Root 3

 

Who said approximating can't be fun?  This is an activity that I designed to get geometry students to come up with estimates for the height of a hexagon.  The tool I asked them to use was Tileland.  Tileland is a world of polygon paths (well known to those who follow this blog regularly) that doesn't allow for polygons to overlap (the top green squares replaced vertical squares that were there previously).  The following polygon path demonstrates an estimate for the height of a hexagon with unit length sides that is pretty good.  Here the height of four hexagons is slightly less than seven squares; this means that the height of one hexagon is slight less than 7/4 = 1.75.  Using the diagram of the hexagon composed of equilateral triangles and the Pythagorean Theorem, it is easy to show that the hexagon's height is actually root 3. Thus root 3 must be slightly under 1.75.  It is pretty close since root 3 is about 1.732--so within 0.02.

The reason I prefer to use types of approximations like this has to do with having a concrete physical focus to talk about.   It allows for variations and better discussions.  These two diagrams are more of the conclusions to the discussions and experiments.  Also I like reducing and/or delaying  using the magical calculator.  To me, it seems to be an ender rather than a promoter of thought.  In fact, I prefer to have students multiply 1.75 by itself to show how close it is to 3.

For better estimates, you can construct paths corresponding to these fractions 12/7, 19/11, or 26/15 fairly easily in Tileland.  To obtain 45/26, you have to use a more wiggly approach....