I was tutoring a kid the other day. I'm introducing her to Algebra2, and we spend the hour talking about relationships between numbers. Once we've got a few of these relationships pinned down, I tell her that they're called functions, and then we talk about some other functions. She asks, "So Algebra2 is pretty much about functions?" Yep, that's right.
She pauses, and thinks. "It's weird. When I took Algebra the first time it was just all these random topics that we needed to know, and I knew them, but they were all different. I thought that Algebra 2 would be the same, but I guess I'm wrong, it's all about functions."
That's satisfying. Because she knows what the question is she knows what's important (general foundational stuff about functions, stuff that relates to the nature of these functions) and what, relatively, isn't. She knows what she's studying, she'll know how to integrate the new knowledge. We'll introduce each new function with a similar "big" question ("When will the missile fall?"; "Are we going to overpopulate Earth?"; "Why does Albany want us to study DMS notation?")
So, what's the question for Algebra 1? At first I thought Algebra 2 was the challenge, but now I'm having trouble constructing a meta-narrative for Algebra 1 and teasing out a question that introduces that narrative. Clearly a lot of the course is driving towards the concept of a function/2-variable equation. But at the beginning of the year we're still doing arithmetic, so how do I describe the endgame early on?
The best I'm doing right now is thinking about the question, "What counts as a number?" I'm imagining this as a mini-arc that develops, with care, the concept of what we're going to treat as a number in Algebra while also giving me a chance to brush up their arithmetic skills. I'd like the answer of this question to involve integers, fractions, properties of real numbers, square roots, expressions and variables. I could add a historical subplot to the story, revealing info about when this stuff was thought up ("People invent math? WTF?").
Here's what I have so far: