I would start pre algebra when the student is really ready for pre algebra -- and see where that takes you. (The AoPS site has pretests for every book. I strongly recommend using them if you're not sure when to start a book.) It's really impossible to tell how fast or how far a young child, even one who's very good at math, will go. At every level, other things being equal (health, attitude towards math, relationship with teacher), the student should be doing work that's challenging and interesting but not overwhelming. If it gets too easy, speed up; it it starts to become overwhelming, try something different, slow down, or take a break. The goal is not just to finish calculus at a certain time - there's much more math "after" a first year of calculus, and much "before" it, too. Also - with AoPS there are some books one could skip to get to Calculus faster -- such as Number Theory or Counting and Probability. I wouldn't advise doing that, but it could be done, for instance if needed for a calculus-based physics course.

If you have a student who is ready for the AoPS course and whose goal is to qualify for one of the USA(J)MO exams (I assume this is what you mean by "pass" the AIME, since it isn't really a pass-fail exam), then definitely I would choose AoPS over the EPGY course. Your student is in the target group for AoPS. Just make sure the student understands that the challenge problems are supposed to be hard and it's OK not to get them all -- many of them will take some time, work, and insight. In the AoPS sequence, the Intermediate Algebra (or Algebra 3) course covers much of the algebra that would normally be covered in a precalculus course in a great deal more depth than usual. The Precal course covers trig, complex numbers, and linear algebra. (My homeschooled son was a five-time USA(J)MO qualifier. He gives AoPS books and courses much of the credit. But definitely no guarantees -- neither AoPS nor the AIME is for everyone.)

My older son used Saxon Algebra 1/2 when he was 7-8 and it worked well for him -- the only time we've ever used a Saxon math book. He was also a strong reader and did it mostly on his own with me giving the homeschool packet tests after every 4th lesson. He corrected his own problems and figured out if his mistakes were conceptual or careless (usually they were "copied wrong"). We took breaks and did scattered chapters from Harold Jacobs's Mathematics: A Human Endeavor, and that way stretched pre-algebra out to 1 1/2 years. AoPS didn't have algebra, much less prealgebra back then. He did VideoText Algebra next, then when he was 10 he started taking AoPS online courses. If you want to introduce problem-solving along the way you might look at some of the MOEMS books -- they sell them on the AoPS website. They introduce strategies for solving various types of problems that come up in contests and AoPS courses. I'm using the AoPS Prealgebra book with his younger brother. He's not as mathy as his older brother (he's almost 11) but he really likes it. Likes the videos, too.

I really like MEP (downloadable from the website cimt.plymouth.ac.uk/) and am using it for the third year with my now-third-grader. It's a British adaptation of a Hungarian school math curriculum. When used with the accompanying lesson plans, it's a fun, fast-paced course with 175 lessons/year (counting the 5th day reviews). I like the tone of the lesson plans -- they're specific enough that you can follow them as written, but the "official" advice to teachers is to adapt the LP's to their own teaching style and students, so if you're the type who prefers to do things your own way you can do that and have a lot to work with. From the third grade there's an online interactive version of the student workbook, though we haven't used that yet. One of the latest learning theory articles I've read suggests that moving fairly quickly from one aspect of a subject to another improves learning by increasing the number of connections the student makes. The MEP lesson plans, which consist of 7-10 short activities or exercises on a related topic each day, totally feed into this approach to learning. Each exercise is corrected/discussed before moving on, and mistakes are approached as a normal part of learning new math and opportunities to clear up misunderstandings. There are mental math problems, word problems, and logic puzzles. Singapore Challenging Word Problems can be a good supplement for some kids. (A downside for homeschoolers with large families is that it's definitely teacher/parent-directed, with lessons lasting 25-45 minutes.) MEP is an extremely thorough program that includes discrete math (counting and probability, number theory), set theory, and so forth even in the early years. I like the way that topics are integrated -- inequalities are taught along with equations and so forth. There's a fair amount of "fact" practice -- probably too much for some kids and not enough for others but that's something you can supplement on your own if you need. Students are taught several different ways to think about each operation and can use the one that makes most sense to them at a given stage. Although there are a very few things that seem unique to MEP, in general the program teaches standard notation and "proper" mathematical language from the beginning. I think this does a lot to make high school topics seem less strange when a student gets there. A student who masters Y1-Y6 should be ready for algebra, but there are optional Ys7-9 that review prealgebra topics in depth for students who start later or otherwise aren't ready to move to high school math after Y6. I do recommend that MEP users print out the "scheme of work," a week-by-week list of topics, and refer to it often. It helps to keep track of where you are if your child moves more or less quickly than the lesson-a-day pace. (While the LP's make it clear a child doesn't need to have mastered everything to move on, in the homeschool setting it seems to work best if the child achieves a fairly high degreee of mastery.) Some of the resources on the MEPhomeschoolers Yahoo group files are also useful -- especially the videos of Hungarian or British classrooms.

