I think if they compare the transcripts and notice that you left it off to raise the GPA, it may raise the question of the legitimacy of other grades.

I think part of it is that tests in quantitative subjects, or subjects where there is a clear "wrong", are intrinsically more easy and rapid to grade. I still remember one of my undergrad profs saying "Well, I was a double major in English and Math, and I was thinking about graduate school ... and I thought "Do I want to grade papers, or problem sets?" and after that the choice was clear." Personally, I usually get test grades posted the same day, unless I have multiple students who are taking a delayed test -- I grade each problem together for consistency and fairness. Quiz grades are usually posted within an hour or so. But I'm in math. That being said, I completely agree that not getting some kind of feedback on paper 1 before being required to submit paper 2 is utterly ridiculous.

This reminds me of a student that my undergrad adviser told me about. They were using a classic book on mathematical probability/statistics -- the issue was that sometimes for 2 SD they'd use 95% and sometimes they'd use 95.45%, and so he wouldn't get exactly what the back of the book said. It drove the student nuts.

So I've just gone and checked the TOC. (went with this one -- https://www.sjaweb.org/ourpages/auto/2014/5/27/63078308/Precalculus with Limits _ 2E 2010 ISBN 9781439049099 _ Ron Larson_ Robert P_ Hostetler.pdf -- in case your numbering is different). First of all, he won't be crippled in calc if you just stop where you are. Other people may vary, but I'm going to give my opinion just as someone who sees what my calculus students struggle with. I assume (based on what you say about him) that he is solidly competent in basic algebra skills such as negative/fractional exponents, factoring, rational expressions/complex fractions, and graphical representation of functions. If he weren't, my advice would be more to review those. I've listed each chapter individually below, but I've summarized all the sections here. The ones I'd really hate to miss are 9.5 - 9.7, 10.1 - 10.4, 10.6, 11.1. The last two, just so that there is some basic exposure to coordinate systems beyond the standard cartesian plane. When I say "skim", I mean read the section, work through the examples and try to understand the big picture, and do at least the beginning computational problems. Chapter 9: I'd look at arithmetic + geometric sequences and series -- they'll be re-done in calc 2 as part of a much broader overview but calc 2 students find sequences + series horribly difficult in general. Some base knowledge is a good idea, especially becoming familiar with the sigma notation for series. Very few precalc students are going to get anything out of the mathematical induction section -- I'd look through it (it's an early exposure to proof), but not worry about doing a lot of the homework. You can return to this one later if there's time. Binomial theorem I *would* do, and counting + probability -- it's not so necessary in calculus, but imho every functional adult should have some basic understanding of counting and probability. The binomial theorem is actually a huge help if you need to expand something like (x + y)^4. I also think I'd skim the "proofs in mathematics" at the end of the chapter. Chapter 10: For analytic geometry, I would definitely cover lines, parabolas, ellipses (I assume circles fit in there as a special case?), and hyperbolas. Lack of understanding of these really hurts a lot of calculus students. Parametric equations and polar coordinates are also something that I'd like to include although the calculus textbook should also teach it. I'd at least skim the graphs of polar equations. I wouldn't worry about the rotation of conics and the polar equations of conics -- again, you can return to these. I don't even know if I'd skim them. We don't do a lot with them in calculus class, other than 90 degree rotations and polar equations of circles, and those usually aren't terribly difficult for people who understand the basic levels. Chapter 11: It's good to have some exposure to 3d geometry but it is customarily taught from scratch in calc 3 and/or linear (whichever comes first at the school) because many high schools don't teach it or don't teach it well. I would make sure to do the first section just so he has an idea that the cartesian coordinate system goes beyond two dimensions, and after that I would skim. Chapter 12: The limits + intro to calculus will be completely retaught in calculus. This is the last chapter in the book. I think at least skimming it would be a good idea just to ease the transition to calculus, but if by then you are just ready to be Done it won't hurt him to stop. It'll just make the beginning of calculus a little bit easier if he's seen it once before. HTH.

The only way that I get 1239 is if I round r^2 and h two decimal places prior to substituting into the final formula. Which is WRONG. ARGH.

I guess that I'd think of it as ... does she buckle down and do her math? Is she a "just get it done" type of person? Saxon may work. But otherwise, I'd be far more inclined to go with something a little less tedious. Saxon is kinda like digging into a plate of boiled broccoli because you're eating your veggies. It gets the job done, but ...

