A fair amount but it's usually not super explicitly spelled out. There is one scene though where someone is quite clearly getting his jollies while listening to someone else being boiled alive, but this is me looking at it as an adult. I would not have a problem with it but I would really think that it might be a good idea for you to at least skim that scene. I think that is the scene in the book most likely to be objectionable.

Oh, I got reminded. Power basics also has a "basic math" book -- intended for 9th graders who are behind, goes through arithmetic. It *IS* intended for students with learning difficulties, and this is reflected in the instruction being short with simple sentences, but I do not think this is necessarily a bad thing. I'm giving you the Rainbow Resource link to the student book because it includes the TOC: https://www.rainbowresource.com/viewpict?pid=033179

One of my adult friends did Fred trig, loved it, did Fred calc, hated it. He went to Stewart and did better with that.

So someone I know was in this situation. He was very bright and highly literate, but had not received instruction in math, and at 18 was at about the same level as your student. He had placed into "arithmetic" at the community college. He did not start with a college text; rather, he got the supplementary workbooks from the store (golden step-ahead, if you are familiar with those -- the yellow ones with the happy kids with backpacks on the front) -- he worked through the 2/3 book in a couple of days and the 4/5 book in a couple more, then read through/worked through life of fred: fractions. After that, he retook the CC placement test and placed into beginning algebra, which he took and received an A. BCM is a good idea, but it tends to introduce a lot of stuff fast and can be visually kinda overwhelming for someone who has not used a math textbook before. I had a similar book from Bittinger, but it was such a big book he got intimidated. He just found the workbooks less threatening, and they were pretty cheap overall. I don't think they're still published in that brand specifically, but there are definitely loads of them at the dollar stores/bookstores. There was not a lot of instruction, but the very brief demonstration (one per page) was sufficient. I mention this not because there's anything wrong with BCM, but because it may be a good alternate plan -- I would honestly present both as an idea and see which he likes. He may find the juvenile workbooks insulting, or he may find them amusing. Either way.

Lial's or another author's BCM would be my choice, too. I'd probably supplement with something online for practice. But I'd also ask -- has he been instructed and simply not learned, or has he just not been instructed? Is there a chance of learning disabilities?

Aww thanks. Unfortunately I really don't have a single text. I haven't gotten this yet (my professional development fund is limited) and it's sufficiently advanced that there are bits and pieces not covered in standard precalculus textbooks. Most of the trigonometry stuff and some of the linear algebra and complex numbers stuff will be covered in a standard precalculus textbook, but for a fair amount of the linear algebra material (chapters 9-13) you'd really need a proper linear algebra book, and much of the complex number stuff (7.3 onwards) will be covered ab initio in a complex variables textbook -- and the complex variables textbooks usually assume you've had at the very least calculus 3. There are other people who've actually worked through it and I'll defer to one of them to answer. (I *want* the book, but then I want a million books -- if I had a million bucks I'd spend them on a million books)

I think Axler is a good book but too theoretical for some. I've never seen Swokowski but I believe it's similar. Both of them will expect you to extrapolate a bit in the homework problems. Axler's a very good mathematician who wrote his own linear algebra textbook following a different paradigm than the normal. Swokowski (deceased) was better known as a calculus textbook author and the pre-calculus book was written based on aiming the student at calculus. Lial, on the other hand, was a mathematics educator through and through; her books started with arithmetic (bcm) and continued higher. They're very well done for self-instruction and expect less extrapolation. I found Lial *can* be a little bit cookie-cutter as far as "follow the examples" although it's certainly not a bad textbook. A student who worked industriously through the Lial series would be quite well-prepared for most university calculus classes, other than highly theoretical ones at some colleges. They'd definitely be prepared for engineering calculus. Other popular books: I used Cohen's precalculus when I learned it and again in graduate school when I taught it. It fits again in the slightly more theoretical explanation category, similar to Axler and Swokowski. Blitzer's books are also very popular. I loathe their explanations (they are very monkey-see-monkey-do in MY opinion) but their word problems are really great. Some might find them a bit too social-justice oriented but for me (and most of my students at the CC) this is a plus. I mention it only so that noone is surprised. One of the things I really DO like about Blitzer's is the cumulative reviews. I find that this is often lacking. Opinion: I think it'd be great to have more than one book (and these are all sufficiently well-aged that you can get an older edition cheaply). For example, the 5th edition Lial (the one that I have taught out of repeatedly) is available used on Amazon for $4.94 (and there is absolutely no reason to go with a newer edition ... I hate Pearson and their predatory edition changes ... sorry ... what the heck happened to the rant on/rant off emoji). Ideally, I'd get one of the more theory-oriented ones, one of the slightly more hand-holdy "this is how you do it" ones, and something like "precalculus demystified/precalculus for dummies" when I just needed a ***RRRGGGHHH WHAT IS GOING ON*** explanation (I know the name is insulting, but I learned some great teaching techniques from calculus for dummies), and I'd buy them all used. The topics/order are fairly standardized so it's easy to check explanations in multiple textbooks (plus index). Ideally, I'd work the problems out of the one that requires more extrapolation in the homework problems, but if the student is not there developmentally, it should not be forced. However, frankly, getting *any* standard pre-calculus textbook with 3.5 or higher stars on amazon, working through it, and understanding it, and if something is not understood, possibly moving to a different chapter (for example, if someone is just NOT GETTING logarithms, there is nothing wrong with going and doing a trig chapter and then coming back to continue work on logarithms -- they're a hard topic) and then returning to make sure things are *thoroughly* grokked, will result in above-average preparation. Sorry that I wrote a book instead of simply answering your question. Hope you find something useful there anyway ?

