It also brought to mind a possible new species of Math Talk/3 Act. One could present a scenario/video/picture, and ask students about the relationships between the quantities. I have seen questions like this in other materials, like the qualitative graphing in the Shell Centre materials. But I realized you could revisit this kind of qualitative analysis regularly using a 3 Act type of presentation. Some possible roads of inquiry:
-What quantities do you see? Which are important?
-What quantities do you see that are changing?
-Which quantities are related, and how are they related?
-Take a pair of related quantities and sketch a graph of their relationship.
-If you were to write a formula for this, what variables would you use? What operations do you think might be involved?
Once you have introduced and worked with some formulas, you could start to expose the students to formulas from various fields and they would have a better appreciation for what they represent and how they came to be. The students can explore and get a feel for the difference between direct and indirect variation, and they could begin to see why some formulas included constant coefficients. And, perhaps most important, these kinds of discussions would help build their sense of what a variable is all about, why you sometimes have more than one, and why we bother with variables, functions, and equations in the first place.