The “take everyday stuff and bring it into the classroom” shtick just doesn’t work for me when I’m preparing Algebra II lessons. And I think it’s because by the time we get to Algebra II we’ve reached a new point in the education of our students. We’ve exhausted the material that we think everybody out on the street ought to know, and we’ve started introducing specialized mathematics that not everyone needs to know. That is, our broader goal in Algebra II isn’t to provide people with the math they need to be average working folks, but rather to make more specialized education in the maths and sciences both attractive and feasible. That is, we teach it so that we attract kids to more math and science, and also so that it’s possible for kids to be prepared for more math and science.
As far as I can tell there are two reasons why we teach kids stuff:
(1) We think that they need to know it, even as a non-specialist. So comfort with percentages, ratios, rates, averages, comfort with numbers, abstract thinking--these are all skills that we want our students to have no matter what they do in life.
(2) We want to recruit and prepare students for a specialty. We, as a society, need mathematicians, physicists, doctors, engineers, accountants and all sorts of other professions that require more mathematical comfort than your average citizen, and therefore need more training. If we don't teach higher math, then our students won't be prepared for the training requisite for these jobs. Further, part of our job is to make working with math enticing enough that we're able to recruit workers into fields that require a good deal of number work. So our job is dual when we're in this mode: to prepare and recruit.
Basically, I think that Algebra I mostly falls into Category 1, and Geometry is half and half, but Algebra 2 is firmly in Category 2. So much of that curriculum is either preparatory for Calculus or of application only to scientists. Which is NOT a knock on it. But it just means that we can't use the same approach to teach Algebra 2 as we do Algebra 1.
So we need to think about the best way to do Algebra 2. I think where we end up is what so many teachers are already doing: integrating scientific material into our Algebra 2 courses. This is difficult for me, since I don’t have a great physics background beyond mechanics. But I think that this is the direction where I’m heading: our job in Algebra 2 is to make math, and its applications to science, seem attractive while simultaneously preparing students for their future.
Not sure exactly how this shows itself day-to-day, but I think we need to present material with scientific motivations. For instance, maybe the proper way to introduce complex numbers isn't as most general solutions to polynomial equations, but rather scientifically. Maybe we integrate complex numbers into our trigonometry so that we can ask "How can we model trigonometric fluctuations algebraically?" or something. (Truth is, I'm just learning about complex integration now, so I don't understand the applications of complex numbers at a depth greater than wikipedia browsing).