Base 12

A few days ago I started a unit on angles, and one of the questions we asked in class was why there are 360 degrees in one full revolution.  The answer has something to do with how ancient astronomers noticed that the stars move approximately 1/360th of a rotation each day.  But it also relates to the fact that some ancients used a sexagesimal number system (base 60).  That is, they had 60 different number symbols compared to our 10 (decimal).

Seeing as how we brought up the concept of different bases, I thought it was a good chance to introduce them to the following math joke: 

"There are 10 kinds of people in this world: those who know binary, and those who don't."

This is funny only if you have some familiarity with binary (base 2), where the symbol 10 means "two".  In any case, after this discussion about different base systems, I recalled a technique for teaching fraction multiplication that I had seen in my math curriculum class.  The method has students represent fractions as a piece of a double hexagon. 

A double hexagon is useful because you can easily divide it into 12 pieces.  That is, it is a base 12 system.  A base 12 system is good for doing fractions because 12 is divisible by 2,3, 4 and 6.  This means it is easy to represent the fractions 1/2, 1/3, 1/4, and 1/6.  A base 10 system is less desirable because 10 is only divisible by 2 and 5.  So to do fraction multiplication with a double hexagon, one first divides the shape into the number of pieces represented by the first fraction, then divide that piece by the denominator of the second fraction, then decide what fraction of the whole shape remains.

When I first saw the double hexagon, I was confused because I didn't immediately recognize that both hexagons were supposed to represent one unit.  The students in the case study we read for class had the same problem.  Although I like the base 12 idea, this confusion can muddle the process of trying to teach fraction multiplication. An alternative would be to use a dodecagon, but the 12 sides can get confusing as well.  So what would my solution be?  Use a 3 x 4 rectangle:

There are some disadvantages to using the rectangle, as well.  Some may not find it convenient to represent 1/6 x 1/2, for instance.  Also, in order for the rectangle to make sense you have to have it pre-divided into 12 pieces whereas the double hexagon has 12 "implied" pieces.