There are lots of great reasons to use this task from NCTM’s Illuminations site, which asks students to derive an algebraic function from a problem situation. But one of those reasons isn’t “to show students why they should derive algebraic functions.”
It’s a real world problem, by most definitions of the term. But let’s not let that fact satisfy us. It’s possible for math to be real-world, but also unnecessary. For example, I can ask students to use trigonometry to calculate the height of a file cabinet. But that math isn’t necessary when a measuring tape would suffice.
The same is true here. I can find Stages 1 through 5 by multiplying by three successively. So why did we invent algebraic representations? Life would be so much easier for both the student and the teacher if we relaxed that condition.
But if we added the question, “How long would it take the entire world to experience a good deed?” we will have both a) identified the need for algebraic functions – to calculate outputs given any input, even distant inputs – and b) put students in a position to experience that need.
That’s a two-step process. With the line, “Describe a function that would model the Pay It Forward process at any stage,” the author satisfies the first step. He understands the value of algebraic functions, himself. But without our added question, that’s privileged knowledge and we’re hoping students infer it. Instead, let’s put them directly in the path of that knowledge.
Real world, and also necessary.