Download Fibonacci sequences and the golden ratio

Maths lesson
Exploring sequences and the Fibonacci sequence
Learning objectives
1. Explore the exponential sequences, leading to negative powers.
2. Discover the Fibonacci sequence and the Golden Number or Golden Ratio.
3. Connect this number to nature – sense of awe and wonder!
4. Make a Golden rectangle.
Knowledge
♦
To develop understanding of Fibonacci sequences and application within nature.
♦
To create a Golden rectangle.
Skills
♦
Mathematical skills.
♦
Enquiry skills – investigate, research and evaluate.
Resources
♦
Power point presentation – PHI – The Golden Number.
♦
Calculators, rulers, a pineapple.
♦
Strips of paper, graph paper.
Main activity
Introduction to sequences (25 minutes)
Write some sequences on the board, look at next terms and nth terms. Try some arithmetic
sequences, geometric sequences, quadratics, etc. Use some ‘tricky’ examples that could have
been generated by more than one rule.
Suggested examples:
♦
Easy: 3, 6… could be 3, 6, 9, 12… or 3, 6, 10, 15, 21…
♦
Harder: 2, 4, 8… could be 2, 4, 8, 16, 32… (powers of 2) or 2, 4, 8, 14, 22… (differences
2
go up by 2 each time, or—equivalently—nth term is n – n + 2)
♦
Even harder and kind of obscure: 1, 2, 4, 8, 16… could be 1, 2, 4, 8, 16, 32, 64… (powers
4
3
2
of 2) or 1, 2, 4, 8, 16, 31, 57… (nth term is (n – 6 n +23 n – 18n + 24)/24 —try it, it
works!)
Brief discussion: How many terms do we need to be sure we’ve got the right rule? (Strictly, we
can never have enough terms!)
Look at powers of 2 and extend to the left to introduce negative powers. Can a number raised to a
negative power ever be negative? What is (-1) to the power of 100? Etc.
PowerPoint – journey of discovery (10 minutes)
Use this is a stimulus to start discussing Fibonacci and where one finds sequences
Fibonacci sequence (15 minutes)
The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa or
Leonardo Pisano in Italian since he was born in Pisa (Italy), the city with the famous Leaning
Tower, about 1175. Leonardo'
s father was called Guglielmo Bonacci. He called himself Fibonacci:
a shortening of the Latin "filius Bonacci", which means "the son of Bonaccio",
The sequence
Start with two 1’s, add them to give 2, add 1 and 2 to give 3, then 2 and 3 to give 5, then 3 and 5
to five 8…………………..
1
1 2 3 5 8 13 21 34 55 89 144 …………………………..
Continue the sequence to 20 terms at least.
Calculate fib(n)/ fib(n-1) (i.e. one term divide the term before it). What do you notice?
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Start with any two numbers, not 1 and 1. Do the same, find 30 terms and calculate fib(n) / fib(n1). Share your results with the group, what do you notice? This number is so important that it has
a special name, a Greek letter Phi written as . Choose any large number term and divide by the
term before it, what do you get?
Group work (35 minutes)
Activity 1
Fibonacci numbers and the Golden Number or Golden Ratio
If we take the ratio of two successive numbers in Fibonacci'
s series, (1, 1, 2, 3, 5, 8, 13…) and we
divide each by the number before it, we will find the following series of numbers:
n
fib(n)/fib(n-1)
2
1/1=1
3
2/1=2
4
3/2=1.5
5
8/5=1.66667
6
13/8=1.625
7
21/13=1.61538
Plot the ratios on a graph. What do you notice
The value that the ratio seems to be settling down towards is called the golden ratio or the
golden number. It has a value of approximately 1·618034
Try this… (need a calculator)
Pick any positive number (like 10 or pi or .78 or whatever), add 1, take the square root, add 1,
take the square root, add 1, take the square root, etc. forever. What happens?
