I read this in a book by Mike Ollerton some time ago, I think it was his book called “Mathematics teacher handbook”. One of the activities in the first chapters of the book was this one:
Imagine a rectangle on a square grid, say a 9 x 3 rectangle. Draw the diagonal in the rectangle. How many of the squares within the rectangle will be crossed by the diagonal?
It turned out to be a very nice activity, and it can be attacked in different ways, as well as making a good arena for mathematizing and exploring. The aim being, of course, to see the connection between the size of the rectangle and the number of squares that the diagonal passes through. It is not too difficult, and not too easy either and everyone can understand the question.
I made this GeoGebra file to help explore the question. You can also see it embedded in a post on my Norwegian mathematics blog on blogger.com.