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

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

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

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

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

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

7. 7 §8-4 Fourteenth Century Student Discussion.

8. 8 §8-5 Fifteenth Century Student Discussion.

9. 9 §8-6 Early Arithmetics Student Discussion.

10. 10 §8-7 Algebraic Symbolism Student Discussion.

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

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

14. 14 §8–9 François Viète Student Discussion.

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

16. 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 1 2 + f 2 2 + f 3 2 + f 4 2 +... + f n 2 = f n · f n + 1 3. f n 2 = 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 n 2 + 4 · ( –1) n is a perfect square.

17. 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. 18 Fibonacci 3 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 510,... 10. f 2n 2 = f 2n + 1 f 2n – 1 - 1 Area is 64Area is 65 3 3 3 5 5 5 5 3 8 8 3 3 5 55 5 8 8 13 ? ? ? ? ? ? ? ? ?

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

20. 20 Fibonacci 5 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... 1 1 2 3 5 8 13

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

22. 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 = 1n = 2 n = 3 n =4... f n,f n+1,f n+2,f n+3 1, 1, 2, 3 1, 2, 3, 5 2, 3, 5, 83, 5, 8, 13... f n · f n+3 = a 3 5 16 39... 2f n+1 · f n+2 = b 4 12 30 80... f 2n +3 = c 5 13 34 89... K  ABC f n · f n+1 · f n+2 · f n+3 6 30 240 1560...

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