I agree. I'd skim through it and if it piqued interest, go for the intermediate -- if not, choose another topic.

There are different kinds of applied math. Someone who's looking at engineering would be looking more at the calculus side. Someone who's looking at actuarial science would need the probability side (and the calculus side, but). University-major applied math is still pretty theoretical. I went back and looked at the requirements for one school I attended. They required calc 1-3, linear, diffeq, modeling, advanced calculus 1-2 (this is a theory class, calculus with proofs), a survey of probability and statistics, and three electives. Their electives all included significant quantities of theory. Examples were differential equations with proofs, complex analysis (calculus with proofs and complex numbers), numerical analysis (again, proofs). They also needed to pursue an approved minor. Someone who was looking at actuarial science, for example, might choose something like economics or finance. Someone else might choose physics because it's more in line with their interest.

re: calc 3 and linear algebra. It is quite uncommon to require linear algebra as a prereq for multivariable calculus; it is generally taught on an as-needed basis. Calculus is usually required as a prerequisite for linear, not because the linear algebra content itself requires calculus, but because a certain level of mathematical maturity is required. Calculus examples are used in some of the popular linear algebra textbooks (Anton, for one) but they are asterisked and could be omitted. There are other courses (modern algebra, for example) where technically they could be done after precalculus (there are modern algebra texts that are structured for intro to proofs) but it would be rare indeed for a student to have that level of maturity and not had calculus. re: op, if she thinks calculus is easy and boring I would not do diffeq or mvc, I think it would be more of the same. I think that a discrete math course would be a wonderful idea. I had taken calc 1-3, linear (computationally oriented), diffeq, and stats, all of which were easy and boring. I took discrete, which at my college was intro to proofs, but it was also the last class required for my math minor, and halfway through the semester I changed my major to math. I think I'd consider the following: The aops discrete math classes -- the intro level might not be very challenging to someone who has already had calculus, but the topics are fun and it certainly will fill the bill of a senior math class. If nt/combinatorics are completed I'd say 1cr discrete math. For each one I'd say a semester's credit would be totally reasonable. The stats II -- AP stats often doesn't transfer in very well, and especially if she took stats at the CC. Stats II should transfer as something. A CC liberal arts math class -- again, it will be a bunch of fun topics that won't be especially challenging, but will be generally chosen to be of either high interest or high applicability. It'd probably end up transferring and counting for a general elective. Doing math for liberal arts on your own with Lippman's math and society (free) and/or jacobs MHE, choose your own units based on interest. Many of these would be great one semester courses, so you could mix and match. I've headed towards the easier and of more interest to the humanities side because you mentioned her desire to git'r'done and study languages.

I came in here to recommend MHE as well. A lot of people have done that prior to algebra. You may also have some luck with finding interesting units from MEP (free).

Jann is right and I'm just going to add some reasoning. A PS textbook is frequently not designed to be completed in its entirety, but rather the teacher or school would choose the "core" sections and then add in other sections as required. They often include what would be more than a year's work for an average student. An excellent example would be alg 2/trig, where the trig chapters need not be done if the student is going into a standard one-year precalculus class the next year, but a student who was going to take an accelerated precalculus class that actually got through a significant amount of calculus (a friend of mine teaches at a school where the accelerated track is alg 2/trig, precalc/calc a, calc bc) would need to complete the whole thing. The flexibility in textbooks allows different courses at different levels to be run using the same text, which is a strong consideration for a text that is expected to be adopted widely and taught by a teacher who is experienced with the material. Many textbooks designed specifically for the homeschool market are designed for self-study, for video learning, or to be taught by someone who does know the material but is not necessarily at a level where they would know what is "core" and what is "advanced/optional". As such, they are designed to include a year's worth of material and maybe a little extra. If material is extra, it is usually indicated. TT is designed for the homeschool market (and specifically for standard-level classes) and as such does not include all of the extra work, and again should be completed. Saxon, while not designed for the homeschool market, is designed to be completed in its entirety and includes a standard year's worth of work; the review at the beginning of the next course may suffice for a bright student, but I would recommend against it. I hope this adds a little bit of clarification as to why your program would say it's okay in general. I'm not sure if summer schooling is an option, but one good option would be to cut math back to three times a week (distributed through the week, not mon/tues/weds) and just spend more weeks on it. This would help reduce summer "brain-drain" as well. You may find that you can move more rapidly through the review at the beginning of next year as a result. It is generally better to move more rapidly through the review than to rely on the review to introduce new material.

