here (I hope) is a link to euler's book: https://archive.org/details/elementsalgebra00lagrgoog

you have a real challenge. what to do for a kid who has been so successful that he thinks everything should be easy? hats off to you in this situation. this was my own problem and it threw me for a loop in college after being a star in high school. somehow we need to remind our kids that real science means facing hard challenges and grappling with problem that are hard for us, i.e. the point is not to breeze through every curriculum, but to challenge ourselves until we find the one that is hard for us. real character is built by finding our own challenge and facing it. good luck and god speed.

Our older child started jacobs algebra at about age 11 or 12 and did it easily. the younger one was capable of doing the jacobs geometry at age 8 when we briefly experimented, but we resisted because the older one had objected to being taught extra math at home in addition to school. 7 years later, when the younger one finally got to geometry in school, I regretted not introducing him to jacobs earlier at home, since by then his school had switched to saxon. I believe there is no particular age or grade at which a child can handle a certain topic, if he/she is interested and it is introduced with words and terms he/she can understand. E.g. not every child should have to wait until age 15 to contemplate a circle, or a triangle, or an icosahedron for that matter. A very young child may appreciate, by playing with straws, that a triangle is rigid but a 4 sided plane figure is not, whereas all convex solids are rigid, no matter how many faces they have. i.e. you can squash a square made of straws but not a triangle (this is called the side -side -side congruence theorem), but you can't squash either a tetrahedron or a cube. It is true that today many geometry books, including jacobs, assume a child already knows about real numbers, and base geometry on measurement. Although I like Jacobs' book, I find this approach somewhat artificial. In particular it departs totally from Euclid's original naive treatment which assumes only a knowledge of adding and subtracting integers, barely even that. Euclid does everything using only the simplest notions, such as comparing two things to see which is greater. Jacobs' approach to algebra is also very elementary and intuitive, using empty boxes to represent unknown numbers. thus again essentially one only needs to know about adding and subtracting and multiplying. my apologies as this is fast becoming another one of my pointy headed professor soapbox speeches.

We had two mathematically gifted children with different characteristics. One had a steel trap memory while the other caught on quickly and forgot as quickly. The first was taught algebra at home from Jacobs Elementary Algebra and benefited enormously. The second was taught at school from Saxon and may have benefited in retention from the repetition, but found it dull. Both scored very high on SAT math tests. The first became a math major in college, the second dropped out of math in college. Unless a child has retention issues, I would tend not to encourage use of Saxon materials in the case of mathematically gifted kids, due to what I perceive as lack of creative stimulation, understanding and depth. Indeed some scholarly studies I have read show that even the claims made for increased standardized tests scores and increased math interest by the publishers of Saxon are unjustified. To me, Jacobs excels in creativity, challenge, and the element of fun, combined with solid mathematical content. For profoundly gifted kids, the classics such as Euclid and Euler may be worth a try for geometry and algebra.

if you used the 3rd edition of jacobs, you will find considerably more logic in the 1st edition. the book that opened my eyes on logic 55 years ago in high school, was the early chapter in "principles of mathematics" by Allendoerfer and Oakley. I still have a copy. http://www.abebooks.com/servlet/SearchResults?an=allendoerfer&bi=0&bx=off&ds=30&recentlyadded=all&sortby=17&sts=t&tn=principles+of+mathematics&x=59&y=13 sorry, i didn't process your previous post saying you were not interested in math oriented logic. would you be interested in works by lewis carroll? those were featured a lot in jacobs geometry, in the first edition anyway. i'm not quite sure what you are interested in. in teaching math, the logic i focus on is essentially about how to tell what a statement means, and how to tell if you have done what you were supposed to do to solve a problem. We focus on "if - then" statements, and emphasize the difference between "A implies B" and "B implies A". It sounds simple but almost all logical fallacies encountered daily in the newspaper concern someone confusing the two. e.g. to say that all successful businessmen are hard working implies nothing at all abut what percentage of hard working businessmen are successful. in high school math, we often worked backward in an algebra problem to find X. But the statement that "if f(X) = 0, then X = 3", does not imply the desired result: "if X=3 then f(X) = 0". I used an illustration in my class from a sign on the road leading to campus: "bicycles only in right lane" what does it mean? does it mean: if you are in the right lane you better be a bicycle, or if you are a bicycle you better be in the right lane. i.e. it could mean either "only bicycles allowed in right lane", or "bicycles must use only the right lane". and other fascinating examples....

have you tried introducing him to an actual mathematician? i.e.someone who both loves and knows a lot about math?

