The Fibonacci Numbers, Phi and the Golden Ratio Activity Ws

8/2/2019 The Fibonacci Numbers, Phi and the Golden Ratio Activity Ws

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Name:_______________

Date:_________

The Fibonacci Numbers,

Phi and the Golden Ratio

Were going to do a series of exploratory activities, drawings, and video watching to learn about the irrational

number Phi. In many ways, it is even more natural than Pi, and its certainly more beautiful. If youre curious, here

are the first few hundred digits of Phi:

1. 6180339887 4989484820 4586834365 6381177203 0917980576 2862135448 6227052604 6281890244

9707207204 1893911374 8475408807 5386891752 1266338622 2353693179 3180060766 7263544333

8908659593 9582905638 3226613199 2829026788 0675208766 8925017116 9620703222 1043216269

Activity 1: Spirals and the Golden Ratio (which is also known by the Greek letter Phi) are intimately connected.

1.) There are three types of spirals. Try drawing some spirals below to see if you can figure out what the three

types of spirals are. (Note: keep your spirals round and circular. The three types of spiral have nothing to

do with the types of lines you use.)

2.) What types do you think you found?

Type 1:_______________________________

Type 2:_______________________________

Type 3:_______________________________

3.) List the types we came up with together as a class:

Type 1:_______________________________

Type 2:_______________________________

Type 3:_______________________________

4.) Try drawing something that has each of the three types of spirals

a. Type 1 b. Type 2 c. Type 3


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Activity 2: For this one, youre going to create the perfect spiral for yourself. Use the graph paper Ill hand out and

carefully follow the instructions. Ill help you along and do an example with you using the elmo.

1.) Look at the top right hand corner of the graph paper. Count down 12 squares from the top, then 12

squares to the left- into the piece of paper. Now outline that square that you land on.

2.) Outline the square directly above the first square you outlined.

3.) Now you have a rectangle. Record the dimensions of this rectangle in your table

4.) Construct a square that has side lengths equal to the first two squares you just made to the left of the

rectangle.

5.) Now you have a new, bigger rectangle. Record the dimensions of this rectangle in your table

6.) Now construct a square below this rectangle that has the same side lengths as the length of the rectangle.

7.) Now you have a new, bigger rectangle. Record the dimensions of

this rectangle in your table

8.) Now construct a square right of the biggest rectangle that has the

same side lengths as the length of the rectangle.

9.) Now you have a new, bigger rectangle. Record the dimensions of


10.)Now construct a square above the biggest rectangle that has thesame side lengths as the length of the rectangle.

11.)Now you have a new, bigger rectangle. Record the dimensions of


12.)Now construct a square left of the biggest rectangle that has the


13.)Now you have a new, bigger rectangle. Record the dimensions of


14.)Now construct a square below the biggest rectangle that has the


15.)Now you have a new, bigger rectangle. Record the dimensions ofthis rectangle in your table

16.)Now, try to draw a spiral by connecting the corners of each of the

squares you have drawn.

17.)Youre out of paper, but pretend we could keep going. Record the

width and length for two more rectangles in the table above.

18.)What patterns do you notice in the table?

19.)For each rectangle in the table, try dividing the length by the width. Record the decimal you get below.

Rectangle

#1 2 3 4 5 6 7 8 9

Ratio of

length to

width

20.)What number is this ratio approaching?

Rectangle

#

Width ofrectangle

(shorter

dimension)

Length ofrectangle

(longer

dimension)

1

2

3

4

5

6

7

8

9


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Activity 3: Come grab a pine cone and an artichoke from up front and two different colors of glitter glue. Look at

your pinecone from the top. Can you see spirals? You should be able to.

1.) With glitter glue of one color, trace out spirals going to the right. How many spirals did you find?

2.) With glitter glue of a different color, trace out spirals going to the left. How many spirals did you find?

3.) Do the same thing with the artichoke. Record the number of spirals going left and right. Once youre done

with finding these four numbers, come up to the board and record your findings on the table on the board.

4.) What do all these numbers have in common?

5.) Write down the name of these numbers and please write out the first 10 of them.

Activity 4: Were going to watch some videos. As you watch, please answer the questions below.

1.) Why do plants want to angle their leaves?

2.) Why are nice fractions not good enough?

3.) Draw the symbol for Phi

4.) Look at the fractions shes using to find Phi. What do you notice?

5.) What are Lucas numbers?

6.) Once the videos are finished, give a summary as best you can as to why plants grow this way.


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Activity 5: The Golden Ratio and the body.

Many architects and artists find the ratio of Phi beautiful. Maybe its because the human body is also built with the

ratio of Phi.

1.) Partner up. Have one person measure the other persons height in

inches. Then measure the distance from the persons feet to their

navel. Divide their height by the distance from their feet to their navel.

Record all measurements in the table.

2.) Switch roles and try it again.

Activity 6: The Golden Ratio in Art and Beauty.

The Golden Ratio has been used by artists and architects for centuries. See if you can find where the golden ratio is

in each of the following pictures. Use a centimeter ruler.1.) The Parthenon. Measure widths and lengths

and write below which measurements, when

divided give the golden ratio.

2.) Measure different parts of Mona Lisas face and record

below which measurements, when divided give the golden ratio.

Person 1 Person 2

Height

Distance

from feetto navel

Ratio of

height to

distance


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3.) Notre Dame Cathedral. Measure widths and

lengths and write below which measurements,

when divided give the golden ratio (Note: the

Golden Ratio appears many times in different

measurements)

4.) School of Athens Painting by

Rafael. Measure widths and

lengths and write below which

measurements, when divided give

the golden ratio (Note: the

Golden Ratio appears many times

in different measurements)

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The Fibonacci Numbers, Phi and the Golden Ratio Activity Ws