Right. But linear algebra does not require calculus itself. The calculus prerequisite is to ensure a sufficient level of mathematical maturity. Some of the linear algebra classes with a calculus prerequisite will apply it to calculus or use examples from calculus, but many don't. Calc 4 is very non-standardized. Most schools don't have a class called that, so if you see it, you should be very cautious with transfer equivalences. In some cases it's a differential equations class, in some cases calculus is spread over four semesters instead of three so it's multivariable calculus, and in some cases it's because the school is on quarters rather than semesters. I haven't seen linear algebra called that though. Calc AB corresponds to two quarters of a three-quarter calculus sequence (calc A, B, C), which is why credit is not given for more than one semester no matter how high the score is. Calc BC corresponds to all three quarters, and three quarters is equivalent to two semesters.

I like the self-teaching or reading idea. Some subjects are highly conducive to this. There are also some books designed for self-study where there are fully worked solutions available. History of Math (there are some really good resources out there now) might be interesting. There are some hoary old classics, such as E.T. Bell's Men of Mathematics, which (as a product of its times) has problems with several -isms, and yet has inspired multiple highly regarded mathematicians. There are many newer books as well. Stephen Hawking has "God Created the Integers", which leads interested readers through several genuine masterpieces (full disclosure -- this is still on my amazon wishlist). Ian Stewart has a LOT of good books. Kline's Mathematics and the Physical World would be an interesting segue into physics before you go for a full-fledged course -- they're beyond the math in it, I'm sure, but it goes through a lot of applications. It's also a Dover book so it's super cheap. Kline's Mathematical Thought from Ancient to Modern Times is also a great mathematical history. I'm going to shut up now but I could keep dumping for ages. In case it's not obvious, I'm kind of a book hoarder. A local math circle might be interesting if you can find one. You'd probably have to find an out of level one. Long-distance mathematical penpals might also be interesting. From where you are, you might also be able to find a local math professor or graduate student who would be interested in weekly tutoring sessions, with the idea that they'd supplement between -- even if you can't afford it, you might be able to snag someone who's willing to do it just because it's a thrill to have highly talented students who like math. You could call a local university's math department and ask. When you do decide to return to proofs, I'd look for something where you can get an instructor's solutions manual. One that's now open-source would be a good choice -- https://aimath.org/textbooks/approved-textbooks/ has several textbooks that were originally published but are now out of print and copyright has reverted -- and you might well be able to get an ISM from the author by explaining your situation. But with all that I've said above, I don't think it's going to do them any harm to take some time *off* math. I still think that I'd make sure that there were interesting math books around. Again, you might contact a local university and ask if any of their professors have old books floating around that they'd recommend. I picked up some of my most interesting supplemental reading books from a professor who was retiring and "get this out of here".

I think going through Foerster's in a diagnostic-prescriptive manner would be a good fit. Roughly, pre-test each chapter where she feels she knows all of it(chapter test is fine for this), look for anything NOT understood and work relevant problems , continue through. If she knows she's missing a section already, work that before you take the test. That way both you and she could be certain that she wouldn't be spending a lot of time drilling problems that she already knows how to do instead of moving onwards through the curriculum. It's a good solid curriculum that includes great word problems and will prepare them for any career pathway that they want.

When you say P(T/M'), do you mean probability of rain on Tuesday given no rain on Monday? (not familiar with this notation) If so, then the answer would be: When you use this, you are already assuming that it has not rained on Monday. Monday has not happened yet, so you do not know whether it will rain or not. This is not yet given. They are two separate events. So you need to multiply by the probability that it will rain on Monday. With the dice-rolling example, they are the same event. Overall, the sums of your four possibilities should add to 1. I'm going to label them for easy reference. This would be better done in a 2 x 2 table, btw, but that's a pain to format. A = P(rain on monday and also on tuesday) = 0.21x0.83 = 0.1743 B = P(rain on monday and not on tuesday) = 0.21x0.17 = 0.0357 C = P(no rain on monday and rain on tuesday) = 0.79x0.3 = 0.237 D = P(no rain on monday and no rain on tuesday) = 0.79x0.7 = 0.553 If you add A + B, you will get 0.21, the probability of rain on Monday. If you add C + D, you will get 0.79, the probability of no rain on Monday. Notice that these sum to 1. If you add A + C, you will get 0.4113, the probability of rain on Tuesday If you add B + D, you will get 0.5887, the probability of no rain on Tuesday. Notice again that these sum to 1.

