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

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 & 13 th Century Student Discussion.

6 §8-3 Fibonacci & 13 th 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” DCLXVIRoman number of the Beast High-precision beast Millibeast Binary beast iComplex Beast Area code of the Beast Zip code of the Beast Live Beasts! One-on-one pacts! Only $6.66 a minute!

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.

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 f 1 + f 2 + f 3 + f f n = f n f f f f f n 2 = f n · f n f n 2 = f n + 1 f n ( - 1 ) n – 1 for n > f m + n = f m - 1 · f n + f m · f n · f n · ( –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 f 50 = 12,586,269, f 1 + f 3 + f f 2n - 1 = f 2n 8. f 2 + f 4 + f f 2n = f 2n The sum of any ten consecutive Fibonacci numbers is divisible by 11.

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

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

20 Fibonacci

21 Fibonacci 6 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610,... f i f i Sum adjacent f n f n f n f n f n f n 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 = 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, f n · f n+3 = a f n+1 · f n+2 = b f 2n +3 = c K  ABC f n · f n+1 · f n+2 · f n

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