I thought Khan did an excellent job of explaining basic calculus in the few videos I watched. His lectures seemed exactly what my college non honors classes would have benefited from.

I love "Calculus made easy", ever since being introduced to it in 1960 during honors calculus at Harvard. Thanks for mentioning it. I apologize for my obsessive behavior, but I like to give credit where it is due, and for the other acadenic historians among us, I must say that this book is written by Silvanus P. Thompson, dating from 1910, and was only superficially revised by Martin Gardner much more recently, and perhaps not much improved.

here are the notes i wrote this summer on algebra, geometry and calculus, for my epsilon campers, aged 8-11. http://www.math.uga.edu/~roy/epsilon13.pdf

One always needs to know what is the definition of "gifted" that is being used in a program. If kids with 120 IQ are classified as gifted, then those with IQ > 150 are as far out of the scale in a gifted class as gifted kids are in an average class. To put it another way, 99th percentile sounds high, but the 99.99th percentile is in the 99th percentile of the 99th percentile, (if i did the math right). But I don't think you have to worry if you monitor the level of effort. As long as your child is challenged, that is what determines adequate progress.

As a little kid i felt i was often the best at lots of things (ok, not sprinting, not music, not basketball, not writing,...ok, you get the idea, I was a bit deluded), but I got somehow the impression the goal was to always be the best. then i went to a good college where I was nowhere near the best, and I tanked. Some 10 years later, I realized I had no chance at achieving my dreams unless I acknowledged that even when I tried my very best I still might well not be the best. Indeed that fear had kept me from trying my best for a long time. So I began to try as hard as I could, and I moved to another level; I became not more gifted, but much more accomplished. I was still not anything like what I had dreamed about, but much closer than i had been while simply fantasizing. So Donna, what I suggest is this: ask your DD what she really wants to do. Then tell her honestly that she may not be the best at her goal, but support her in her attmept to reach it. Please forgive my naive advice. By the way, I think you are doing a wonderful job!

i also recommend harold jacobs' book on algebra as self teaching. if it covers the required stuff. what is algebra 2 anyway? quadratic equations? i.e. what sort of question is causing the trouble? maybe we can address it right here.

some of this discussion reminds me a little of my learning to play snooker well. in high school, i played this guy who was one of the best in the city, several times a day, every day for a year, losing every game. then i finally won one, on a hard bank shot. he complained bitterly about my "lucky shot", and did not want to play me any more. I then discovered i too had become one of the best players in town. we also had a situation like the one in this thread at my uni, where a few professors always won the "best honors professor" award every year. so they created a higher category, for multiple repeat winners. instead of just excluding them from the competition they gave them a special permanent recognition that they had achieved another level of expertise above the usual, and then they felt good about it. Of course then there were also larger venues to compete in, like state level, national level,... it is very unusual for there to be truly no competition for someone at any level. has everyone seen chariots of fire?

this is the main skill gifted kids need. it is of course arguably much more blessed to have to learn to pretend one is less gifted than one is than to be not so gifted?

in some sense it is sort of like paternity testing - i.e. what will you do with the information/

Others here are more current than I, but I will try to start things off and inspire their input. Caveat: although the sentences are written declaratively, these are just opinions and anecdotes. First of all since there are many definitions of "gifted", there are many tests for detecting the various types. If your goal is to gain entrance to some specialized programs for your children, then you need to document giftedness in the way those programs require. When we were in your situation, we lived in Athens, GA, where the late Paul Torrance, a famous academician specializing in "creative giftedness" was head of an institute. So we went there and one of his students gave our children a test, probably a Stanford Binet type IQ test, with some questions designed to measure also creativity. Since the student wanted to practice on our kids the cost was minimal. The output was a numerical score and a comment on the creativity of the child, relative to others tested there at about that same time. Since we were doing it to establish their eligibility for Duke TIP, this was all we needed in addition to an SAT score, or maybe we only needed the SAT score. Please remember that even if you child does not receive the blessing you expect from the tester, that they are probably still gifted, since you have observed this. It only means the method of measuring it differs from yours. E.g. while being tested at the Torrance institute, one of our children gave one of the most creative answers to a problem I have ever heard, but it did not count because he did not follow the rules correctly while giving it. More recently one of our grandchildren, who we think is certainly brighter than the rest of us, who are all labeled "gifted" by someone, received a much lower IQ score than expected by me, on a professionally administered test for similarly meaningless reasons of not working rapidly enough (too perfectionist). So don't get too invested in the results. Remember you are the best judge in most cases, but the official recognition is useful as entree to some good programs. Here for your information, is the eligibility criteria for epsilon camp, a program for "profoundly gifted" children whatever that means. Well for this purpose I guess it means exactly what is on their web page. (From my personal experience there, it means children of age 8-10 who are better candidates for a typical college math course than is an average college student.) These of course are not to be taken as examples of typical gifted students, since they fall in a tiny percentile of the most exceptional students, and then also primarily in math, which some very gifted students may not be inclined to at all. http://www.epsiloncamp.org/who_is_eligible.php There is also a link on the epsilon web page to a discussion of giftedness in the sense that they understand it, which is much more comprehensive. Here is a related link: http://www.acsu.buffalo.edu/~mjh7/marjijonesadvocacy.pdf

