Why can't you flatten a sphere?

Solution 1:

Experimentation is fun!

Give him a globe, some string, a ruler, a compass, some paper, and a pencil. Then have him try to map 4 different cities from the globe onto the paper, keeping ALL the distances between them the same as on the globe.

Solution 2:

Show him a triangle on the sphere with 3 right angles. He knows that it is impossible in a plane right ?

Solution 3:

One simple way is to think of an orange peel. Have the 10-year old peel an orange nicely and tell him to try to flatten the peel pieces completely without breaking or streching any part of the peel.

Solution 4:

I take it that you will be explaining this to a group of kids. Why don't you try several approaches? Some will probably appreciate the orange peel explanation simply because it gives them something to do. Others may appreciate the following. Mathematically it is based on the fact that the circumference of a circle on a spherical surface is shorter than the circumference of a circle (with same radius) on a flat plane (see also rschwieb's suggestion of calculating the ratio of the length of the equator to the distance to a pole). So changing the radius by a fixed amount will not result in a fixed change in the circumferemce.

I suspect some of the kids in your audience will have seen just that while crocheting. If you are crocheting a circular piece (like this), then you have to be careful to add just the right number of links per round. Add too few, and the end result will have a positive gaussian curvature (a bag-like shape) - add too many and you get a warped surface of a pseudosphere (was still good enough, when yours truly handed such a potholder to his dear Grandma 40+ years ago).

Undoubtedly you will have shown how to flatten a cylindrical surface prior to this (also suggested by rschwieb).

Also make them gift wrap both a cylindrical tube and an inflated basketball.