I think some give up on BA/AOPS too soon. What is sometimes identified as not having a solid foundation is just not memorizing facts at the same pace as old fashion school learning. A good example of this is multiplication. BA moves on from basic multiplication long before most kids have memorized many multiplication facts. The important multiplication lesson is what multiplication is, how it works, and how to figure out what 6×3 is without resorting memorization. Memorization comes with use through the months or years. In fact, one of the first mutilation lessons builds the multiplication table that is ment to be available for use from then on. I also don't understand what people mean when they say a kid is good at math but doesn't like the inductive approach. Inductive thinking it what mathematics is. You can memorize algorithms, formula and arithmetic facts without inductive thinking, but thats as much mathematics as engineering is physics, or technical writing is poetry. The goal in teaching mathematics is to teach logical inductive critical thinking skills. We often fail at this and fall back on the minimum skills skills in applying facts to get by in life. To be clear, all this is not to say that BA in the only way, or even the best way, but to say BA's intentions are in the right place. We shouldn't give up on teaching the heart of mathematics before we even try. What most of us had in school shouldn't be good enough for our kids if we have the resources to provide better.

I am not familiar with BJU at all, but many Pre-algerbra texts/programs don't necessarily introduce a lot of nee material, but instead focus on developing mathematical maturity, more adult learning styles and developing problem solving skills.

I doubt your daughter will get a "black mark" for not responding, but opening a conversation opens the potential for her is to stand out when it comes time to apply. Universities generally treat their students and perspective students as adults. They have little interst in dealing with parents, or, if it is a selective school, with students who rely on their parents to ask questions and advocate for them. Selective schools are interested in independent self motivated students interested in taking control of their future. As a homeschooling parent it will be an uphill battle for them to see you in a similar role as a high-school counselor, and nor as a parent advocate for your child. When asking about for advice on things that have no publicly available written policy, lower level employees will be hesitant to put anything in writing. I would just call the admissions department (or better yet have your daughter call).

If he is great a math, but dislikes it, it might be time to consider different curricula. Saxon can be great at stomping out any love of of math for those that are good at it. (I know there are exceptions). I suspect there are many adults out there who think they hate math, but actually just hat the plug and chug and repeat Saxon method. (I know it works well for some). It sounds like he is intelligent and motivated. Check out a curriculum designed for that kind of kid.

AOPS is written to the student, so should be fine on its own. The student should be doing the problems, but that doesn't mean that you can't talk them through together if the student is leading the conversation. One of the points of AOPS is to teach "constructive struggling". If you don't allow the student to struggle, they aren't getting the benefit of the curriculum.

Although it is counterintuitive, some children who struggle a little with math because they forget procedures do better with a deeper, more challenging curriculum. When they have to focus more on the why, and work out procedures for themselves (with help), they "own" them and math becomes less about memorizing procedures and more about establishing a logical framework.

NT and C&P are not needed for the traditional U.S. education system core sequence. So if your goal is to check the required boxes and be succefull on standardized tests than they are not needed. They are however, just as important for achieving a thorough understanding in the portion of mathematics accessible to a pre highschool graduate student. I would encourage you to fit them in if you can.

It depends on what you see the core purpose of a pre-college education to be. If you think the driving focus should be on narrowly defined career readiness, then you are right, there is little need for most to go very far in math (even most engineers only need to know how to calculate things). If you belive in a liberal education (liberal as in the classical sense, not in the modern American political sense), that has a goal of producing productive and empowered citizens instead of a democracy than the picture looks different. Our public education system has been modeled on the liberal model but has been failing. It has shown to be more able to achieve good results when career focused.. In other words I don't know whether your misgivings are right or wrong...I countiue to teach my kids higher math, because they like it, and to keep all their options open as adults.

Statistics is a good option, but so are number theory, probability, and "pre-calculus", which despite its name covers plenty of stuff that is very useful in own right (trigonometry and analytic geometry).

It sounds like you should give AOPS a shot then. Although math is always puzzleish, there are no, or extremely few, problems presented as puzzles after BA 5.