Games in Context Jim Diederich UCD Math Project

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Games in ContextJim DiederichUCD Math
ProjectMath Dept.dieder_at_math.ucdavis.edu

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CONTENTS

• Games in context - a philosophical view
• The Survivor Game - an experimental view

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Interesting Source

• The Puzzle Instinct by Marcel Danesi
• Indiana University Press 2002

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Give your best guess in of your class who would
enjoy

• a mystery
• a puzzle
• a game
• a math problem

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How can we differentiate among

• Mysteries
• Puzzles
• Games
• Math Problems

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Historically there seems to be a need for
mystery, puzzles, games

• 11,000 year old dice
• Riddle of the Sphinx - first intelligence test

What is it that has four feet in the morning, tw
o at
noon, and three at twilight?
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• Greek dramas by Aeschylus, Sophocles, Euripides
  address the mystery of existence - 525-406 B.C.
  
• Need for catharsis - feeling of suspense that
  calls for relief
  
• Puzzle fixation puzzle depression - an
  irrational craving for them
  

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Rhind Papyrus - Egyptian

• About 1650 B.C. - a collection of problems in
  puzzle form plus tables for area, fraction
  conversion, algebraic problems, etc.
  
• They converted fractions of the form 2/(2n1) to
  sums of unit fractions like
  
• 2/5 1/3 1/15
• where 5 2n 1 101,

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Unit Fractions Problem 1

Write 1 as a sum of unit fractions without d
uplications
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A Solution

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Unit Fractions Problem 2
Write 1 as a sum of unit fractions without dupli
cations and such that all unit fractions are less
than 1/2.

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A Solution

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(No Transcript)
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Problem 63 - Rhind Papyrus
Directions for dividing 700 breads among four pe
ople, 2/3 for one, 1/2 for the second, 1/3 for t
he third, and 1/4 for the
fourth
Add 2/3, 1/2, 1/3,1/4. This gives 1 1/2 1/4.
Divide 1 by this. This gives 1/2 1/14. Now
find 1/2 1/14 of 700. This is 400.
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• Charlemagne (742-814) was a puzzle addict
• Alcuin was his mentor - est. effective education
  program affecting the Western World
  
• 56 puzzles in Problems to Sharpen the Young

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Problem to Sharpen the Young
When 100 bushels of grain are distributed among
100 persons so that each man receives 3 bushels,
each woman 2 bushels, and each child 1/2 bushel
, how many men, women, and children are there?
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• Some of the most popular works of the medieval
  era were puzzle collections including the Greek
  Anthology from A.D. 500.

I desire my two sons receive the thousand starte
rs of which I am possessed, but let the fifth pa
rt of the legitimate ones share
exceed by ten the fourth part of what falls to
the illegitimate one.
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• Fibonaccis Liber Abaci, A.D. 1202
• It established the Hindu-Arabic numbering
• It had many puzzles

A snake is at the bottom of a 30-foot well. Each
day it crawls up 3 feet and slips back 2 feet.
At that rate, when will the snake be able to
reach the top of the well?
A certain man put a (new born) pair of rabbits,
male and female, in a very large cage. How many
pairs of rabbits can be produced in that cage in
a year if every month each pair produces a new
pair which, from the second month of its
existence on, also is productive?
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• Movable type in 1400s made puzzle books more
  available. Two who employed puzzle format to
  illustrate mathematical concepts
  
• Robert Recorde 1510-1558
• Tartaglia Cardano 1499-15571501-1576 solved
  the cubic
  
• By the seventeenth century puzzles were widely
  accepted both for pleasure and for illustrating
  mathematical ideas

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• Euler 1707-1783
• Aroused interest in combinatorics problems and
  puzzles - 36 officers problem
  
• Konigsberg Bridge Problem his most famous

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Konigsberg Bridge Problem
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What is a mystery?

• It often has the connotation of being unsolvable
  
  
• Who killed JFK?
• Why did ancients use unit fractions like
  1/101 or 1/300?
  
• The more information you obtain the deeper it
  becomes!

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(No Transcript)
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What is a puzzle?

• Conceals an answer but cries out to be solved
• Pits the solver against the constructor
• Are structures of the imagination, a region
  called Wonderland - Lewis Carroll.
  

