The great comments about Algebra 2 have got me thinking about relationships, descriptions of “having to do with”.
Every pair of people that have anything to do with each other is a different case, yet each relationship is pretty unambiguously father-son, or business partners, or best friends, or lovers, or whatever it is. We all relate to each other in a small number of very common ways. The examples of each category contain a dazzling, unending variety–but in spite of that, every variation on grandmother-granddaughter is still an example of grandmother-granddaughter. The grandmother and the granddaughter have something to do with one another, and it’s a specific something.
Variables–anything we can count or measure; any quantity that changes, or could–can be related, or not. They can have something to do with each other, or not. The period and length of a pendulum have something to do with each other. The height of a projectile and time have something to do with each other. We need not be applied, we can stay pure: x^2, the product of a number with itself, certainly has something to do with how big x is. 2^x also has something to do with how big x is, but it’s a different something. There are a few, very common ways things can be related (along with lots more less common ways). Is the goal of Algebra 2 to find out what those common ways are?
What are the different kinds of relationships among variables? What can we infer about one if we know something about the other? If Mike is my father, then I must be Mike’s … what? If y=10^x, then x must be … what? What is the only number that has the following complicated relationship to 3? What makes a relationship quadratic? If two people are seen at the movies kissing, what can you infer about their relationship? What can’t you? If two variables take on values together in the following pairs, what kind of relationship is it? What does that tell you about the other ways those variables will be seen–excuse me, observed–together?
This might work.