@Jennyfr1
Our target was $25k per child. We found a decent number of possible choices/chances for this level of funding outside of our in state schools. Some are guaranteed $, others competitive. A few public out of state examples:
Purdue: https://www.admissions.purdue.edu/costsandfinaid/freshman.php
COA: $41.8 k, trustees + national Hispanic, 13 to 17 k/year
Miami Ohio: https://miamioh.edu/admission/merit-guarantee/
COA: $50k, $20-36k/year for 1450+ SAT and 3.5 GPA ,
Texas Tech: https://www.depts.ttu.edu/scholarships/incFreshman.php
COA in state $26.7k but if win presidential merit scholarship get in state tuition from out of state + $6 to 8.5 k/year for SAT 1400 – 1500+ range
Oldest ended up national merit scholar, youngest is semifinalist, should be named finalist. This opens up some significant aid (free to $20k /year net cost) at a couple dozen out of state publics: our favorites were U Florida and Texas A&M. We passed on these for fit reasons for the oldest, who is now a freshman at a different out of state flagship with effectively “½ ride”. Our youngest, high school senior, is emphasizing smaller privates in his search for fit reasons, and a couple out of state publics. I will post in spring how it turns out.
Note, New Jersey is one of hardest states to get national merit from, 223 index score in 2019
Privates: hard to estimate chances, but $25k and lower per year is potentially doable:
I listed them before: Lafayette, Richmond, Washington and Lee, Davidson, Wake Forest (only 3% get merit $ at Wake but they seem to be actively seeking more diversity),Emory, Furman. If can expand westward a little, suggest Vanderbilt, Wash U, Denison. Some of these are only a handful of scholarships at the level needed, but they are there and potentially worth a shot. Another is college of Wooster, can get that down to $27 to 28/k per year current on merit aid last I checked (pick up a few smaller outside scholarships and can get to $25k/year)
Couple other thoughts: if your son co-ops, that can help with costs and get him critical job experience.
Also, perfect to near perfect test scores and grades not enough to get merit aid in many cases. Need strong leadership and extracurriculars as well to differentiate.
For math, some extracurricular activities to look at (already suggested by others here):
See below competitions and resources, all focused on math prior to calculus. Excelling in those will put you in position for primes, etc. AMC, AIME, USAMO, USAMTS, USACO, etc
https://www.maa.org/math-competitions
https://www.usamts.org/
http://www.usaco.org/
https://artofproblemsolving.com/
https://www.awesomemath.org/
Also check out the small, select summer math camps, SMaRT camp Texas A&M https://www.math.tamu.edu/outreach/Camp/ (free)
And Williams college math camp https://sites.williams.edu/mathcamp/ (reduced cost). The AMS website for others
http://www.ams.org/programs/students/emp-mathcamps
These and a few others are highly selective and limited to about 20 participants, they seem to be viewed highly for college admissions (more so than pay, for profit camps) but more importantly a great chance to see how much your child likes math, and give him a chance to meet other math focused students.
Good luck!