February Above & Beyond Due Friday, Feb. 27, 2015 Name: Directions: Complete the following problems. Show your work and justify your answers. 1. Fibonacci Sequence The Fibonacci sequence is a famous list of numbers named after the Italian merchant (c. 1170 CE-c. 1250 CE) who popularized the Arabic numerals (0-9) that we use today. The sequence appeared before Fibonacci in ancient Indian writings. The sequence starts with two 1’s, and then every other number is formed by adding the two preceding numbers. So the sequence begins 1, 1, 2, 3, 5, 8, . . . and continues forever. (a) Write the first ten numbers in the Fibonacci sequence (including the numbers above). (b) Fill in the chart below: Sum of first two Fibonacci numbers: Fourth Fibonacci number: Sum of first three Fibonacci numbers: Fifth Fibonacci number: Sum of first four Fibonacci numbers: Sixth Fibonacci number: Sum of first five Fibonacci numbers: Seventh Fibonacci number: (c) Look for a pattern in part (b). The numbers in the first column should relate to the numbers in the second column. Looking along each row, what pattern do you see? (d) Use your discovery in part (c) to find the sum of the first 8 Fibonacci numbers (without adding them up). Explain how you got your answer. The Fibonacci numbers are not just some weird mathematical pattern. They show up in nature: in the spiraling scales on pineapples and pinecones, the spiraling “florets” on the face of a sunflower, and the petals of daisies and roses! Why do Fibonacci numbers show up in nature like this? Often, it occurs because plants, through natural selection, have come up with the most efficient organization of their parts. 1 2. The Golden Ratio The ancient Greeks were geometers, which means they did essentially all of their math- ematics through geometry (they did not have an algebraic system like we have today). As such, they often viewed numbers in a geometric way (i.e., as lengths or areas of various shapes). Suppose you have a wooden board that you want to cut into two pieces. The most aesthetically pleasing way to do this, according to the ancient Greeks, is to cut it according to the Golden Ratio: cut the board into two pieces, one longer and one shorter, so that the ratio of the longer to the shorter is equal to the ratio of the total board length to the longer. In symbols: Golden Ratio = Total board length Longer piece = Longer piece Shorter piece (1) Let’s figure out the value of this Golden Ratio using algebra. (a) You want to cut a board into the Golden ratio. To make our lives easy, let’s assume that we start with a wooden board that is 1 meter long. Let’s say the longer piece has length x. What is the length of the shorter piece in terms of x? 1︷ ︸︸ ︷ cut short piece long piece x Longer board length: x Shorter board length: (b) Now rewrite equation (1) in terms of x, using your answers to (a). Total board length Longer piece = Longer piece Shorter piece 1 = (2) (c) S ...

1. February Above & Beyond
Due Friday, Feb. 27, 2015
Name:
Directions: Complete the following problems. Show your work
and justify your answers.
1. Fibonacci Sequence
The Fibonacci sequence is a famous list of numbers named after
the Italian merchant
(c. 1170 CE-c. 1250 CE) who popularized the Arabic numerals
(0-9) that we use today.
The sequence appeared before Fibonacci in ancient Indian
writings.
The sequence starts with two 1’s, and then every other number
is formed by
adding the two preceding numbers. So the sequence begins
1, 1, 2, 3, 5, 8, . . .
and continues forever.
(a) Write the first ten numbers in the Fibonacci sequence
(including the numbers
above).
(b) Fill in the chart below:

2. Sum of first two Fibonacci numbers: Fourth Fibonacci number:
Sum of first three Fibonacci numbers: Fifth Fibonacci number:
Sum of first four Fibonacci numbers: Sixth Fibonacci number:
Sum of first five Fibonacci numbers: Seventh Fibonacci
number:
(c) Look for a pattern in part (b). The numbers in the first
column should relate to
the numbers in the second column. Looking along each row,
what pattern do you
see?
(d) Use your discovery in part (c) to find the sum of the first 8
Fibonacci numbers
(without adding them up). Explain how you got your answer.
The Fibonacci numbers are not just some weird mathematical
pattern. They show up
in nature: in the spiraling scales on pineapples and pinecones,
the spiraling “florets” on
the face of a sunflower, and the petals of daisies and roses! Why
do Fibonacci numbers
show up in nature like this? Often, it occurs because plants,
through natural selection,
have come up with the most efficient organization of their parts.
1
2. The Golden Ratio
The ancient Greeks were geometers, which means they did

