old books written before some of those topics were taught in high school, or maybe before everyone took them there? Then in the upper reaches of undergraduate school there began to be classes on intro to analysis, apparently for people who were not ready for just plain analysis. (Some elementary high school books also use the word analysis, quite incorrectly by mathematicians' standards, for precalculus topics.) Later we introduced courses in college called "introduction to proof" for students who had not absorbed logic and proof techniques in the natural sequence of courses on geometry, calculus, etc.... I had almost never heard of pre algebra until now, but logically speaking it should be arithmetic in my simple world By the previous posts, when well done, it serves its student population with whatever needs exist at an introductory and preparatory, and sometimes exploratory level. So some of us are answering just what topics are likely included in a book with this on the title, some of us are addressing what topics should precede algebra, and some of us are trying to explain why this name exists, I think. Forgive me if this analysis (!) is unhelpful.

30 years ago I enjoyed teaching my kids from Harold Jacobs' books Elementary Algebra, and Geometry. More recently, with PG kids and grad students in education, I especially enjoyed teaching geometry from Euclid, with Hartshorne's Geometry: Euclid and beyond, as a guide. I finally felt I understood Euclidean geometry, and the famous "shortcomings", but more significantly the strengths of Euclid's original approach. Another more advanced one I have enjoyed teaching from to future mathematicians is Michael Spivak's book Calculus. I like books that are enjoyable to read and that teach me something, or make me look at the subject from a new perspective. I like books by masters, that make me say "Of course, why didn't anyone say it that way before? It seems so clear and simple now." As long as something seems hard, I am not satisfied. I want to eventually see why something should be easy, even obvious, or at least natural. for example: Archimedes says a spherical ball should be thought of as a pyramid, where the surface of the ball is the base of the pyramid, and the center of the ball is the vertex. If one can see that, it becomes clear why the volume of the ball is (1/3) radius, i.e. (R/3) times surface area, because the volume of a pyramid is (1/3) base times height. So once you know the volume of a ball is (4/3)πR^3, it follows that the surface area is (3/R) times that, or 4πR^2. When I was in high school, I don't recall learning the connection between those formulas. To help see it as Archimedes did, start from an (upside down) ice cream cone, and imagine that as a pyramid with a rounded base. Then let the ice cream part get wider, and the angle at the point of the cone get broader, like an umbrella being opened and then blown around backwards by the wind, until the "cone" is finally a full ball of ice cream, and the base of the "pyramid" has become the surface of the full ball of ice cream, with the point of the cone at the center. Geometry in Euclid and Archimedes is very interconnected, so that understanding one figure leads to understanding another one. Euclid talks about using triangles to study circles and vice versa. Archimedes goes on to cones and balls, and their similarity to pyramids. His proof that in terms of volume, a ball plus a (double) cone equals a cylinder, is simple and beautiful, depending only on the Pythagorean theorem. Then since the cylinder of height 2R and base πR^2 has volume 2πR^3, the cone has (1/3) that volume and the ball has 2/3 that volume or (2/3)(2πR^3) = (4/3)πR^3. This simple approach was not explained in books I had seen, and I learned it from a picture Jacobs' Geometry, not that long ago. As an old calculus professor, I was quite embarrassed not to have known how easy Archimedes made this, when I was making it look much harder in my calculus classes. Another volume problem we consider very difficult in calculus classes is that of a "bicylinder", the intersection of two perpendicular cylinders, something difficult even to visualize at first. (It looks sort of like a rounded pagoda plus its reflection in the water.) Using Archimedes' approach it can be seen to equal a cube minus a square - based pyramid, so that again its volume is (2/3) that of the enveloping cube, exactly similar to the case of the ball. I.e. he shows that the bicylinder, the cube and the square based pyramid, are exactly analogous to the ball, the cylinder and the circular based pyramid. So if you can do one you can do the other. I have never seen this explained in any calculus book. It is described in this paper, which I don't find super easy to read, but could be interesting to someone, in which Tom Apostol and a collaborator generalize Archimedes' idea to a wide class of solids.. https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Apostol496-508.pdf There are many other amazing things, in this book by the same two authors: http://www.maa.org/publications/books/new-horizons-in-geometry