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• Not solved by accurate reckoning alone - requires
  insights, i.e.informed hunches, imagination
  memory.
  
• Aha! or Eureka! effect
• No extensive mathematical training is required to
  solve puzzles, they come by flashes of insight -
  Martin Gardner
  
• Puzzle solving does not always correlate with IQ

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What is a game?

• Usually has a set of rules
• Often has some physical components
• Usually has various strategies
• Generally pits two or more players against one
  another
  
• May have a limited time and a defined end point
• May involve a reward or payoff

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What is a math problem?

• Sometimes its a mystery
• Sometimes its a puzzle
• Sometimes its a game
• But in our teaching we usually want to eliminate
  the mystery and the puzzlement and reduce insight
  to routine

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The Dilemma

• How do you keep math interesting as you try to
  make it routine?
  
• Games!!!

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Math Survivor III (2002)The Mathematical
Outback

• Mathematics and Computation

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Rules of the Math Survivor III

• 1. There are two competing tribes
• the ARCHS and the EUCS
• 2. No one gets voted out of their tribe.
• 3. The only thing you may have to swallow is
  your pride.
  
• 4. No one wins 1,000,000, maybe some candy.
• etc.

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Math Survivor IV (2007) New Rules

• There will be 6 tribes (teams) with 6-7 members
  each
  
• In each game, 4 members will play and the
  remaining will observe with roles rotating
  
• Each tribe will select its own name and write it
  on its flag
  
• When you have all of the answers for all problems
  in a game, raise your flag and call out your
  tribes name
  

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Math Survivor IV - New Rules Cont

• A tribe must stop working when its flag is
  raised
  
• Once three flags are raised Ill call stop and
  all work must cease while answers are checked
  
• No tribe can raise its flag more than twice in a
  game
  
• The first tribe to finish with all correct
  answers comes in first, the second to finish will
  all correct answers comes in second, up to third
  place
• A time limit may be set for a game

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8 - the magic number

• In the following you will be given three
• sets of numbers such as
• 2, 4, 6 1,2,2 1,5,6
• For each set find a mathematical expression that
  produces the number 8.
  
• For example the following would work
• 6 (4/2) 2(12) (1.6)5

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• You may use any of the following mathematical
  operations , -, , /.
  
• You can use parentheses and decimal points.
• You can use the given numbers in exponents.
• Each of the three numbers must be used once and
  only once in the expression.
  

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• 2, 7, 1
• 2, 7, 2
• 2, 7, 3

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• 2, 7, 4
• 2, 7, 5
• 2, 7, 6

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• 2, 7, 7
• 2, 7, 8
• 2, 7, 9

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A Local Teachers Experience

• She modified survivor for her elementary class
  room in a low performing school
  
• She had six pre-established cooperative groups
• She used decks with various problems relating to
  their curriculum and solutions. For example
  division decks had 4 cards with division
  problems, and 8 cards with answers and
  distractors
• She had codes on the cards and an answer card
• Her rules for managing the game were similar to
  the ones we are using today
  

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• (reminder pass out A,, D station cards)

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A Local Teacherss Feedback Experience

• Students loved Math Survivor and begged her to
  let them play it
  
• Students begged her to review material
• Students coached other students on their teams
  during recess to bring them up to speed
  
• She moved to a better performing school and
  stopped using the method
  

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Possible Alternative to Decks

• You will see a slide with problems labled A, B,
  C, and D.
  
• The person in your group at Station Card A must
  solve Problem A, etc.
  
• There are 8 given answers. Write the number of
  your answer on a piece of paper and place it next
  to your Station Card.
  
• Each station must work only on their problem

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Group or Multigroup Problem

• In the following you will be given a long
  division problem
  
• Some of the digits in the problem are missing and
  indicated by s
  
• Find the correct digits
• The whole group may work as one and teams can be
  formed from several groups

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Determine the s

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INTERSECTION GAME

• If A 3,4,6,10 , B 4,5,10,11 then
• A Ç B 4,10
• You will be given two decks of index cards with a
  number on one side of each card.
  
• One deck is set A, the other deck is set B
• The first tribe to correctly determine A Ç B
  wins