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Cultural Connection

Cultural Connection. Serfs, Lords, and Popes. The European Middle Ages – 476 –1492. Student led discussion. 8 – European Mathematics. The student will learn about. European mathematics from the dark ages through the renaissance. §8-1 The Dark Ages. Student Discussion.

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Cultural Connection

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  1. Cultural Connection Serfs, Lords, and Popes The European Middle Ages – 476 –1492. Student led discussion.

  2. 8 – European Mathematics The student will learn about European mathematics from the dark ages through the renaissance.

  3. §8-1 The Dark Ages Student Discussion.

  4. §8-2 Period of Transmission Student Discussion.

  5. §8-3 Fibonacci & 13th Century Student Discussion.

  6. §8-3 Fibonacci & 13th Century More Later.

  7. §8-4 Fourteenth Century Student Discussion.

  8. §8-5 Fifteenth Century Student Discussion.

  9. §8-6 Early Arithmetics Student Discussion.

  10. §8-7 Algebraic Symbolism Student Discussion.

  11. §8-7 “The Beast” DCLXVI Roman number of the Beast. 666.0000 High-precision beast. 0.666 Millibeast. 1010011010 Binary beast. 666 + 666i Complex Beast. 1-666 Area code of the Beast. 00666 Zip code of the Beast 1-900-666-0666 Live Beasts! One-on-one pacts! Only $6.66 a minute!

  12. §8-7 “The Beast” $665.95 Retail price of the Beast. Phillips 666 Gasoline of the Beast. Route 666 Way of the Beast. 666k Retirement plan of the Beast. 6.66% Beastly interest rate. lbraden@sps.edu

  13. §8–8 Cubic & Quartic Equations Student Discussion.

  14. §8–9 François Viète Student Discussion.

  15. §8–10 Other Mathematicians of the Sixteenth Century Student Discussion.

  16. Fibonacci 1 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 510, . . . f 1 = f 2 = 1 and f n = f n – 1 + f n - 2 1. f 1 + f 2 + f 3 + f 4 + . . . + f n = f n + 2 - 1 2. f 12 + f 22 + f 32 + f 42 + . . . + f n2 = f n· f n + 1 3. f n2 = f n + 1 f n - 1 + ( - 1 )n – 1 for n > 1. 4. f m + n = f m - 1· f n + f m· f n + 1 5. 5 · f n2 + 4 · ( –1)n is a perfect square.

  17. Fibonacci 2 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 510, . . . f 1 = f 2 = 1 and f n = f n – 1 + f n - 2 6. f 50 = 12,586,269,025 7. f 1+ f 3 + f 5 + . . . + f 2n - 1 = f 2n 8. f 2+ f 4 + f 6 + . . . + f 2n = f 2n + 1 - 1 9. The sum of any ten consecutive Fibonacci numbers is divisible by 11.

  18. Fibonacci 3 3 13 3 8 5 5 3 3 8 5 5 5 3 5 5 8 3 5 8 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 510, . . . 10. f 2n2 = f 2n + 1 f 2n – 1 - 1 Area is 64 Area is 65 ? ? ? ? ? ? ? ? ?

  19. Fibonacci 4 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 510, . . . 11.

  20. Fibonacci 5 1 1 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 . . . 2 3 5 8 13

  21. Fibonacci 6 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, . . . f i 1 1 2 3 5 8 13 21 34 . . . . f i2 1 1 4 9 25 64 169 441 1156 . . . . Sum adjacent 2 5 13 34 89 233 610 1597 . . . . f n + 1 - fn 3 8 21 55 144 377 987 . . . . f n + 1 - fn 5 13 34 89 233 610 . . . . f n + 1 - fn 8 21 55 144 377 . . . . Etc.

  22. Fibonacci 7 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 510, . . . Given any four consecutive Fibonacci numbers - n = 1 n = 2 n = 3 n =4 . . . fn,fn+1,fn+2,fn+3 1, 1, 2, 3 1, 2, 3, 5 2, 3, 5, 8 3, 5, 8, 13 . . . fn· fn+3 = a 3 5 16 39 . . . 2fn+1· fn+2 = b 4 12 30 80 . . . f 2n +3 = c 5 13 34 89 . . . K  ABC fn · fn+1 · fn+2 · fn+3 6 30 240 1560 . . .

  23. Assignment FALL BREAK Paper presentations from chapters 5 and 6.

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