So for a), they've given you the conditional probability -- so you need to multiply (probability of rain on monday) x (probability of rain on tuesday, given that it rained on monday) For b), I think it's easier to think about it this way. You have two choices for rain on Monday -- either it will rain on Monday or it will not. You also have two choices for rain on Tuesday -- either yes or no. This means that overall there's 4 possibilities. You want P(rain on monday and rain on tuesday) + P(no rain on monday, rain on tuesday) You already know the first one because that's the answer to part a. What is the probability that it will NOT rain on Monday AND it will rain on Tuesday? You'll want to start with the probability that it will not rain on monday (1 - (0.21)) and multiply by the conditional probability of rain on tuesday (that's your 0.3).

She could always go back and do chemistry later, couldn't she?

Sorry that I'm late, but this is actually not as bad if you use trig identities. The easiest way to do this is to recognize this as one of the formulas for tangent (x/2), which can be directly evaluated at zero. If this doesn't pop into mind, you can also get there by multiplying by the conjugate of the numerator. Multiply numerator and denominator by 1 + cos x. Numerator is now 1 - cos^2 x, which is equal to sin^2 x. Denominator is (1 + cos x)sin x. Cancel sin x from the numerator and denominator, and you now have sin x / (1 + cos x), and you can directly evaluate this at zero. This trick (conjugates) can often pop up in trigonometry problems -- if you see anything that looks like a fraction with 1 +/- sin x or 1 +/- cos x involved that is not immediately solvable by other means, see if this helps.

Yes ... but I wouldn't assume that it was going to be boring/tedious without at least *trying* to make it interesting :) I'm also basing my comments on your statement about your own experiences in algebra through calculus. I had the same experience as you did in calculus, and also in linear and diffeq, FWIW. It was easy, but rather uninteresting and tedious. Then I took discrete and discovered how much fun there was and changed my major that semester. But going back through, and teaching these topics (algebra/precalculus/calculus), and researching alternative ways of doing and explaining, and looking at the fun theoretical problems in my calc textbook (it's not AOPS but Thomas is a pretty decent book, much better than what I used), well, there's a lot more to it than I thought there was, and a lot more that can be done at a relatively elementary level, and I very much wish I'd seen all of my math that way. There are also a lot of very interesting links to other math topics within elementary algebra that kids can totally work with but we omit because ... well, that would lead into a rant, but ... anyway, we don't do them. That being said -- if the pre-algebra/algebra books just aren't for them -- I would totally switch away. I would just try it first.

I agree that I'd try to make algebra exciting instead of treating it like "eat your boiled spinach and then you can have your fun stuff". There are so many interesting things that are totally within the scope of the standard curriculum but are omitted because the focus is on dragging everyone through the basic level.

I really find "doesn't matter" somewhat specious. It does, somewhat. But I would restate it as "may matter less than you think". For example, if you want to be a researcher in a subject, it isn't necessary to go to the Ivy Leagues, although it's nice if you can. But it's important that you attend a school with a sufficient student population to support graduate-school-bound students. If your school only offers barely enough classes to complete the major, or if they don't teach the classes at a sufficiently rigorous level, it matters. Major state college? Probably fine. Compass direction State College? Probably not the best choice. This is especially relevant for junior and senior years, unless you are a freshman who's entering with a couple of year's credit in your major (in which case you're probably looking at flagship state colleges anyway). On the other hand, if you want to be an elementary school teacher, you are not expecting to be paid well. You should probably choose the most affordable program that has a good passing rate on certification exams and job placement. If you want to go into politics or some kinds of business, the connections are super important. If you want to go into performing arts, you really, really need an institution that can support your learning and provide you with the exposure that you need. Here, though, it's actually pretty common for regional schools to have some absolutely top-notch teachers/practitioners/programs (for example, one of the CC's has a just amazing art program), but you do need to do your research. And so on.

There's also no reason that he *has* to do biology next year. You mentioned conceptual physics -- that would be a completely reasonable 9th grade class, look fine on a transcript, but have sufficient overlap with a rigorous physical science class that it should be a bit lighter for him and give him some wiggle room on the rigorous classes. It also leaves breathing room later on if he wants to reduce science.

