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Math in the Modern World - Module 1 (Sequences)
Nursing (University of Eastern Philippines)
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Module
Section 1: The Nature of Mathematics
GE 1 – Mathematics in the Modern World
Author: Mary Jane B. Calpa
1.1 Mathematics in our World
(A Study of Patterns)
Overview
Welcome to the first module of GE 1 (Mathematics in the Modern World)!
This course begins with an introduction to the nature of mathematics as an
exploration of unseen patterns in nature and environment, a rich language in itself
governed by logic and reasoning, and an application of inductive and deductive
reasoning.
Section 1 is composed of the following: 1.1 Mathematics in our World; 1.2
Mathematics Language and Symbols; and 1.3 Problem Solving and Reasoning. These
topics will allow students to go beyond the typical understanding of mathematics as
purely a bunch of memorized formulas and duplicated mathematical computations, but
as a powerful tool used to understand better the world around us. Moreover, we will
discuss and argue about the nature of mathematics, what it is, and how it is expressed,
represented, and used. We will study mathematics as a language in order to read and
write mathematical texts and communicate ideas with precision and conciseness. We
will also justify statements and arguments made about mathematics and mathematical
concepts using different methods of reasoning.
Mathematics has always been perceived as a study of numbers, symbols, and
rules. It is an art of geometric shapes and patterns, a tool in decision-making and
problem solving. It has a language that differs from the ordinary speech. It is done with
curiosity, with a penchant for seeking patterns and generalities, with the desire to
know the truth, with trial and error, and without the fear of facing more questions and
problems to solve. The following diagram shows the very nature of mathematics.
study of
patterns
art
Mathematics
language
is a/an …
set of
problemsolving tools
process of
thinking
Nocon, R. & Nocon, E.
In this module, we will focus on Lesson 1.1 - Mathematics in our World (A
Study of Patterns). The lesson is anchored by the following core idea: Mathematics is a
useful way to think about nature and the world. Our intention is to observe things, in
both in nature and the world, through pattern-seeking, understand the substantial
interconnection and relationship of the mathematics and the world, and appreciate
mathematics as a discipline full of essence and beauty
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Module
Section 1: The Nature of Mathematics
GE 1 – Mathematics in the Modern World
Author: Mary Jane B. Calpa
Learning Outcomes
After working on this module, you will be able to:
1. identify patterns in nature and irregularities;
2. articulate the importance in mathematics in one’s life;
3. argue about mathematics, what it is, how it is expressed, represented, and
used; and
4. express appreciation for mathematics as a human endeavor.
Activities To Do
1.) Watch the video “Nature by Numbers” by Cristóbal Vila
(link:
https://vimeo.com/9953368) and write one (1) sentence that describes your
impression after watching the video.
2.) Identify pattern/s observed in the pictures.
(a)
(b)
(d)
(e)
(c)
Sources: (a) https://www.library.illinois.edu/mtx/2018/10/09/mathematics-in-nature/; (b); https://www.weareteachers.com/teacherdresses-ms-frizzle/ (c) https://www.smithsonianmag.com/science-nature/science-behind-natures-patterns-180959033/;(d) and (e)
http://mustafacil-online.blogspot.com/2015/08/manmade-patterns.html
Questions To Ponder
The video and the pictures leave us questions to think about.
• What are the different kinds and forms of patterns you have seen in the video
and/or pictures?
• How does these patterns help us understand the connection between our world
and mathematics?
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Module
Section 1: The Nature of Mathematics
GE 1 – Mathematics in the Modern World
Author: Mary Jane B. Calpa
Patterns and Numbers in Nature and the World
When we buy clothes, accessories, furniture, house decorations, and other similar objects,
we tend to look for beautiful geometric designs or patterns. We appreciate the patterns seen in the
colorful wings of butterflies, the arrangement of flowers and leaves, the reflection of the mountain
tops to the clear waters of lakes, the different shapes of clouds in the skies, and other patterns seen
in the nature. In the busy streets of the cities, we are impressed by the intricate but well-designed
modern homes and high-rise buildings. We are wowed by nature and man-made creations because
of these repeated designs of geometric visuals.
Repeated ways or occurrences that happens or was done are also considered as patterns.
For example, the cycle of the moon, the changing seasons, and even the transmission pattern of the
COVID 19 pandemic.
Patterns surround us. It is everywhere and are in every people’s task or activity.
Mathematics, developed by human mind and culture, is a formal system of thought for
recognizing, classifying, and exploiting patterns. (Stewart, I.). Mathematics is indeed a study of
patterns. Results in mathematics are brought by the generalizations of patterns. The study of
patterns allows us to observe and identify relationships, discover logical connections, and make
generalizations. Moreover, the use and study of patterns allows us to be logical thinkers and better
problem solvers.
Now, let us take a look of some of these patterns.
Example
1
Logic Pattern
Choose the figure that completes the pattern.
