I wonder how you teachers walk the line between justifying mathematics because of its many—and sometimes surprising—applications, and justifying it as the study of one of the great intellectual and creative achievements of humankind?
I have quoted to my students G.H. Hardy's famous line,
The Theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics.
and then contrasted this with the role number theory plays in contemporary cryptography. But I feel slightly guilty in doing so, because I believe that even without the applications to cryptography that Hardy could not foresee—if in fact number theory were completely "useless"—it would nevertheless be well-worth studying for anyone.
One provocation is Andrew Hacker's influential article in the NYTimes, Is Algebra Necessary? I believe your cultural education is not complete unless you understand something of the achievements of mathematics, even if pure and useless. But this is a difficult argument to make when you are teaching students how to factor quadratic polynomials, e.g., The sum - product problem.
So, to repeat, how do you walk this line?