How to explain integrals and derivatives to a 10 years old kid?

This is a question for a long car journey - both speed and distance travelled are available, and the relationship between the two can be explored and sampled, and later plotted and examined. And the questions as to why both speed and distance are important can be asked etc

Explain derivatives using the speedometer ! Ask her how one could find the speed of something , and if she goes saying the average speed , ask her how one could calculate the instantaneous speed. And boom , there comes your derivatives.

Isn't there a way to describe with drawings that the derivative is the limit of the slopes of the secants, that the integral is the area under a curve, and that the derivative of the area is the height of the function at the point at which you compute derivative? Sure, she might not know what "slope" means, but certainly you can explain that to her if she has the patience ; she's not too far away from it mathematically.

If you make "clear enough" drawings, I think geometry is a good enough analogy to explain what you do. If you put in contexts where derivatives apply (physics/chemistry/economics), I think your little sister will just look at too many pictures at the same time and get lost in what you're saying. Try to be patient and stick to the "closest-to-the-real-picture portrait" of derivatives and integrals.

Oh, and one more thing ; if she isn't interested in what you do, don't lose your time. Only do it if she has the courage to listen to you, remember that mathematics are pure pain to the closed ears.

Hope that helps,