I always thought fractals and similar were great introductory programs for children.

The fractals need only a few lines of code, with a bit of extra boiler plate. Tweaking the formula can give dramatically different results. You get to use a variety of programming concepts in a short program.

Relevant books are

Dynamical Systems and Fractals (Karl-Heinz Becker, Michael Dörfler) (This book has extensive Pascal code) http://www.amazon.co.uk/Dynamical-Systems-Fractals-Computer-...

http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511663031

Fractals - Images of Chaos (Hans Lauwerier) (This book has an appendix of BASIC code) http://www.amazon.co.uk/Fractals-Images-Chaos-Penguin-Scienc...

Sadly neither book is suitable for general 11 year olds, but a modern version certainly could be.

And A.K. Dewdney obviously has a bunch of books which could be used to inspire young programmers. The Armchair Universe, or The Magic Machine and similar launched many programmers, with "Core War", "Wator", "Bugs", etc

If you want books you can try these old books:

Dynamical Systems and Fractals (http://www.amazon.co.uk/Dynamical-Systems-Fractals-Computer-...)

Mathographics (http://www.amazon.co.uk/Mathographics-Robert-Dixon/dp/048626...)

Computers Pattern Chaos & Beauty (http://www.amazon.co.uk/Computers-Pattern-Beauty-Clifford-Pi...)

Fractals Images of Chaos (http://www.amazon.co.uk/Fractals-Images-Chaos-Penguin-Scienc...)

I suggest these because they all contain some kind of code.