Kathy, Thank you. This is extremely helpful -- and encouraging. My last experience with AP exams was taking them myself 30 + years ago. I'm looking at the resources you've recommended. Just out of curiosity, what was the text your children used? Grace

My son (14, 9th grade) is planning to prepare for the BC calculus exam on his own this year. He's quite capable -- took Advanced Precal last year at one of the top public high schools in the country and did very well. For several years before last year he'd mainly done AoPS online courses but can't schedule the AoPS Calc class this year on account of a standing time conflict. He plans to use David Patrick's AoPS text as his main resource, but he will need practice problems specifically to prepare for the AP exam as well. I'm looking for a book with good problems and an available solutions manual -- one I can use to quiz him on the basics, to keep him on track, and to make sure he gets plenty of experience working the kinds of problems that show up on the test. I've had three semesters college-level calculus and four of physics with calc -- but that was decades ago. I expect to be able to give him guidance if he gets stuck on a concept, but not really to be able to teach him from the ground up. I'm looking at Forester's -- I know it's pretty basic, but also that it's designed to prepare kids for the test, and that's what I need. I know that other texts such as Larson's probably do better presenting the theory -- but he'll get that from the Patrick book. Key Curriculum press sells teacher materials for both the old and new editions, the old one being much less expensive, of course. I think that part of the difference between old and new may be that the new includes more graphing calculator applications. Does anyone have experience with the different editions? In my mind real-world and graphing calculator experience are good (this kid has a bit of an anti-tech "purist" streak, and he's won three lovely programmable graphing calculators in math competitions) but I'm wondering what the real difference between these books is. Does the new one do a much better job of preparing students for the portions of the test where graphing calculators are allowed? Or is the main difference that it has even more easy problems to lead students into the harder material (we don't need that)? I'd appreciate comments from anyone who's seen the two editions -- as well as anyone who has successfully coached a child through the BC Calculus exam without an external class. Thank you.

I saw the author of "Math Doesn't Suck" etc. speak at MATHCOUNTS nationals a couple of years ago -- preaching to the choir, and a heavily male choir at that, but she was quite entertaining and enthusiastic about kids doing math. Haven't read her books, but I wasn't under the impression that they're just for girls, though I'm sure girls are a large part of her intended audience. A young, inspiring woman mathematician that some girls (and parents) might enjoy reading about is Melanie Wood, the first U.S. girl to compete in the International Math Olympiad and now I believe a professor at Stanford. The Wikipedia article about her has some good interviews and articles listed in "external links." I like the one from the "Girls' Angle Bulletin" -- as well as the article by someone else that precedes the interview. Melanie answers questions about doing work in a heavily male field and what that's like. (She was a cheerleader in high school -- so definitely not one to avoid "girly" things.) Grace A

MEP is a British program, and in Britain kindergarten starts at 4, first year at 5. BlueMorpho

My son used VideoText after Saxon Algebra 1/2. It took about 12 months, but he finished shortly after turning 10 (very mathy kid). He took the author's advice and did a few problems at the beginning of each problem set, then just did the odd problems. Sometimes he went back and worked the evens if he needed to. Overall we found this to be a very solid program. It doesn't have the most challenging problems, but my son feels very well-prepared for the kind of non-standard problem solving he's done since in the Art of Problem Solving courses, AMC tests, and MathCounts. As others have said, Module A is essential. In fact it's likely that some of the language introduced early in the module will not be familiar to your daughter. Encourage her not to get frustrated because she thinks the first module is supposed to be "easy." The rest of Module A is a review of things like commutative and distributive properties -- but these really need to be second nature for an algebra student. I'll mention that before starting VideoText I had acquired a used Lials Introductory Algebra and found that the order of presentation was very similar to VideoText (after module A and up to somewhere in module E). At a couple of points when my son had trouble really catching on to a new concept (as evidenced by not performing up to par on a test), I read through the presentation in Lials with him and had him work extra problems. He always did much better on retaking the test after this. For later on in the course, when we got to conics and functions, I found a free online graphing calculator (Googled "online graphing calculator"). We haven't seen the need for a handheld graphing calculator (with a tiny screen), though the schools here use them from algebra I or II on. But it was instructive for my son to see what these calculators can do and to see the graphs of equations and how modifying coefficients or constants altered the graphs. Good luck, BlueMorpho