of course, standardized tests! that explains this whole exercise. thank you, I get it now.

"Teacher says the independent variable = # kittens in the basket Dependent= total weight in basket" Let me pose one further question. Is a "function" from a given domain to a given range supposed to have just one value at each domain point? (This is the usual meaning of "function" in mathematics.) If so, then the answer given above by the teacher, cannot be correct unless all kittens have the same weight. I.e. if e.g. some two of the kittens weigh different amounts, then just knowing the number of kittens in the basket does not determine their total weight, for any number less than 5. thus the total weight of the kittens in the basket, even if proper subsets of the litter are allowed, is not a single valued "function" of the number of kittens in the subset. of course i wasn't present in the class, so maybe multiple valued "functions" are allowed? of course i realize the word "function" has not occurred in this problem, but that is the usual context for the words "domain" and "range". is this possibly a problem about multiple valued correspondences rather than functions? or to put it in the less precise language of variables, is the dependent variable in this problem allowed to take more than one value when the value of the independent variable is fixed? that would seem to be needed to rescue the teacher's answer. I.e. the teacher's answer is conceivably correct, but only with somewhat unusual assumptions. I have thought of a way to rescue it another way - let the function have as input the number of kittens, and as output the largest possible weight of that number of kittens from the litter. then again we have a single valued function. I want to emphasize again that the key information is "what is the function?" only then can one decide what is the domain and range.

with all due respect, this is not what i expected. If that interpretation is considered correct, then there are many other equally valid ones that must also be considered correct. perhaps the teacher just wanted any conceivable interpretation that involved numerical domain and range? But this cannot possibly be argued to be the one correct answer, unless there were more instructions we do not know about. my idea that i logged on here to offer as an attempt to justify what i assumed to be the teacher's answer, was to let the independent variable be the average weight of the kittens, so that then the function was y = 5x, with domain [10,12] and range [50,60]. but when the basket is supposed to be chosen to hold the weight of the entire litter, as was specified, ("Jill chooses a basket that can support the total weight of the litter.") then I do not see how one can say that the number of kittens it should hold is somewhere between 0 and 5. this is becoming not so much a math problem, but a debate with imprecise conditions, and arguable conclusions. please consult euler's elements of algebra for some algebra problems I would recommend more highly.

notice that until you describe what function you are considering, it is impossible to determine the domain and range. your child gave as domain the litter and as range the interval [50,60]. Ok that is correct if the corresponding function is the one assigning to the collection of kittens in the litter, their total weight. If the teacher wanted as an answer that the domain was (5 copies of) the interval [10,12], then that would be correct for the addition function assigning to 5 individual weights from the interval [10,12], their sum. However if the teacher wanted as an answer the domain as being the interval [10,12], and the range being the interval [50,60], then in my opinion technically this can only be the domain for one of the variables, say K5. Hence this can only be correct for a somewhat artificial partial function such as the one which assumes that say the first 4 kittens' weights are already known, and then we look at the function assigning to the fifth kitten's weight K5 say, the total weight K1+...+K5. I.e. if the domain is a single interval, like [10,12], then the function can only have one variable. If the function has 5 variables, as seems the case here, then the domain must consist of 5 tuples of numbers, so it should be a product of 5 intervals, even if those intervals are the same. (This agrees with what Kendall said.) So I am kind of wondering what your teacher thinks the answer is here, and why. It is necessary to read carefully the description given in the book or course of "domain" and "range" and "variable", (and hopefully "function").

aha! actually there are one or two more functions involved in this problem. the third function is the "maximize" function, that has domain the total weight interval [50,60] , and out put the maximum of these values, or 60. Then the fourth possible function has domain the one number 60, and has output an actual basket which will hold that weight! this last function is not necessarily single valued, since many baskets may work, so is not called a "function" in elementary math books, which pretend that such animals do not exist. I.e. the goal is to choose a basket, and the steps involve 4 stages, kittens, individual weights, sum of weights or total weight, maximum total weight, and finally a choice of basket! So there are 4 functions, 4 domains, and 4 ranges, and at each stage the role of independent and dependent variable undergoes an interchange. I.e. dependent variable in each function becomes the independent variable in the next function. So you see why these terms are misleading and the driver of the process is the function being considered at each stage.

