<p>Your list is good but considering your math classes (self study for calc BC is impressive and should definitively be mentioned) and general application criteria, I think you should absolutely apply to Harvey Mudd (they’re very supportive of women scientists and I’m sure you’d thrive there) and MIT.
<a href=“http://www.math.hmc.edu/[/url]”>http://www.math.hmc.edu/</a>
I understand the cost issue but when colleges really, really want someone (ie a young woman who has a gift for math) they may find ways. Not always but often so it’s worth a try.</p>
<p>To increase your odds look at schools outside your geographical area (500+ mi away).</p>
<p>If you can pull A’s in these math classes at college, it’ll trump your AP scores. Alternatively, in addition to your math classes you could take one gen ed college class in the social sciences or humanities and get an A in it. To be better for you it’d have to be a seminar-style class so that you could get a recommendation from the professor that would show your 3 on the AP test does not reflect on your ability to think critically outside your specialty field.</p>
<p>Regarding ECs, you’re in a particular category due to your family situation and gift for math that makes you enroll at a university rather than high school. You <em>could</em> conceivably try to get involved in a club at your DE university, if only the math club or even the math honors society or something (not sure if it’s possible but it’s worth asking.) As a DE student you should be allowed to participate in most activities at your college. However due to your special gift for math, your (very relative) lack of ECs would not play against you as much as for others.</p>
<p>However don’t assume that, just because a university has a grad program and a LAC doesn’t, that the classes will necessarily be more advanced at the university.
Some LACs essentially offer grad-level classes to juniors and seniors because their 1st and 2nd year classes are more advanced and faster paced.
At some schools, students start with Calc I. At others, they start at Precalc because most students have done Algebra 2 only and only in some cases a course higher than that. At other schools, almost everyone’s done Calc BC and starts at Calc III except that in these schools what is Calc III elsewhere is just called Calc II. The pace can be different too from school to school (cf. the calc sequence done in 3 semesters, but can be found in 2.) </p>
<p>In short, your work on a college list is not finished yet 
since it means going to each school’s math dept page and looking at their course offerings, then emailing the dept. chair to let them know what classes you’ll be taking senior yearas DE and what would your freshman sequence would be at their school, then point out 2-3 advanced classes that look interesting and ask how often they’re offered (every year, every other year, less than that?)
You can be SURE that at many LACs this will ring a sweet bell in the Department Chair’s ear (“we want this student”) which can only help 
if you eventually decide to apply.
If you like the answer regarding the advanced classes, look into the merit aid + email about it to the relevant person, asking also if they require non custodial parent information. Don’t forget to email back to ask the dept chair whether there is merit aid for math students (= “showing interest”, #2).
Then get into the Common Data Set and/or email to ask whether being first gen is taken into account in the admission process. If yes, it will boost your odds. (Do this for every school on your list, BTW).
This should produce a list of 3-5 LACs to which you can apply which offer interesting aid and have the math offerings that would “fit” your needs.</p>
<p>You could also make a thread titled “which LACs in the top 100 have the best math programs” and schools ranked 40-75 would be likely to offer merit aid and admit you so you would look at their math program opportunity.</p>
<p>Oxford/Cambridge: it’s $22,000/year for tuition for math, and if you score high on their special tests and interviews you CAN get scholarships (for internationals, they’re based on merit only). However the Further Math A Level is pretty advanced, so you’d have to be further than that and show “merit” at a higher level than what you’d have to show in the US - cf.
[A</a> & AS Level Mathematics - Further](<a href=“Cambridge International AS & A Level subjects”>Cambridge International AS & A Level subjects)
[AQA</a> | Mathematics | A-level | Further Mathematics](<a href=“http://www.aqa.org.uk/subjects/mathematics/a-level/further-mathematics-6360]AQA”>http://www.aqa.org.uk/subjects/mathematics/a-level/further-mathematics-6360)
[Further</a> Mathematics - Wikipedia, the free encyclopedia](<a href=“http://en.wikipedia.org/wiki/Further_Mathematics]Further”>Further Mathematics - Wikipedia)
Ex. of admission odds at Cambridge:
[Application</a> statistics](<a href=“Study at Cambridge | University of Cambridge”>Study at Cambridge | University of Cambridge)
You’d do math only and all the time so it depends if you want a well-rounded education or not. Also, I agree with ClassicRockerDad that the AP scores will hurt you since they want 3-5 5s.</p>
<p>Don’t know if you know that but it may be fun for you
<a href=“http://chmmc.caltech.edu/[/url]”>http://chmmc.caltech.edu/</a></p>
because seriously you’d have a shot, you’d be supported, there’s no shortage of math classes, and their liberal arts classes are strong too. They’ve got a special CS class for people who have NEVER tried or done anything with CS and other sections for those in CS camps etc. Heard that on NPR the other day