Triangles and Bridges

The kids and I have been working on some geometry.  One part of this is showing how polygons can be subdivided into triangles.  I had T work out the formula t = s - 2; the number of triangles, t, needed to build a polygon with s sides.

Also I showed the connection between reducing a square into two triangles to reducing a square number, 2x2=4, 3x3=9, 4x4=16, ... to two triangular numbers 4=3+1, 9=6+3, 16=10+6, where 1, 3, 6, 10, ... are triangular numbers (can be arranged in a triangle, see my earlier post about pairwise comparisons). 

I also tried to illustrate how a triangle is more stable than a square to bending.  This led to a bridge building experiment the next day.  I picked up two bags of gum drops and a box of toothpicks from the grocery store.  I gave the kids 100 toothpicks each (midway we increased this to 110) and told them to build a bridge between two blocks (borrowed from F) one foot apart.  Then we would test how much weight they could hold before buckling.  (As a kid I tried to do something like this in my grandma's kitchen with bread dough and toothpicks; I refined my initial idea by looking online and seeing the post about gumdrop bridges at Beck Logan's STEM Blog.) M immediately said she would use triangles because that's how the Eiffel Tower is built. 

The kids really got into it.  This was a lot of fun.

Getting Started.  M has a pyramid base up already. 

Some more shapes appear. 

Taking Shape

T went for a Warren Truss design, a weight efficient pattern used in airplanes.  He also has a stick and gumdrop bystander.

M went for a series of interconnected pyramids with a complex base on each block. 

Measuring the fit. 

T's completed bridge, using 103 toothpicks, supporting four pencil weights.

T's bridge supporting 10 pencils.  At 10 it is starting to bend under the weight, but is still holding. 

Structural failure at 11 pencil weights. 

M redesigned her bridge a bit with anchors and weights at the ends to try to keep it from sagging so much as it pulled toward the middle. 

Placing the fifth pencil. 

Failure at six pencils. 

I didn't want to be left out of the fun so I made a bridge that combined the two kids designs and used exactly 110 toothpicks.  A Warren Truss with a row of pyramids on top to support against compression of the top edge. 

The combined design holding 12 pencils.

Structural failure at 16 pencil weights!