Honestly, the fact that they aren't willing to reassess after the summer would have me quite concerned. It makes it seem incredibly rigid, and this was one of the problems with tracking that led to it being abolished -- someone being put on a low track would have little to no opportunity to jump up if they matured later or turned out to have been misplaced. If we (at the college) have someone who is trying to challenge a class, we usually just give them the final. Most of them are not successful, but at least both we and they are convinced that they need the class. The very occasional student who actually does do well on a cumulative final clearly shouldn't have to take the class. What do they have to lose by redoing the assessment at the end of the summer? At worst, he would score the same.

Right, but I've accidentally hit a livestock fence while I was standing up to my knees in water/mud, and while it definitely provoked a scream it was in no way life-endangering. This article has a picture of the bridge: https://www.kcra.com/article/teens-killed-trying-to-save-dog-solano-county/27012461 This article has some speculation about how it got electrified. https://www.sfchronicle.com/bayarea/article/Teens-electrocuted-in-freak-accident-in-Dixon-13737085.php

Personally, if I were aiming for a PhD in pure mathematics (which is what I ended up doing) from early on, I would go to a good undergrad school for math (it doesn't have to be top-line, but not some random state college, which is where I went -- it should probably have a graduate program) where I could take all kinds of upper-division electives and enroll early in graduate courses, and instead of applying for credit and accelerating graduation, get them to let me skip classes I'd learned through self-study and substitute graduate classes or advanced electives in the same area. For example, if I had already learned undergraduate abstract algebra, it would be perfectly reasonable for a college to let me skip it and take the graduate sequence instead. This would show up on a transcript just fine, and even if another graduate school did not transfer it in as equal to theirs, it would DEFINITELY satisfy requirements for admission. That being said, research was very different from the parts of math that I really loved. I loved solving problems that could be done in a day or two, but not working on the same problem for months. I think it might have been different if I had had my ADHD treated much younger, but I just got bored working on the same thing. However, I'm just getting back into a bit of research now (after starting treatment), and well, we'll see. Does he know what being a math professor is like? He should really look into seeing if he can job shadow a few.

This was posted when I was on break and traveling, so I missed it. I am not sure if you're looking for formal classes, or for books, but there are so many interesting and accessible discrete math books that would probably work fine for self-study. Many of them will have a significant overlap with AOPS, but for self-study this can be very easily dealt with by skimming the chapter and only working the interesting problems. I'm especially going to give a shout out to the Aimath free open textbooks (https://aimath.org/textbooks/approved-textbooks/) which include three discrete math textbooks, three combinatorics textbooks, three number theory textbooks (one of which includes some abstract algebra), and three intro to proofs texts. The "math in society" liberal arts text, while significantly easier, has some very interesting chapters on mathematics applied to social sciences. Abstract algebra isn't usually considered part of discrete math, but it would definitely be interesting for someone interested in discrete math, and there are three texts on that as well.

I see no pitfalls to this in a homeschool environment with a parent who understands math. If you realize you needed something from an earlier chapter you can always go back and work on it then. For some highly talented students I have actually seen some impressive results from diving in headfirst to something where they were NOT prepared but highly motivated and learning the prerequisites on-the-fly. Edit: I don't know if your boys enjoy actual competition, but it might be fun to work on the starred problems for each chapter individually in a timed manner, then compare. I know I would've found it fun as a kid UNLESS I lost all the time or creamed my partner all of the time. MAN, I wish AOPS had been around then. It would have been SO perfect for me.

Design houses and/or barns? You can incorporate area and surface area for sure, figuring out paint needed, carpet needed ...

No -- the reason that radical equation could result in extraneous solutions which are still in the domain of the radical is because squaring (or raising to an even power in general) is not a one to one function. If you substitute an extraneous solution into a radical equation at the point at which you squared both sides, you will end up with something like -1 = 1, which of course is true when both sides are squared. The exponential function, however, is one to one, so it is impossible to have such things happen. If f(x) = a^x is an exponential function, and x is not equal to y, then there is no way for f(x) to equal f(y). The way an extraneous solution will happen is if you have something like log (x + 2) + log (x - 2) = log 5 and solve by combining to get log (x^2 - 4) = log 5. In this case, the extraneous solution x = -3 gives log (-1) + log (-5) = log 5, and you cannot use the logarithmic addition formula for inputs that are not in the domain. Multiplication is also one to one as long as whatever you are multiplying by is both defined and nonzero. With a rational equation that has an extraneous solution x = a, at some point you have multiplied both sides by (x - a), which is invalid when x = a -> x - a = 0. That being said, it's in the texts because it's a good habit to catch boneheaded mistakes 😛 Some students also find any kind of case-based "if a, do x, if b, do y" very confusing.

