1 cultural connection serfs, lords, and popes student led discussion. the european middle ages –...
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Cultural ConnectionSerfs, Lords, and Popes
Student led discussion.
The European Middle Ages – 476 –1492.
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8 – European Mathematics
The student will learn about
European mathematics from the dark ages through the renaissance.
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§8-1 The Dark Ages
Student Discussion.
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§8-2 Period of Transmission
Student Discussion.
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§8-3 Fibonacci & 13th Century
Student Discussion.
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§8-3 Fibonacci & 13th Century
More Later.
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§8-4 Fourteenth Century
Student Discussion.
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§8-5 Fifteenth Century
Student Discussion.
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§8-6 Early Arithmetics
Student Discussion.
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§8-7 Algebraic Symbolism
Student Discussion.
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§8-7 “The Beast”
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§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.
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§8–8 Cubic & Quartic Equations
Student Discussion.
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§8–9 François Viète
Student Discussion.
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§8–10 Other Mathematicians of the Sixteenth Century
Student Discussion.
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Fibonacci 11, 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 2
2 + f 32 + f 4
2 + . . . + 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.
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Fibonacci 21, 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.
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Fibonacci 31, 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
3
3
3
5
5
5
5
3
8
8
3
35
5 5
5
8
813
? ? ? ? ? ? ? ? ?
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Fibonacci 41, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 510, . . .
11. ...618033989.12
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f
flim
n
1n
n
...618033989.02
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f
flim
1n
n
n
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Fibonacci 51
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
. . .
11
23
5813
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Fibonacci 61, 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.
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Fibonacci 71, 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 . . .
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Assignment
FALL BREAK
Paper presentations from chapters 5 and 6.