alt Hacker NewsI did this when I taught my third grader calculus on the train, when she asked a question about the train accelerating faster sometimes than other times. She loved it, but I was just taking advantage of children's natural curiosity.
Do you have some examples that the adult could instigate, rather than waiting for the child to express curiosity?
I've used filling a tank, balloon, or bucket (rate of flow, can be subdivided to teach limits, and use weird shapes for teaching area under curve and interpolation en route to integrals), or the classic throwing a ball back and forth and trying to describe the shape, the distance it flies, peak speed vs peak height, figuring out how hard you are "actually" throwing instantaneously. Honestly as soon as you start thinking about bulk substances moving around (gravel piles! fuel tanks!) it's easier to find examples than you'd ever reckon. Rate of change is everywhere.
Seems like I start by asking "how do we know how much this tank holds?", or "how fast does this line go up on the side of the tank?" and curiosity goes from there usually.