Cuboctahedrons: A Perplexing Polyhedron Probability Problem

Due to the Thanksgiving holiday, nosotros exclusively had 2 days of schoolhouse this week.  Needless to say, my students were non real excited close having to come upward to schoolhouse on Mon together with Tuesday.  Several of the schools about us took the entire calendar week off, together with thence that made the calendar week seem fifty-fifty to a greater extent than torturous to my students.

My statistics students were particularly restless.  We're inward the middle of our unit of measurement on probability.  Monday, nosotros looked at some probability intelligence problems.  On Tuesday, I wanted to practise something fun together with interesting but nevertheless related to probability.  I did a quick google search of probability activities, together with I ran across a internet of a cuboctahedron.  Isn't that simply a fun word?  Cuboctahedron.  Cuboctahedron.  Cuboctahedron.  It simply makes me smile.

The activeness instructed students to gather their ain cuboctahedron.  (The internet is on page four of the linked PDF document.  I'm also intrigued past times the probability activeness on page five that involves acting out a Russian fable that predicts who volition larn married inside the side past times side year.)  Then, they were to toss the cuboctahedron 100 times together with count how many times it landed on a foursquare human face upward together with how many times it landed on a triangular face.

Assembled Cuboctahedron together with Net Pattern

I allow my students each alternative a sail of card stock from the cabinet, together with I chop-chop ran off nets for them to cutting out together with assemble.  The cutting together with gluing procedure was to a greater extent than fourth dimension intensive than I realized.  This activeness took the entire 50-minute period.  Since it was the solar daytime earlier Thanksgiving break, this was perfectly fine.  Most of my students ended upward opting for record because the internet was together with thence difficult to set together.  I used glue, together with it industrial plant fine if yous bring plenty patience to allow the mucilage dry out a picayune betwixt steps.

My Class' Finished Cuboctahedrons 

The cuboctahedron consists of six foursquare faces together with 8 triangular faces.  Students were asked to predict the probability that the cuboctahedron would province on each type of human face upward BEFORE tossing it 100 times.

As a class, they decided that 6/14 of the faces were squares.  Therefore, the probability of landing on a foursquare human face upward was closed to 0.43.  8/14 of the faces were triangles.  Thus, the probability of landing on a triangular human face upward was closed to 0.57.  '

Our Class Data 

In 299 trials, the cuboctahedron landed on a foursquare face.  In 91 trials, the cuboctahedron landed on a triangular face.  So, the experimental probability of landing on a foursquare human face upward was closed to 0.77, together with the experimental probability of landing on a triangular human face upward was closed to 0.23.

My students were intrigued past times this data.  I'm non certain what the authors' motivation was inward writing this activity.  Were nosotros supposed to larn these surprising results?  We had a give-and-take of the departure betwixt theoretical together with experimental probability.  What is the argue behind this discrepancy?  Is it related to the differing areas of the faces?  Or, is it every bit 1 pupil suggested related to the agency that the cuboctahedron lands?  It frequently hits on a corner, together with when this happens it almost e'er favors the foursquare faces for landing.

I liked this activeness because it got my students thinking together with talking close math on a solar daytime when they didn't experience similar doing whatever math.  It's a rare affair when I seat my students a occupation I don't already know the answer to.  I demand to practise this to a greater extent than often!  Does anyone know to a greater extent than close this perplexing polyhedron probability problem?  

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