1.
?
_________
A
B
C
D
A
B
C
D
2.
Sources: (1) and (2) hhtp://www.jobtestprep.co.uk;
Solution
1. D
The sketch is being built stage by stage. A new line is added in each stage and it never
touches the last line added in the previous stage.
2. B
Each figure consists of 3 shapes; namely: external shape, middle shape, and inner
shape. Notice that the external shape appears to be the middle shape of the next figure.
The middle shape disappears in the next figure. While the inner shape appears to be
as the external shape of the figure two steps forward. For example:
The external shape of the 4th figure is a circle,
The middle shape is a pentagon, and the inner
shape is a hexagon. The circle becomes the
4th figure
middle shape of the next figure, the pentagon
disappears in the next figure, and the hexagon becomes the external shape of the figure
two steps forward from the 4th figure.
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1
Module
Section 1: The Nature of Mathematics
GE 1 – Mathematics in the Modern World
2
Number Pattern
Find the next number in the sequence.
Example
Author: Mary Jane B. Calpa
1. 3, 8, 13, 18, 23, ?
2. 1, 4, 9, 16, 25, 36, ?
Solution
1. For this sequence, the difference between each term is 5.
3, 8, 13, 18, 23, __
⋁ ⋁ ⋁ ⋁ ⋁
5 5 5 5 5
Thus, the next number is 28 (23 + 5).
2. The next number in the sequence can be determined in two ways:
(1) The sequence 1, 4, 9, 16, 25, 36 can be written as 12, 22, 32, 42, 52, 62. Thus, the next
number is 72, that is, 49.
(2) For this sequence, the difference
1, 4, 9, 16, 25, 36, __
⋁ ⋁ ⋁ ⋁ ⋁ ⋁
3 5 7 9 11 13
3 + 2 = 5; 5 + 2 = 7; 7 + 2 = 9; 9 + 2 = 11; 11 + 2 = 13
Thus, the next number is 49.
Examples 1 and 2 are usually seen on aptitude tests. Before we determine the next shape or
number, we have to observe the objects, look into their properties, and their relationship on other
objects. In such a way, we are allowed to hypothesize, predict, and construct generalizations based
on the observed patterns.
Patterns, such as geometric and word patterns, are also very common to us. Word patterns
focused on the morphological rules in pluralizing nouns, conjugating verbs for tense, and metrical
rules of poetry.
Examples:
baby: babies
buy: bought
trolley: trollies
bring: brought
catch: ?
ally: ?
answer: allies
answer: caught
https://newsinfo.inquirer.net/941295/bat
ok-tattooing-tattooing-mambabatok
https://www.our7107islands.com/basey-samarthe-new-banig-capital-of-the-philippines/
http://alvicsbatik.weebly.com/
mindanao-accessories--page2.html
While geometric patterns are designs that depict geometric shapes like lines, circles, and
polygons. Geometric patterns are observed in nature. These patterns are also associated to the
identification of a particular country and culture. Below are samples of geometric patterns that are
associated to Philippine ethnic groups and local regions.
(1)
(3)
(2)
(1) Tattoos in the Cordillera
(2) Woven mat “banig” in Basey, Samar
(3) T’boli belt made of beads
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1
Module
Section 1: The Nature of Mathematics
GE 1 – Mathematics in the Modern World
Author: Mary Jane B. Calpa
Self-Assessment Activity 1
I. For each set of figures, what comes next?
1.
A
B
C
B
C
D
2.
A
D
3.
A
B
C
D
II. What is the next number in the series?
4. 3, 6, 12, 24, 48, ?
5. 1, 4, 10, 22, 46, ?
6. 4, – 1, – 11, – 26, – 46, ?
Sources: (1) and (2) hhtp://www.jobtestprep.co.uk; (3) www.psychometric-success.com
5
Department of Mathematics, College of Science, University of Eastern Philippines
Answers to SAA 1:
1. C, 2. B, 3. C, 4. 96, 5. 94, 6. –71
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GE 1 – Mathematics in the Modern World
Author: Mary Jane B. Calpa
1
Module
Section 1: The Nature of Mathematics
The world and the universe are full of beautiful patterns and designs that are mathematical
in nature. Let us take a closer look on some of these patterns in nature and the world.
Symmetry and Angle of Rotation
Consider the figure below.
Leonardo da Vinci’s Vitruvian
Man is an image of proportion
and symmetry of the human
body.
https://sotafoundations9.wordpress.com/2018
/03/08/transferring-your-design-using-radialsymmetry/#jp-carousel-89
http://www.bio.miami.edu/dana/226/226F
09_22.html
https://leonardodavinci.stanford.edu/submi
ssions/clabaugh/today/health.html
When you draw an imaginary line across an object and the resulting parts are mirror images
of each other, we have shown a symmetry. The A figure above is symmetric about the axis indicated
by the broken line. This is called as line or bilateral symmetry and is common to animals and
humans.