@Mike: Nice argument! it can also be done "by inspection" as follows: the remainder of (16)!+1 upon division by any integer from 1 to 16, is visibly 1 (why?). so the smallest prime factor is at least 17. (hint if wanted by anyone: "the 3 term principle" says: if n divides a and a+b, then n also divides b.)

is it 17? (took me a few minutes of thought.) hint: factor it. quickly: why is the smallest prime factor of 16! + 1, at least 17?

intelligent memorization is always useful.. i used to memorize the formula 1^2+ 2^2 + 3^2+....+n^2 = (1/6)(n)(n+1)(2n+1), at least i think that's it, to use in my calculus classes to integrate the area under the graph y = x^2. then i realized that the only part i needed for that was the first term: (n^3/3 +......). then i realized that all the formulas are like that: i.e. 1^3+2^3+3^3+.....+n^3 = (n^4/4 +.......), and 1^4 +2^4 +3^4+....+n^4 = (n^5/5 +......), and that you never need the part where the dots are..... Very few calculus textbooks point this out, most just bash out the result using the full formula, showing to me their lack of appreciation of what matters. always try to find the understanding that makes the memorization easier. if someone has trouble with (A+ B )^2 = A^2 + 2AB + B^2, show her/him the picture of a square with sides A+B, and how it decomposes into an A square and a B square and two AB rectangles. that's how Euclid did it in the Elements. No one forgets it after that.

apologies if i have told this story too often. i am reminded that when my younger son started a new prestigious private school in 2nd grade, I made him learn his multiplication tables up to 12 x 12 before showing up the first day. when he came home all smiles i asked how it had gone and he said fine. i asked if his multiplication tables were up to snuff, and he began laughing uncontrollably. when he calmed down he said in this class they did one multiplication table per month, and the first month was the zeroes times table. it was a really good school in lots of ways, but somewhat math phobic, and i thought about how much tuition i was spending. i have noticed a lot of mathy kids enjoy memorizing things like a lot of digits of pi, so if they enjoy it, let them. I agree too it can be useful to know several squares and cubes. (I myself probably know squares up to 17^2.) this includes the expansions of things like (a+ b )^3, (a+ b )^4, and the factorization of (a^3-b^3), ((a^3+b^3), (a^4-b^4). these latter are actually more important. i see kiana beat me to these. amazingly, in the age of dependence on calculators i have had students in calculus who did not know the cube root of 8, nor how to multiply 2 digit numbers even using pencil and paper. a propos of nada, i happen to know the random 14 - symbol alpha numerical password for my wifi connection, which always amazes guests, but i find it useful when they ask what it is, instead of turning the modem over and squinting.

I read that book in college over 50 years ago in a class on the nature of prejudice. I don't remember any of the things you are worried about, only his stark first experience of the "hate stare". Another thing I remembered, his reported death from skin cancer as a result of his experience, seems not to be true.

recalling helping my own kids prep for the sat: i would let them take old tests from the practice book, then i would go over them, grade them and point out the right answers they missed, in my opinion, or maybe they had answers in the book. one thing that came up in our practice was that kids can think of reasons several possible answers could be correct, and my experienced perspective was useful to explain to them why the test maker wanted only one of those answers. i.e. there were often more than one technically "correct" answer, but I usually knew which one the test maker wanted and why. this sort of test taking savvy is helpful to youngsters, and it takes an older head to be aware of this. so i recommend discussing with your kids why certain answers were likely chosen. you'll probably see something they won't.

as others have said, it is lucky for her that this is happening now before she goes to college, and she is lucky you are available now to help! if there is some way to give her an advance glimpse of what college will be like, it might help it sink in. maybe a visit to a college with a view of the schedule, the workload, and the effort being expended by the students to stay abreast. good for you!