3. essentially all of their math-
ematics through geometry (they did not have an algebraic
system like we have today).
As such, they often viewed numbers in a geometric way (i.e., as
lengths or areas of
various shapes).
Suppose you have a wooden board that you want to cut into two
pieces. The most
aesthetically pleasing way to do this, according to the ancient
Greeks, is to cut it
according to the Golden Ratio: cut the board into two pieces,
one longer and one
shorter, so that the ratio of the longer to the shorter is equal to
the ratio of the total
board length to the longer. In symbols:
Golden Ratio =
Total board length
Longer piece
=
Longer piece
Shorter piece
(1)
Let’s figure out the value of this Golden Ratio using algebra.
(a) You want to cut a board into the Golden ratio. To make our
lives easy, let’s assume
that we start with a wooden board that is 1 meter long. Let’s say
the longer piece
has length x. What is the length of the shorter piece in terms of
x?

4. 1︷ ︸︸ ︷
cut
short
piece
long
piece
x
Longer board length: x
Shorter board length:
(b) Now rewrite equation (1) in terms of x, using your answers
to (a).
Total board length
Longer piece
=
Longer piece
Shorter piece
1
= (2)
(c) Solve your equation from part (b). (You’ll probably have to
use the quadratic

5. formula at some point.) Show your work. Simplify your
answers, but leave your
answers in exact form.
2
(d) [Calculator allowed] In part (c), you got two solutions.
Approximate these so-
lutions to three decimal places and plot them on the number line
below.
−2 −1 0 1 2
You have two solutions, but only one makes sense in this
context. Which solution
makes sense? Why?
(e) Using parts (c) and (d), determine the length of the two
pieces of the board below.
In the second column, determine the exact length of each piece,
using your answers
from part (c). In the third column, approximate the lengths to
three decimal places
(using part (d) and a calculator).
Piece of board Exact length Approximate length
Longer piece
Shorter piece
(f) Now we can find the Golden ratio. From what we know
above, the Golden ratio
will be:

6. Golden Ratio =
1
Longer piece
i. Find the exact value of the Golden Ratio using the exact value
from (e).
Rationalize the denominator and simplify.
ii. Now approximate the Golden Ratio to three decimal places,
using a calculator
and the approximate value from (e)
3
3. Interesting Connections Though it is not obvious, the Golden
Ratio and the Fi-
bonacci sequence are related. Let’s explore some of those
connections.
(a) Fill in the chart below, with the help of a calculator. In the
left column, write
the ratio of consecutive Fibonacci numbers (with the larger
number on top). In
the right column, write the corresponding decimal
approximation (round to three
decimal places when appropriate). The chart has been started
for you.
Ratio of consecutive Fibonacci numbers Decimal
Approximation
1/1 1

7. 2/1 2
3/2 1.5
5/3
8/5
(b) The ratios in part (a) are approaching some special number.
Looking at the second
column, can you guess what special number they are
approaching?
(c) On a computer, navigate to wolframalpha.com. In the search
field, enter exactly
the text in each line below (before the colon). The program
knows the value of
the golden ratio and computes the value of your entry. Write the
corresponding
answers below.
(golden ratio^1 - (-1/golden ratio)^1)/sqrt(5) :
_______________
(golden ratio^2 - (-1/golden ratio)^2)/sqrt(5) :
_______________
(golden ratio^3 - (-1/golden ratio)^3)/sqrt(5) :
_______________
(golden ratio^4 - (-1/golden ratio)^4)/sqrt(5) :
_______________
(golden ratio^5 - (-1/golden ratio)^5)/sqrt(5) :
_______________

8. (golden ratio^6 - (-1/golden ratio)^6)/sqrt(5) :
_______________
(golden ratio^7 - (-1/golden ratio)^7)/sqrt(5) :
_______________
(golden ratio^8 - (-1/golden ratio)^8)/sqrt(5) :
_______________
(golden ratio^9 - (-1/golden ratio)^9)/sqrt(5) :
_______________
(d) What is the name of the sequence of numbers that you get in
part (c)?
4
http://www.wolframalpha.com