After AOPS intro books
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pettifogger
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While starting to learn the beginning of AoPS Precalc would not be a bad idea, I would personally not recommend it until they have a very good grasp of algebra. Despite its smaller size, the precalc book has more difficult material than Intermediate algebra (for the most part). The book is very different from a typical K12 precalculus book. It essentially teaches three topics in great depth, connecting them all together: 1) Trigonometry in the first third of the book, 2) Complex Numbers in the second part of the book, and 3) Fundamentals of Linear algebra (vectors, matrices in both 2D and 3D) as well as a final chapter on solving tough geometry problems using vectors and other tools learned in the book. Some of the problems (not all) can be very difficult, and a few have appeared in past high school Olympiads such as the USAMO, etc. I think the most important principle is that your DC should enjoy math (CMMS is a lovely little book, but I don't think it was a great idea to have made them repeat problems they missed 4 times, as that is not a recipe for enjoyment...) One way to do that is to solve lots of interesting problems at an untimed and relaxed pace. A few ways that could be done is by perusing the AoPS site and cherry picking problems that look interesting to them (i.e from the past AMC 8 or even AMC 10 questions) or perhaps via playing Alcumus (which itself contains a very large collection of problems, many from past contests). Earlier in this thread I have recommended for you Anna Burago's Mathematical Circle Diaries. If your DC finished the Intro series, Year 2 (the second book in the two book series) would be more appropriate. It contains many interesting problems organized by various topics, that are excellently curated, similar to AoPS. It introduces students to some difficult ideas that most don't see until a discrete math course in college (i.e Pigeonhole Principle, Invariants, Parity, Combinatorics, Graph Theory, etc) via approachable problems aimed at advanced middle schoolers. Some of these topics are really very lovely, but sadly AoPS did not include them in their books (other than perhaps in their Intermediate Counting and Probability book, which is an amazing but challenging book, on par with their Precalculus or Calculus book). |
Anonymous
| Although a student taking AoPS precalc without the intermediate algebra course would be unprepared, the same doesn't apply for more ordinary precalculus textbooks like Axler or Cohen. A student who has completed AoPS' introductions to algebra and geometry should be prepared for them |
Anonymous
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Thanks! I will check Mathematical Circle Diaries Year 2, and a precalculus book by Axler or Cohen.
Yes, I just want DC to enjoy math. No need of competition. |
Anonymous
I wouldn't abandon the AoPS Intermediate Algebra book altogether. There's a lot of great material there, and it would help your child get a great foundation. Instead, view it this way: The problems they work out in the chapter + the Review problems would alone act as a pretty comprehensive Algebra II course. There's no need to even do the challenge problems unless your child really wants to look at them. There's no need to take a cover-to-cover and all problems approach. |
Anonymous
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It's true thay AOPS Intermediate Algebra has a lot of more abstract math (like factoring and solving polynomials, conic sections, and functional equations), that isn't relevant to the STE part of STEM.
If you aren't focused on math for math's sake, you can skip a lot of the AOPS material and problems, and stick with Khan and Brilliant type stuff which focuses more on the math for engineering and technology. |
pettifogger
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Polynomials are not relevant in science and engineering?? I beg to differ. |
Anonymous
"Polynomials" are covered in Intro Algebra. Professionals use calculators and computers for numerical methods, not Vieta's Formulas and rational roots. |
Anonymous
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Doing AOPS intermediate algebra book by skipping challenge questions sounds a good idea.
Maybe I can get math circle book, precalculus book, and intermediate algebra book, and let DS choose which to do next. |
pettifogger
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Your comment is quite misleading. Polynomials were barely touched upon in the Intro book; there were only two sections of very basic material. The core and heart of polynomials, namely polynomial division, polynomial roots, including the Fundamental Theorem of Algebra, is all covered in the Intermediate Algebra book. In fact, a big portion of this material is standard K12 school material, required in an algebra 2 or precalculus class and is critical to STEM. This material is of course also found in Khan and any other algebra textbook or algebra course (though taught much more procedurally, and/or by rote). Where I do agree with you is that yes, in addition to what I just described, The AoPS Intermediate Algebra textbook goes way beyond a typical Algebra 2 course, even delving into high level math contest topics with chapters on advanced inequalities and functional equations. The Intro to Algebra book, while very good, does not come close to covering all the algebra students need to be successful in STEM, that's why it is an intro course. While it may be ok to take a school based Precalculus course after taking the AoPS Intro series books, (largely because many algebraic topics are sometimes taught in a K12 typical precalc class), I would not recommend directly going into AoPS Precalculus just from the Intro Series. |
Anonymous
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If you want a more "normal" path through the books:
Intermediate algebra textbook: Ch 1 - 4: review Ch 5 - 6: all Ch 7.1 - 7.5 Ch 9.1 - 9.2 Ch 10.1 - 10.5 Ch 11.3 -11.4 Ch 13.1 -13.3, 13.5-6 Ch 14.1 Ch 15.1 - 15.3 Ch 16.1 - 16.3 Precalculus textbook: Ch 1 - 2 Ch 3.1 - 3.4 Ch 4 - 7 Ch 9 - 11 |
pettifogger
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- I'd very strongly recommend 12.1 - 12.3 (first half of chapter) in Intermediate Algebra -- this is very important and fundamental material on inequalities. - (Also, it's hard not to recommend Vieta's formulas in 8.3 as they're just too beautiful) - Chp 12 in Precalc is pretty fundamental 3D material that builds on all the work done in the prior vector chapters... very useful later in calculus and physics (and it's usually good to see it more than once as it's not easy to learn it well the first time one is exposed to it). |
Anonymous
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I'm OP.
We have been doing Intermediate Algebra book, and now at Chap. 14. We are skipping all challenge questions and Chap. 5, and will do up to Chap. 16. I originally thought this book is too difficult, but actually it works for us. |
Anonymous
I can understand skipping ellipses and hyperbolas, but why skip 5.1-5.3? |
Anonymous
If Khan Academy / school math is too easy, and AoPS is too hard / unmotivating, maybe your kid is good enough at math for their level of interest, and should focus on other interests like science. |
Anonymous
Because the book preface says chap. 5 is less important, I thought. We may revisit it later. By the way, we did Alcumus intro to Algebra section before this book, and for some questions we needed to look at the intermediate book. By doing so, we thought that the intermediate book is doable. |