Eating under a certain amount for a prolonged period makes it very hard to get sufficient quantity of nutrients. A few low days here and there won't hurt you in the slightest (assuming a generally healthy person without other medical issues who is not overly thin). Eating at too much of a deficit will have some very negative effects on your health in the long run -- but it's important to note that it's the long run. So if you find yourself eating under 1000 a day for several days in a row, it isn't good. But it's pretty normal to eat different amounts on different days. If you don't feel hungry, and you haven't been deliberately undereating or distracting yourself from hunger, you're probably fine.

No place that I have worked has required students to use the accommodation. Even if I have sent a student's test to the center, I am legally required to have enough in the room that if s/he has a last minute change of mind, the test is available in the classroom. Talking with the DSS office or whatever they call it will probably be a good idea.

Usually at colleges, you get a standard "Disability sheet" and either hand it to each professor or it is emailed out automatically. There is usually a "testing center" for proctored quizzes and tests with extra time, and it is the professor's responsibility to send them the tests. It is not usually the responsibility of the student to arrange proctors, and frankly if it is I would transfer to a school that took accessibility more seriously. For pop quizzes, I do them at the end of class; I have my students with accommodations leave when I hand out the quiz, and then they will have the rest of the day (or the next day, if it is a late class) to go over there and take the quiz. Sometimes, people choose to just take them in the regular classroom and stay late; this is up to them. Sometimes, for a private room, you need to schedule. You definitely need to schedule if you require a scribe or a reader. But to just take a test, you do not. There's usually a couple of large rooms where people take tests for all different kinds of classes, and a few private rooms. At my current college, we have a couple of private rooms and one "quiet room" which has a cap on the students. Your dd MAY run into a professor who is not enthusiastic about the accommodations. She will probably get a feel for this when she hands the sheet over -- there's usually a small meeting so that you can discuss how best to make the accommodations work. I have had some people who had professors say pretty rotten things to them, like "well boy I bet EVERYONE would like extra time". But it is rare, I assure you. Quite honestly, if she runs into someone like this, I'd drop the class and take it with someone else.

Oh man, that's rough. I mean, I love math, but I too fall victim to wanting to study everything at once. I know he doesn't want to cut anything, but would he be interested in doing some of the books -- whichever ones you consider more peripheral -- as audiobooks? I'm thinking actually of Cheaper by the Dozen and Gilbreth having his kids listen to language records on the victrola in the bathtub -- he classified baths as "unavoidable delay". Maybe an audiobook while he's in the car somewhere would let him feel like he's cutting less out. Otherwise, there comes a point when you need to say "one of these things has to give, kiddo, which one?" and present him with a list.

Dover has a lot of interesting books at very cheap prices, many of which would be accessible to someone who has just finished algebra. You might order a few -- if they aren't good now, they might be good later, and you won't be out a lot of money. I was writing an amazon list of general ideas for potential use in math for liberal arts classes (in these classes, you can't assume more than algebra 1 if that) and there are a load of books on there, some cheaper than others. I've just gone and looked at my list (it's pretty long) and I started to type in some books but then I said "This is silly" so I just made it public. Try here: https://www.amazon.com/hz/wishlist/ls/WI6UF3LX4VEU. Some of the books have comments; they were more aimed for me personally, but I'd be happy to elaborate if you had any questions. Topics include music, art, geometry, political science, history, game theory, number theory, biographies of mathematicians, graph theory, cryptography, and possibly others that I missed. Another option would be doing a few units from the MEP GCSE level. You might either consider doing some geometry units aimed at prepping for geometry in the fall, doing some stats and data analysis aimed at general cultural literacy, or possibly even the first half of the proofs unit (it's quite short) again to prep for geometric proofs.

Hi mom, I just wanted to say I'm sorry now. No, but seriously, I was this kid and it was all about somehow "look at me I'm smarter than these stupid adults." I am not 100% sure what would have made a difference at that age. Having other high school kids tell me that "everyone hates you because you never stop arguing" did make some difference, but I would not recommend that course of action. I do think explicit instructions or possibly being instructed to write them down might have made some.