John Conway introduced the notion of a rational tangle in a foundational paper on knot theory. He also devised a wickedly cool dance to explain them. This "rational tangle" dance has become a staple of math circles across the country. In one presentation it may be demonstrated as a magic trick. A description of this method by Tom Davis is attached as a pdf.
The following video shows Tom leading this activity.
http://www.youtube.com/watch?v=iE38AXV_dHc
There are other variations. For example one could begin by asking the participants to reduce a fraction with a fairly large prime factor in the numerator and denominator. If the participants do not know the Euclidean Algorithm giving them the hint to flip the fraction is usually enough for people to figure it out. This is a good way to motivate the Euclidean Algorithm so that students will remember it. The second handout and mentor guide are based on this second variation. It was used in the University of New England Julia Robinson Mathematics Festival.
http://youtu.be/fSaI1jQbEfo
Three Math Ed grad students for the 3/24/14 edition of the Westchester Area Math Circle produced a nice Power Point PDF that can go with this activity. It is attached.
There are many directions the activity may be extended. A bit of thought will show that any rational number may be obtained by a finite sequence of the operations $x\mapsto x+1$ and $x\mapsto -1/x$. This is essentially equivalent to showing that these two linear fractional transformations generate the modular group. In order to show that there is a well defined bijection from the extended rational numbers ot the collection of rational tangles, one needs to show that the relations in the modular group act trivial on tangles. This is not often done when this is presented as a math circle activity, but it is fun to do so.
There is a separate lesson on the Euclidean Algorithm as well as a more advanced one that covers the algorithm and the ABC conjecture.
This is related to continued fractions, symmetries of knots, automorphic forms, elliptic curves, and hyperbolic geometry.
See:
Knot Symmetry
More to come.