The Prisoner’s Dilemma

Here are the links to the activities on game theory.

Prisoner’s Dilemma –

You and Lucifer

Five opponents, varying strategies:

Dilemma with coins

The Monty Hall Problem: the Let’s Make a Deal Paradox. Here, it’s explained in the movie ’21‘. This paradox is related to a popular television show in the 1970’s. In the show, a contestant was given a choice of three doors of which one contained a prize. The other two doors contained gag gifts like a chicken or a donkey. After the contestant chose an initial door, the host of the show then revealed an empty door among the two unchosen doors, and asks the contestant if he or she would like to switch to the other unchosen door. The question is should the contestant switch. Do the odds of winning increase by switching to the remaining door?

After you’ve played for a while, check out the ‘logic’ involved.  Still confused?  Watch this.

If you’re looking for something to watch, the Season 1 episode “Man Hunt” (2005) of the television crime drama NUMB3RS mentions the Monty Hall problem.

You might also check out this cool article about some business applications (Thanks, Sophia).  Here’s an interesting fractal video that incorporates a visual exploration of both fractal geometry and the Sierpinski gasket.  (thanks, Gary)

Students need to:

  1. understand the assumptions of the game [and their opponent]
  2. make a prediction about how things work
  3. make and record observations and new learnings

steps 2 and 3 would go into your journals.  Students should use several of the activities and make comparisons as they drew conclusions about the relationship between mathematics and the world.

Your fractal assignment

Before you leave on break, look for (visually recognizable) fractals in your everyday life.  Then, do the following:

  1. take a photograph of the fractal
  2. paste the photograph into a Word document
  3. write a brief description about the fractal explaining both what the context of the photo is as well as the fractal elements in the photograph.
  4. When you are done, add it to your math journal

Here are some supplements for your Fractal research

  1. Mathematics of Pattern – The Mathematics in life
    1. Dynamic Mathematics
      1. Fractals in the natural world –
        1. A basic introduction
        2. An overview with ideas for additional research
        3. A higher-level math introduction
      2. A great resource from Yale for both overview and detailed approach
  1. A TED talk hosted by Benoit Mandlebrot
  2. Application of fractal geometry in multiple fields:
    1. Business and Finance
    2. Market returns
    3. Ecology
    4. Military
    5. Literature – Hamlet
    6. Biology
    7. Ethno-mathematics – African design

Fractal Geometry

We have discussed whether mathematics is invented or discovered, we have considered mathematics as a language, and we have considered the degree to which mathematics can be used to describe the real world.  Today in class, we explored a type of geometry – Fractal Geometry – that can be used to explore the universe.  As Ian Stewart described it, the Mandlebrot set provides “islands of order in the sea of chaos.”

If you missed it, or want to review it, check it out in the link in the general links below, or click here.  Please do take notes on the material in the video and keep your notes in your Journal.  It is likely that 20 or so of the questions on the final exam come from the content contained in the video.

Mathematical musings…

Today, as we explored the ideas in math, we considered its application…  What is with Pythagoras?  (thanks, Austin) We also considered when math is wrong.  Try this on for size (thanks, Amabel).  Or, if you prefer to explore zero, try this one.  If you decided to go on a nature walk to the stadium during 6th period, check out the video we were going to watch at the end of class.