Q: What new skill should Algebra 2 students leave with?
A: The ability to model a system mathematically.
NY's A: Yeah, that. Also, how to solve an absolute value equation, how to employ Degree, Minute, Second notation to represent an angle, how to graph an inverse cosine curve...
And that, basically, is the challenge in reorienting the Alg2/Trig course around a single question or theme.
Still, I've been working on reorienting NY's Alg2 curriculum along these lines. This isn't exactly ground-breaking: Kirk Weiler's e-text, for instance, points at such an orientation. He starts by discussing functions, and then introduces different families of functions that end with a regression and modeling unit.
Here's what I would like to do differently in organizing the curriculum:
1. I want to bring the modeling and regression to the beginning of the function unit, to motivate our study of the function family.
2. I want to discuss a concrete example of a function, such as the familiar linear functions, before talking about functions in the abstract.
3. I want all the other stuff -- and boy oh boy is there a lot of other stuff -- to fit into the larger discussion about modeling.
1 and 2 are doable. 3 is hard. Still, there are some things that can be done to integrate the various skills of the course. For instance, much time is spent in Alg2 solving equations. By the end of the year, students should be able to solve absolute value equations, radical equations, quadratic equations, trig equations, exponential equations, log equations and rational equations.
These sort of skills, however, become necessary when you've mathematically modeled a system, found a representative function, and now wish to extrapolate. You're either going to be evaluating an expression, or solving an equation. If you think about the curriculum in this way, you have functions at the center of the curriculum, the functions are there for modeling, and a clear distinction between evaluating an expression and solving an equation will be constantly reinforced.
Ditto for inequalities.
So functions, modeling, and solving equations are taken care of. They fit into the larger framework. What's left over is all the stuff that has to do with manipulating expressions. For instance: simplifying radicals, simplifying complex exponents, simplifying complex fractions, exponent rules, etc. How do these things fit into the larger framework?
The best that I can do now is to say that these are upgrade packages, so to speak. The ability to manipulate expressions will allow us to have an easier time evaluating function expressions for a value, or expressing answers to function equations. So I think what I'm going to do is be explicit that these areas don't directly fit into our modeling narrative -- they're not used to describe or extrapolate based on data -- but they're excurses, upgrade packages that will allow us to model certain relationships more accurately.
In summary: 1) Bring statistics and regression to the foreground, to motivate the study of functions. 2) Put extrapolation at the center of function units. Extrapolation motivates both the evaluation of expressions and the solving of equations. 3) Explicitly bring out all the leftovers into upgrade packages, that will assist us in our next modeling exercise.
My next post will organize Alg2 standards into this framework. The post after that, hopefully, will reflect critically on this and think about what some of the problems of this will be.