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Activity 2
Fibonacci rectangles and shell spirals
You need some graph paper for this, a pencil and ruler
We can make another picture showing the Fibonacci numbers 1,1,2,3,5,8,13,21…
if we start with two small squares of size 1 next to each other. On top of both of
these draw a square of size 2 (=1+1).
Here is a spiral drawn in the squares, a quarter of a circle in each square. The
spiral is not a true mathematical spiral but it is a good approximation to a kind of
spiral that does appear often in nature
Here is a cross-section of a Nautilus sea shell. It shows the spiral curve of the
shell and the internal chambers that the animal using it adds on as it grows. The
chambers provide buoyancy in the water.
Spirals
Coneflower 34/55
Pine cone 8/13
Cauliflower 5/8
1. Collect some pine cones for yourself and count the spirals in both directions.
A tip: Soak the cones in water so that they close up to make counting the spirals easier. Are
all the cones identical in that the steep spiral (the one with most spiral arms) goes in the same
direction?
2. What about a pineapple? Can you spot the same spiral pattern? How many spirals are there
in each direction?
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Activity 3
Fibonacci fingers?
Look at your own hand
You have...
2 hands each of which has ...
5 fingers, each of which has ...
3 parts separated by ...
2 knuckles
Is this just a coincidence or not?
However, if you measure the lengths of the bones in your finger (best seen by slightly bending the
finger) does it look as if the ratio of the longest bone in a finger to the middle bone is Phi?
What about the ratio of the middle bone to the shortest bone (at the end of the finger) - Phi again?
Can you find any ratios in the lengths of the fingers that look like Phi? ---or does it look as if it
could be any other similar ratio also?
Why not measure your friends'hands and gather some statistics?
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Activity 4
The exact value of
As we’ve seen, the ratios of successive terms in the Fibonacci sequence get closer and closer to
a certain value, which we call . Now, suppose we take a large term in the sequence, fib(N), and
call it a.
fib(N) = a
♦
Explain why fib(N+1) is almost exactly equal to a
♦
What is fib(N+2) equal to? Explain why.
♦
Explain why a
♦
What equation do we get if we cancel the a’s?
♦
Show that = ( 5 + 1)/2 is one of the solutions of this equation. [Note: students who know the
2
=a +a
quadratic formula can get it from that; otherwise, substituting into the equation is fine.]
♦
Work out this value on your calculator. Write down all the numbers on the display
=
♦
Now find the square of
2
♦
and write down all the numbers on the display
=
Now find 1/ and write all the numbers (this is called the reciprocal)
1/ =
What do you notice?
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Activity 5
The perfect rectangle or ‘golden rectangle’
The length is equal to , the width is 1 unit
A
B
AB = 1.618
BC = 1
D
C
AB + BC = AB
Explain this equation in words.
AB BC
_________________________________________________
If BC=1 then this equation becomes
+1=
1
gives
i.e.
+1=
2
-
1
2
–1=0
2
solution is
= -b + (b – 4ac)
2a
where a=1, b=-1, c=-1
Use this formula to show that
= 5+1
Constructing a golden rectangle
Start with a long strip of paper width 10cm
A
B
B
C
1. Fold corner B over to B, make a crease then unfold
2. Fold side AB over to BC, make a crease then unfold
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A
B
B
C
A
Fold long end over like this and make a mark
at D
D
This is a golden rectangle
Measure AD and see!
A
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D
Plenary session
Students feed back what they have learnt.
Questions for focus
1. Why do the Fibonacci numbers often appear in family trees?
For suggested answers:
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#bees
2. Sunflowers can contain the number 89, or even 144. Many other plants, such as
succulents, also show the numbers. Some coniferous trees show these numbers in the
bumps on their trunks. And palm trees show the numbers in the rings on their trunks.
Why do these arrangements occur?
3. Look at this example of Muslim art and explain how the Fibonacci sequence is
implemented.
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