UAH would also be affordable w/their NMF scholarship. (Huntsville is a wonderful city. Cummins Research Parks is the 2nd largest in the country and the businesses there have an excellent relationship with the university opening up research and internship opportunities.) http://www.cummingsresearchpark.com/
@evergreen5, yes, you’re right! So she was very close to a perfect score; likely, the SAT will be an 800.
I confess my view is skewed by my son’s experience as well as my experience as a college consultant.
@ucbalumnus Just to be fair some of the links you listed were for undergraduate courses only. The OSU course listing also seems to include graduate courses. At least I assume that are what 7000 level courses are.
True – but some are LACs that offer only undergraduate courses.
But for a highly advanced in math student who wants to continue in math, the availability of a large selection of upper level undergraduate and graduate courses is likely to be more important than for a student who goes into college having just completed AP calculus AB or BC in 12th grade. It looks like tOSU is the best academic fit among the affordable safety schools.
Here’s a potential problem with that approach @ucbalumnus :
https://gradsch.osu.edu/handbook/4-1-course-credit-marks-and-point-hour-ratio-course-credit
Then the detail is how commonly undergraduate petitions to take graduate level courses are granted, in the likely situation that the OP’s student will be in. (Of course, a school with no graduate level courses means that there is no possibility at all of taking them.)
Regarding “the credit for the [graduate level] course is not used to meet baccalaureate degree requirements”, it is likely that a highly advanced math major would be taking such a course as a free elective, having already completed the usual undergraduate level math major requirements.
But also, tOSU has a rather extensive selection of upper level undergraduate math courses (4xxx and 5xxx course numbers), so even if the graduate level courses are not available, it is a strong choice in terms of academic fit for a highly advanced student in math.
The grad courses as an UG issue is definitely a real problem at many universities. It should be highlighted and investigated. @ucbalumnus is equally correct that most LACs are not going to offer enough courses to make them viable options, either.
When students enter with an unusual number of UG core credits completed, it can make finding options (outside of the elites that teach at a different level) difficult. There are universities, however, that are willing to work with students entering in with this scenario. I think that OU’s HTC is one such program. I can’t remember for sure, though. It has been too long since we made contact. Bama with the NMF scholarship might be affordable. (not sure if they changed those benefits or not.) Bama does allow students to take grad level courses.
My son’s safety schools were UPitt, UT Dallas and Baylor because they offered graduate level math courses and the potential for full ride scholarships.
Wow! This went in all sorts of directions! I hope that by the time I’m helping kid #6 get to college, I’ll be as knowledgeable and valuable as all of you here. 
A few answers to questions above, sorry for the questions I miss:
The scores above are PSAT scores, so 760 is the max on each section. She’ll take the SAT in March. Her SI for NMF makes it highly likely.
Her interest is mainly in pure math with a growing interest in computer science. Engineering doesn’t appeal. Grad school does.
Beyond the basic calc courses, she’s not concerned about any of the other credits transferring as much as she’s concerned about being able to stretch herself in undergrad. She realizes that, say, her Linear Algebra classes at Otterbein are a totally different animal from Linear Algebra at a Princeton or an OSU or others, and she’s happy to take these classes again if they’re going to take her to a deeper level. If her whole curriculum thus far only counts for high school, that’s fine . . . as long as the “retakes” aren’t virtual repeats. (I realize that sounds naive. Darn these delight-driven learners! ) She just wants to do all the math she can now, and keep doing all the math in college. We know this probably means lots of individual contact with the places that look interesting, to see what they’d do with her. Your comments here are really helping us figure out what things to research and what questions to ask.
I’m thrilled to hear about happy endings for homeschoolers and how they got there. It gives us hope!
Don’t necessarily rule out LACs. They offer math majors, and have bright students (my daughter just declared as a math major today, so irrationally proud parent speaking). Many schools offer cross-registration, too. At Bryn Mawr you can take classes at Penn, for example.
The problem with Bryn Mawr is that the maximum scholarship is $30K with tuition of $49K. Unless they qualify for need-based aid, there is no way to bring it down to $10K.
OP, one other thought…make sure that UG scholarships will cover the cost of grad courses. Some will. Some won’t. Grad courses can add up quickly if not covered. Also make sure to check how many cr hrs are required to keep scholarships. 9 hrs of grad courses is considered full-time, so the mix of UG and grad courses to be keep scholarships can also play a factor.
@AmyC, my son was the same way with math (and physics). He was definitely self-directed. When he ran out of classes at the local community college midway through sophomore year in both math and physics, we had to get creative.
For math, we opted to use John Rosasco one-on-one online for 2 1/2 years. We got some financial help through an organization, but it was pretty unschoolish (once a week or once every other week), as my son made his way through upper division differential equations, dynamical systems and real analysis.
My son also was a mathlete, and did AMC/AIME three times as well as the local math olympiad and ARML (his team did really well in his sophomore or junior year).
For physics, I reached out to a professor at the local state university near us, and he invited my son to audit his upper division physics courses (3 semester’s worth) for free. In turn, because my son did really well in his class, the prof invited my son to do some grad level research during the summer and senior year.
Of course none of that transferred, but we were just happy to have found workable solutions to keep my son reasonably challenged without needing to graduate early from high school. I’m guessing it also made him a more interesting candidate.