I've got somewhat an issue: from time to time I have to teach some math to people who either avoided it or got trough by only knowing a few working algorithms. The only thing that unites all those people: they are sure that math is extremely boring and there's no sense an learning anything except for how to solve the tasks they're given.
I want to be sure that my lessons not only provide primitive formulas or concepts, but motivate students for learning more about the topic, thus I need to show how all that 'pointless knowledge' may turn out to be not so pointless.
Doing that I usually find myself troubled by lacking proper examples of why a sophisticated theory can actually turn out to be benifitial for anyone who isn't pursuing a career of a scholar, programmer, nuclear physicist etc.
As for now I'm limited to usual stuff:
1) Showing how some analysis tricks can make computations easier (L'Hopital's rule and all that) or even possible (in case our only option for a function is approximating it with a series)
2) Show how grasping visual ideas can make it much easier to use all monstrous formulas — case of basic linear algebra (determinants, linear transformations, matrix multiplication etc.)
3) Explaining how treating multiplication as area can allow to derive all neede polynomial formulas (including solving quadratics) without knowing anything else.
4) Refering to how going beyond the scope of a usual math course can make life easier while dealing with polynomials — e.g. knowing the Fundamental theorem and rules of polynomial division, rational roots test etc.
In a desperate search of inspiration I've watched some video courses (the best one yet is Revisiting Calculus by Herb Gross) as well as popular math youtubbers (Numberphile, Mathologer, 3Blue1Brown and others), but still feel like I can use a lot of help from people of the Math community who have way more experience in the field.
I don't teach anything fancy: basic school stuff, some trigonometry, calculus, basic linear algebra and all that (no probability theory so far)
Thanks kindly for your insights and suggestions.