53 777 66bf19e55b524 GE 4 MODULE 1 LESSON 2 WEEK 3 .pdf

MATHEMATICS

2. MODULE LESSONS:
LESSON 1: PATTERNS AND NUMBERS IN NATURE AND THE WORLD
LESSON 2: THE FIBONACCI SEQUENCE
LESSON 3: MATHEMATICS FOR OUR WORLD

3. LEARNING OUTCOMES:
AFTER YOU HAVE COMPLETED THIS MODULE, YOU SHOULD BE ABLE TO :
▪IDENTIFY PATTERNS IN NATURE AND REGULARITIES;
▪ARTICULATE THE IMPORTANCE MATHEMATICS IN ONE’S LIFE;
▪ARGUE ABOUT THE NATURE OF MATHEMATICS, HOW IT IS REPRESENTED
AND USED; AND
▪EXPRESS APPRECIATION FOR MATHEMATICS AS A HUMAN ENDEAVOR.

4. Mathematics isuniversal.Mathematical
conceptsandinfluencescanbeseeninboth
natureandhumancreations,suchasart.
Seeingpatternsinnaturehasledmanyto
believethatsuchpatternsareevidenceof
divinecreation.Thislessonwillshowyouthe
mathematicsofart-relatedconceptslikethe
FibonacciSequenceandtheGoldenRatio.

5. LESSON 2: THE FIBONACCI SEQUENCE
AND THE GOLDEN RATIO
IN THIS LESSON, CHALLENGE YOURSELF TO:
a.EXAMINEFIBONACCISEQUENCEINNATUREANDART;AND
b.SOLVEEQUATIONS USINGFIBONACCISEQUENCE AND
GOLDENRATIORULE.

6. ACTIVITY 1: MEASURE ME
Get a ruler and measure the following by
centimeters(cm):
▪Distance from the ground to your navel
▪Distance from your navel to the top of your head
▪Distance from the ground to your knees
▪Length of your hand
▪Distance from your wrist to your elbow
▪Distances A, B, and C as indicated in the figure to the
right

7. Now, calculate the following ratios and write the results in the table
below:
1.
�??????������ ���� �ℎ� ������ �� ??????��� �����
�??????����������??????�����������ℎ������??????���ℎ���
=
2.
�??????������ ���� �ℎ� ������ �� ??????��� �����
�??????������ ���� �ℎ� ������ �� ??????��� �����
=
3.
�??????������ ���� ??????��� ��??????�� �� ??????��� �����
??????����ℎ �� ??????��� ℎ���
=
4.
�??????������ �
�??????������ �
=
5.
�??????������ �
�??????������ �
=

8. Ratio 1 Ratio 2 Ratio 3 Ratio 4 Ratio 5
Write your answer on each box, after you divide the values in activity 2.

9. FIBONACCI SEQUENCE
TheFibonaccisequenceisnamedafter
LeonardoofPisa,alsoknownasFibonacci,
whofirstobservedthepatternwhile
investigatinghowfastrabbitscouldbreed
underidealcircumstances.

10. Fibonacci’s1202bookLiberAbaciintroducedthe
sequencetoWesternEuropeanmathematics,although
ithadbeendescribedearlierinIndianMathematics.
Bydefinition,thefirsttwonumbersintheFibonacci
sequenceare1and1,andeachsubsequentnumberis
thesumoftheprevioustwo.Inmathematicalterms,the
sequence??????
�ofFibonaccinumbersisdefinedbythe
recurrencerelation??????
� =??????
�−1 +??????
�−2,withseedvalues
??????
1=1and??????
2 =1.

11. Starting with 1 and 1, the succeeding terms in the sequence can be
generated by adding the two numbers that came before the term:
1 + 1 = 2 1, 1, 2
1 + 2 = 3 1, 1, 2, 3
2 + 3 = 5 1, 1, 2, 3, 5
3 + 5 = 8 1, 1, 2, 3, 5, 8
5 + 8 = 13 1, 1, 2, 3, 5, 8, 13, …

12. TofindthenthFibonaccinumberwithoutusingtherecursion
formula,thefollowingisevaluatedusingacalculator:
??????
�=
(
1+5
2
)
�
−(
1−5
2
)
�
5
This form is known as the Binetform of the nth Fibonacci number.

13. EXAMPLE:
UseBinet’sformulatodeterminethe25thand30thFibonaccinumbers.
Solution:
a)25
th
Fibonaccinumber
??????
�=
(
1+5
2
)
??????
−(
1−5
2
)
??????
5
??????
25=
(
1+5
2
)
25
−(
1−5
2
)
25
5
??????
25=
(1.61803)
25
−(−0.61803)
25
5
??????
25=
167,761.0000059609
5
??????
25=75,025
b)30
th
Fibonaccinumber
??????
�=
(
1+5
2
)
??????
−(
1−5
2
)
??????
5
??????
30=
(
1+5
2
)
30
−(
1−5
2
)
30
5
??????
30=
(1.61803)
30
−(−0.61803)
30
5
??????
30=
1,860,498
5
??????
30=832,040

14. DID YOU KNOW?
TheFibonaccisequenceoccursmanytimesinnature.Takea
lookatsunflowers.Inparticular,payattentiontothe
arrangementoftheseedsinitshead.Doyounoticethatthey
formspirals?Incertainspecies,thereare21spiralsinthe
clockwisedirectionand34spiralsinthecounterclockwise
direction.
Heart of the Sunflower
You can observe the
phenomenon inthenumber
ofpetalsindaisies,cauliflower
florets,spiralsinpinecones,
andthebandswinding
aroundinpineapples.
Core of a daisy blossom

15. THE GOLDEN RATIO
Inmathematicsandthearts,twoquantitiesareinagoldenratioiftheirratioisthe
sameastheratiooftheirsumtothelargerofthetwoquantities.
Insymbols,aandb,where�>�>??????,areinagoldenratioif
�
�
=
�+�
�
.
ThegoldenratioisoftensymbolizedbytheGreekletter??????.Itisthe
number??????=�.??????�??????�??????….andtheirrationalnumber
1+5
2
.

16. The Mona Lisa
Thegoldenratioshowsupinart,
architecture,music,andnature.For
example,theancientGreeksthought
thatrectangleswhosesidesforma
goldenratiowerepleasingtolookat.
Manyartistsandarchitectshaveset
theirworkstoapproximatethegolden
ratio,alsobelievingthisproportionto
beaestheticallypleasing.

17. REFERENCE:
Noconand Nocon(2016). Essential Mathematics for the Modern World.
Bley, B. (2012, July 21). Sunflower Seed Patterns [Photograph]. Fine Art America.
https://fineartamerica.com/featured/sunflower-seed- patterns-bruce-bley.html
http://eacharya.inflibnet.ac.in/data-server/eacharya-
documents/53e0c6cbe413016f234436ed_INFIEP_8/37/ET/8_ENG -37-ET- V1-
S1__lesson.pdf
Peters, J. (2014, February 1). Golden Ratio [Illustration]. Research Gate.
https://www.researchgate.net/post/Is_the_beauty_of_patterns_or_ar
e_the_structures_in_patterns_the_most_important_source_of_mathem atics