Cultural Connection
Explore European mathematics from the Dark Ages through the Renaissance in this student-led discussion series. Key topics include the influence of Fibonacci and the development of algebraic symbolism, cubic and quartic equations, and notable mathematicians of the sixteenth century. Engage in discussions about the transmission of knowledge during these pivotal eras and the mathematical advancements that shaped modern mathematics. The series offers insights into the historical context and cultural connections that influenced mathematical thought in Europe.
Cultural Connection
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Presentation Transcript
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.
§8-2 Period of Transmission Student Discussion.
§8-3 Fibonacci & 13th Century Student Discussion.
§8-3 Fibonacci & 13th Century More Later.
§8-4 Fourteenth Century Student Discussion.
§8-5 Fifteenth Century Student Discussion.
§8-6 Early Arithmetics Student Discussion.
§8-7 Algebraic Symbolism Student Discussion.
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§8–8 Cubic & Quartic Equations Student Discussion.
§8–9 François Viète Student Discussion.
§8–10 Other Mathematicians of the Sixteenth Century Student Discussion.
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.
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.
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 ? ? ? ? ? ? ? ? ?
Fibonacci 4 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 510, . . . 11.
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
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.
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 . . .
Assignment FALL BREAK Paper presentations from chapters 5 and 6.
