A course ought to tell a story. If it doesn’t, it’s just a collection of topics, not a course.
Honors Precalculus at Weston definitely does tell a story. I was thinking about the themes of that story today, and I realized that a big one is the idea of expanding a domain to go more broadly and more deeply into a topic. We begin the year with a review of right-triangle trigonometry, where the domain of the sine and cosine functions is the interval from 0° to 90° (exclusive). We then expand it to the inclusive interval, then to obtuse angles, and then to all angles. Through the use of the unit circle and the switch to radians we have a domain that consists of all real numbers. At the end of the year we’ll expand the domain to complex numbers, through the use of infinite series.
In the meantime, we are turning to complex numbers. Over the years our notion of “number” has expanded from whole numbers to rational numbers to non-negative rational and irrational numbers to real numbers and now to complex numbers. Eventually we’ll break out of the idea of “number” altogether and will explore different infinities.
In the area of polynomials we’ve moved from linears to quadratics, and later this year we’ll explore cubics and beyond.
I’m sure fractals can fit this theme also, but that’s for another day.