I've come across a "fractal" that have turned out to be popular with teachers on in-service training courses. It is called the "stair-step fractal", and is discussed in detail on the webpage
Studying Dimensionality Through a Stair-Step Fractal.
While it is interesting to let children meet a fractal, I have actually used this as the end of some work on the A-series of paper formats (most notably the A4). There are lots of mathematics included there, of course, most importantly the point that A3 is equivalent to two pieces of A4, A2 is two pieces of A3, A1 is two pieces of A2 and A0 both is two pieces of A1 and has an area of one square meter. The square root of two figures prominently.
The reason the stair-step fractal is fun to use here, is that the way of making it gives rise to small "rectangles" which are A5, A7, A9, A11, A13 etc. Teachers tend to be quite amazed when they realize this (even though they also think it is fairly obvious once they notice it).
The stair-step fractal can then be used to show quite persuasively that when you double all the sides, you do not also double the area and the volume, they are actually 4x and 8x as large, respectively. While there are of course lots of ways of showing that, this is a fun way of doing it.
This was just a little glimpse into what you may see if you join one small workshop with teachers for 10-13-year olds.
