2013 T3 Interna-onal Conference Nspired Probability, Polynomials and CAS Goals 2013 T3 Interna-onal Conference Steve Phelps Madeira High School Cincinna-, OH [email protected] •  Discover the connec-on between probability objects (spinners, dice) and polynomials •  Use polynomial mul-plica-on to model probability situa-ons (the sum of two spins of a spinner, or the sum of the rolls of two dice) •  Use polynomial factoring to model probability situa-ons (construc-ng new dice that meet certain condi-ons) 1 2 Essen-al Ques-on Is it possible to find a different pair of six-­‐sided dice with posi-ve integer face values that have the same distribu-on of rolls as a pair of standard six-­‐
sided dice? Let’s begin to answer this Essen-al Ques-on by playing a game. 3 Spin the two spinners and add the results. What are the possible sums? 4 Possible Sums -­‐3, -­‐2, -­‐1, 0, 1, 2, 3, 4 5 Steve Phelps [email protected] 6 1 2013 T3 Interna-onal Conference Play a Game Seang up the Spreadsheet •  Write down the 10 sums that YOU think will be the first ten sums to appear as a result of spinning the two spinners 10 -mes. •  The first person to cross off all their numbers “wins.” •  We will model the spins using a List & Spreadsheet. 7 From Spinner to Polynomial 8 From Spinner to Polynomial 2x −1 + x 0 + x 2
•  The EXPONENTS are the values on the spinner. •  The COEFFICIENTS are the rela-ve weights of each value (-­‐1 occurs twice as oeen as 0 or 2), x −2 + x −1 + x 1 + x 2
•  The EXPONENTS are the values on the spinner. •  The COEFFICIENTS are the rela-ve weights of each value. 9 Spinning Both Spinners What Does it Mean? •  The EXPONENTS are the SUMS of the two spinners. •  The COEFFICIENTS are the number of ways the sum can occur. Mul-plying the two polynomials is equivalent to spinning the two spinners 11 Steve Phelps [email protected] 10 12 2 2013 T3 Interna-onal Conference Do It With Probabili-es! Let’s work towards an answer to our Essen-al Ques-on by exploring a simpler problem. •  The EXPONENTS are the values on the spinner. •  The COEFFICIENTS are the probabili-es of each value. 13 Two Four-­‐Sided Dice: What is The Distribu-on of The Sums? 14 Two Four-­‐Sided Dice: What is The Distribu-on of The Sums? 15 One way to roll an 8, two ways to roll a 7, three ways to roll a 6, four ways to roll a 5, and so on... 16 Factor Is it possible to construct two other 4-­‐sided dice that have the same distribu-on of sums? There are six factors! 17 Steve Phelps [email protected] 18 3 2013 T3 Interna-onal Conference Rearrange the Factors •  Both dice are 4-­‐sided •  One dice has one 4, two 3’s and one 2 •  The other has one 4, two 2’s and one 0 What About This? •  One die is 8-­‐sided, the other is 2-­‐sided. 19 What About This? •  Both dice are 4-­‐sided •  One dice has one 3, two 2s and one 1 •  The other has one 5, two 3’s and one 1 20 Back to our Essen-al Ques-on Is it possible to find a different pair of six-­‐sided dice with posi-ve integer face values that have the same distribu-on of rolls as a pair of standard six-­‐
sided dice? 21 Rolling Two 6-­‐sided Dice 22 Factor Each dice is represented as a polynomial by x + x2 + x3 + x4 + x5 + x6
23 Steve Phelps [email protected] •  There are 8 factors •  We need to choose factors that will create a polynomial whose coefficients are all posi-ve and sum to 6. 24 4 2013 T3 Interna-onal Conference This Would NOT Work... This DOES Work 25 26 Sicherman Dice Differences •  Other than the obvious, how are Sicherman Dice different from a pair of 6-­‐sided dice Standard Sicherman 27 Extension Ques-ons for Sicherman Dice References •  How would Monopoly change with these dice? •  How would Backgammon change with these dice? •  How would Craps change with these two dice? •  How would Yahtzee change using these dice? 29 Steve Phelps [email protected] 28 •  hop://plus.maths.org/content/os/issue41/
features/hobbs/index •  hop://plus.maths.org/content/non-­‐transi-v-­‐
dice •  hop://en.wikipedia.org/wiki/Sicherman_dice •  hop://mathworld.wolfram.com/
SichermanDice.html •  hop://www.cut-­‐the-­‐knot.org/arithme-c/
combinatorics/Sicherman.shtml 30 5