Is a tutor out of the question? I think it would work a lot better to get whatever math for liberal arts book they use and hire a competent tutor to go through and pick out the prerequisites necessary. There are a LOT of things in the standard algebra curriculum that are not at all necessary for success in this type of class. I wish there were a published pre-lib-arts curriculum. Universities have been experimenting with this (and it's going quite well in many cases) but I don't think anyone has a specifically designed curriculum yet.

Contact the state universities in the same system and see what they recommend for CC transfers. For example, we would recommend someone wanting to come in from a CC for math transfer partway through sophomore year or plan to enroll for the introduction to proofs class in summer session, because if they cannot move into the proof-based classes for junior year, it will be an extra year to graduation. I would suspect that one of the reasons the courses will not transfer in CS is because they are more applications-oriented classes (for an information systems degree or some such) which is still involving computers, but isn't really computer science. Diff Eq is differential equations. It is commonly offered at CC but it is also common for it to not transfer. We don't accept it because around here it is usually taught at the CC's as a rather "cookbook" class. But if someone challenges the transfer and brings in copies of their syllabus and examinations, and we can verify that it was at least at our level, we will accept it. It is wise to save all graded assignments from CC's until you have had your transfer credit evaluated, because it is not at all uncommon for colleges to make exceptions to their transfer policies if the class actually was sufficiently rigorous. If it was not sufficiently rigorous, you really do need to take it again, but often they can explain precisely what your cc was missing.

I had no problems being in college classes at 14/15. I would worry somewhat about maturity of the course content for classes in the humanities, especially art/literature. But not about the guys so much -- unless you have reason to be concerned about your dd returning their interest.

You need to be enrolled in a program that leads to a degree for title IV aid. See https://ifap.ed.gov/150PercentDirectSubsidizedLoanLimitInfo/FAQ.html#MEP-Q11 We have had to recently concoct programs for pre-majors and admit all the undecided students to those majors. We can no longer have (for example) pre-nursing for people who plan to study nursing. Rather, they get admitted to a nondescript "health studies" degree.

Yes. And laying the groundwork for her success in college classes. Don't worry about AP/DE classes. She can prep for medical school just fine starting with college classes in college instead of high school. Normally I'd agree about not doing conceptual physics but I think because of her non-standard situation with math it might be the best solution for 10th grade. Physical science would look weird after biology and I don't think she's ready for a standard chemistry class with where she is in math. Then you could do a standard chemistry in 11th, and make your decision about 12th later after seeing how she's doing in math, and also assuming her goals don't change.

For med school, her college record is going to matter a lot more than her high school record. Her high school record only matters as far as getting admitted to a decent college. So it is even more important that she be prepared to enter college and *excel*. There is more algebra 2 on those tests than there used to be, but not a huge amount. With this in mind -- I would not shortcut. Here's how I'd accelerate until algebra 1 is finished: Work twice a day, for shorter periods of time. I would say 40 minutes twice a day. Splitting it up like that increases retention while your brain is fresh. Work on Saturdays, for math only. If two sessions don't fit on Saturday, squish in one. Every day that you're training helps. It's like practicing the piano. Work through breaks. If breaks get very busy, still try to fit in one session. Going for an algebra text like Jacobs isn't an unreasonable idea. If you go for that, I'd keep the pre-algebra text that you have on hand in case you need extra practice on a specific topic. After algebra 1, I think (assuming she's still on board) you could do geometry and algebra 2 concurrently, just treating each as one class. p.s. yes, untreated ADHD can absolutely lead to anxiety. There's a lot of "why am I so stupid/everyone else does this so easily/RrrRRRR why can't I just make myself work" and very justifiable concerns about "what am I forgetting/what am I missing/what did the instructor say while I was spacing out".

Haste often makes waste with mathematics learning. If she does an accelerated algebra class and doesn't learn it and has to re-take it, she will be further behind than she is now. She just can't learn it faster than her brain is ready to grok it. I honestly don't think working in pre-algebra in 9th grade qualifies as "very behind". I would say someone working in a solid pre-algebra in 8th is "on level", even though it's the current fad to rush everyone through in 8th. I think it would be better to simply continue working through (including summers) until she finishes algebra 1. At that point, you and she can decide whether to try for algebra 2/geometry concurrently (works for many people, but you do need to allocate time for two classes) or move through. If she completes one class per year, she would have 10th grade algebra, 11th grade geometry, 12th grade algebra 2. She would need to do a conceptual chemistry class or postpone it until senior year, and probably wouldn't be able to do anything but a conceptual physics. But if she *understood* what she learned, she'd be ready to move into college algebra and do well. My developmental classes are full of students who "completed" pre-calculus in high school and yet don't understand algebra 1. If four credits starting with algebra 1 are absolutely required, she could double up in senior year with a non-ap statistics class (or a DE class in spring semester for either stats or math for liberal arts; she'll easily have the algebra for it then). If she's not heading for a STEM-heavy major, I would also have absolutely no qualms about using a rather light geometry (MUS springs to mind, but there are other options).