Forgive me, this sort of thing sets off my mathematical nonsense detector. I am a college math professor and this problem makes little sense to me. I am tempted to say it is primarily asking for jargon. I would say this is an estimation problem - the total weight of 5 kittens weighing 10-12 ounces each is between 50 and 60 ounces, so the basket needs to support at least 60 ounces, unless you hold it from the bottom of course. I don't see a need for any variables independent or otherwise. I suppose you could say the desired output is the total weight, so maybe that would be called a dependent variable, but why do that? I mean of course it helps to ask what you are looking for, what quantity you need to estimate, but why give it an odd name? So hopefully the word "range" applies to the key quantity, the total weight, so it would run from 50 to 60. But is it looked at as a function of 5 variables? each kitten's weight counting as one? and each having range (i.e. domain) 10-12? What a needlessly confusing way to look at this. Maybe I can play that game if I say W = K1 + K2 +...K5, where Ki is the weight of the ith kitten, and W is the total weight. Then there are 5 independent variables, the Ki, and one dependent variable W, and the range (domain) of each Ki is 10-12, and the possible range of W is 50-60. But I would never ask it in an uninformative way like this. By the way mathematicians do not always use these terms "domain" and "range" so much any more. Of course we know what the books say they should mean, but actually there is a no complete agreement on what "range" should mean. I.e. is it the interval within which the output values are allowed to lie, or is it the actual set of output values that occur in the given problem. In this problem however only the first possibility is plausible, since the second interpretation would be a single number, the unknown total weight. But for this reason we geometers tend to use more often the more geometric terms, "source", "target", and "image", for these three concepts, at least for geometric problems. Even in analysis the term "codomain" is sometimes preferred (for the possible range of output values) to alleviate the possibility of two meanings of the term "range". Unfortunately it sounds artificial. I would try to find a better math course/book/teacher. Or maybe teach them more suggestive terms like "inputs" and "output". then these correspond to independent and dependent variables, and "possible (values of the) inputs" and "possible (values of the) outputs" should presumably correspond to domain and range. It may help your child to remind that the desired domain and range are probably supposed to be intervals of numbers, and the "variables" are letters corresponding to the actual kittens, but should represent not an actual kitten, but a number for each kitten. So although in a philosophical sense the domain is the litter, they really want us to use the variables to pass from the kittens in the litter, to numbers in the domain, i.e. their weights. so really there are three relevant sets in this problem, and two functions, two domains, and two ranges. " the first domain is as your child said, the litter of kittens. then the "independent variables" K1,...,K5, are themselves functions from that domain to the range of possible individual weights, namely the interval [10,12]. However in this first function these variables actually represent values, hence are the "dependent variables" in this case. I.e. the Ki take their values in the "range" interval, which characterizes "dependent variables". (Confused yet?, I am. But remember, since here the values Ki are the outputs, they are the dependent variables. I.e. here the kitten is the input and its weight is the output.) Then the second function is given by the addition formula W = K1+...+K5. Now in this formula the Ki are inputs, hence independent variables, and the total W is the output, or dependent variable. The domain of this function of 5 variables is the product of the 5 (equal) intervals [10,12]x...x[10,12]. The range of this addition function (for all possible values of the kittens' weights) is then the interval [50,60]. This function may be called W, the dependent variable for the second function. Actually this became interesting to me as I tried to make sense of it, but my sense is probably not what the teacher has in mind. I hope this has some interest for you, as an illustration of just what variables and the other terms actually refer to. I have also tried to see and reveal the correctness of your child's intuitive take on the meaning of the domain, as the actual litter. It was that insight that led me to see that the so called "independent variables"Ki, are actually themselves dependent variables of another more basic preliminary function, from the litter to the interval [10,12].

don't worry. i am a professional mathematician and i forgot my own research after 3 months of inactivity. it does come back with renewed study, but the moral is not to let the math get "cold".

remind the older child of his/her role in inspiring and teaching the younger one.

i love it as well. my own kids were much easier but calvin and hobbes is an iconic delight not just to me but also at least one of the most brilliant mathematicians i know: http://math1.unice.fr/~beauvill/