Any advice on how to develop intuition in math? I've recently started to learn math online (studying high school algebra right now after being 7 years out of school),
and while I understand the formulas I'm learning I feel as though I'm not really getting much intuition about the subjects I learn, which for me is the most important thing about learning anything.
Last year I started to learn programming and pretty quickly I found that the best way for me was to do something "hands on" so to speak, like after learning a new concept to actually build a project using what I've just learned.
I want to do the same thing with algebra, doesn't necessarily means to code something but I have a hard time really grasping something until I actually do something with it other than some short practice questions
If you have a suggestion please post! Anything from an interesting article to maybe a project idea that will force me to use algebra or how you would study math if you had to start from scratch again
 A: I develop my mathematics intuition visually, seeing figures, studying graphs, and so on, more than analyzing equations.  This works better for some branches of math than others:  geometry, topology, knot theory, etc., rather than formal logic, category theory, etc.  Some branches of math can be presented in a formal, algebraic way and a visual, intuitive way, most notably group theory.
Einstein said he preferred to reason mathematically geometrically, spatially, visually, and only then turn to the formal symbol manipulation and proofs.  I and many others are the same way.
Personally, I learned calculus in high-school math class, but I understood calculus in high-school physics class, where the problems related to mass, motion, velocity, acceleration, moment of inertia, and problems that were so visual and intuitive.
There are several books that are great in this regard, my two favorites being Visual group theory by Nathan Carter and An illustrated theory of numbers by Martin Weissman.  You might like Proofs without words:  Exercises in visual thinking by Roger Nelson.
But some of the best approaches are from Grant Sanderson's Youtube channel, 3b1b, which you simply MUST watch.
Another technique, which need not rely on visualization, is to be explicitly aware of how you found the right approach to a problem.  Consider a tricky integral.  You try integration by parts, series expansion, substitution, and so on until you find that trigonometric substitution worked.  Once you've solved the integral, go back and ask yourself:  "What aspects of this integral meant that trigonometric substitution was the proper approach?"  In this way, you'll become more aware and gain intuition, for instance seeing the $\sqrt{1 - x^2}$ is a functional form that at the very least suggests trigonometric substitution.  Ponder that in order to build up your intuition.
Another suggestion that might help.  I once spent an afternoon with the late titan of theoretical physics John Wheeler.  He once said "Never calculate anything until you first know the answer."  Well, that doesn't always help, but sometimes it does:  think about a problem before you rush into solving it with known techniques.  Perhaps you can reason that the answer is $0$, or $\infty$, or some other special answer first.  Even reasoning that the answer has to be positive, or negative, or irrational helps you build intuition.  Once you solve the problem, go back and revisit and refine your intuitions in light of your initial thoughts.
As a general guide, though, practice graphing equations, sketching solids, and so on and visually understanding them first, even before delving into symbol manipulation.