Here are other images showing symmetry.
This flower has a three-fold
symmetry.
This starfish has a five-fold
symmetry.
Images Source:
https://biologydictionary.net/radialsymmetry/
Observe that if we rotate the flower and the starfish by several degrees, we can still have the
same appearance as the original position. This is called the rotational symmetry.
The smallest angle an object can be rotated while it is preserving its original formation is
called the angle of rotation.
1
A figure has a rotational symmetry of order 𝑛 (𝑛-fold rotational symmetry) if 𝑛 of a complete
turn leaves the figure unchanged.
To compute for the angle of rotation, we use
360°
𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 =
𝑛
For example,
This three-leaf clover has a 3-fold
120o
symmetry. The angle of rotation
is
120O.
o
120
90o
120o
This four-leaf clover has a 4-fold
symmetry. The angle of rotation
is 90O.
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Module
Section 1: The Nature of Mathematics
GE 1 – Mathematics in the Modern World
Author: Mary Jane B. Calpa
https://listverse.com/2013/04/21/10-beautifulexamples-of-symmetry-in-nature/
Snowflakes and Honeycombs
Look into a microphotographed snowflake below.
https://listverse.com/2013/04/21/10-beautifulexamples-of-symmetry-in-nature/
https://www.benefits-of-honey.com/honeycombpattern.html#:~:text=Studies%20on%20the%20geomet
ry%20of,and%20square%20makes%20smaller%20area.
Notice that it exhibits a pattern on each arm that repeats six times. This snowflake indicates
a six-fold symmetry. However, many snowflakes are not perfectly symmetric due to the effects of
the different atmospheric conditions such as temperature and humidity on the ice crystals as it
forms when they descend from the skies. The angle of rotation for the snowflake with a 6-fold
symmetry is 60O.
Humans are also marveled with the almost perfect hexagonal shape arrangements in
honeycombs.
Peacock’s Tail
The patterns exhibited in animal’s
external appearance has to do with their
growth; their survival; and even with their
chances to attract their mates.
Symmetric and repeated patterns,
enhanced with bright, beautiful colors, on the
feathers of a peacock’s tail are used to attract
their mates.
https://www.benefits-of-honey.com/honeycombpattern.html#:~:text=Studies%20on%20the%20geomet
ry%20of,and%20square%20makes%20smaller%20area.
The image on the right explains why mathematicians believed that hexagon is the most
effective way of storing honey. The hexagonal formation allows bees to store the largest possible
amount of honey with the use of the least amount of wax.
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1
Module
Section 1: The Nature of Mathematics
https://www.benefits-of-honey.com/honeycombpattern.html#:~:text=Studies%20on%20the%20geomet
ry%20of,and%20square%20makes%20smaller%20area.
Sunflower
Nature has gifted us with beautiful flowers. The brilliant colors, fragrant odors, petal
arrangements, and different sizes and number of petals make flowers very appealing. If we closely
observe these flowers, we can find interesting patterns.
For example, let us take a closer look on the orderly arrangement of sunflower seeds. We
can see clockwise and counterclockwise spirals extending outward from the center of the flower.
Moreover, the sunflower seed arrangement displays a very interesting numerical sequence called
the Fibonacci sequence. The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, … and so on. Each number
on the sequence is determined by adding the two preceding numbers.
The number of seeds spirals in a sunflower adds up to a Fibonacci number. Spirals of many
plants such as pineapple and pinecones also add up a Fibonacci number.
https://www.bigwalls.net/climb/camf/index.html
https://www.benefits-of-honey.com/honeycombpattern.html#:~:text=Studies%20on%20the%20geomet
ry%20of,and%20square%20makes%20smaller%20area.
Nautilus Shell
Another example that shows how nature seems to follow a certain set of rules governed by
mathematics is spiral patterns seen in a shell of a nautilus.
As the mollusk grows inside the shell, the shell also expands and attempts to maintain the
same proportional shape as it grows outward. This growth pattern results to refined spirals on the
shell which is very evident when it is sliced.
The image on the right is called the logarithmic spiral, also known as equiangular spirals.
The image shows a mathematical curve which has the property of maintaining a constant angle
between the radius and the tangent to the curve at any point on the curve. Equivalently, the
property states that as the distance from the spiral center increases (radius), the amplitudes of the
angles formed by the radii to the point and the tangent to the point remain constant.
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Module
Section 1: The Nature of Mathematics
GE 1 – Mathematics in the Modern World
Author: Mary Jane B. Calpa
World Population
According to the estimates of Worldometers (https://www.worldometers.info), the
Philippines will be rank 13 on the Most Populous Countries in year 2050, with population of
144, 488, 158. Questions like “How is this estimate being computed?”; “What are the factors used
in the computation of the estimate?” may arise. These questions can be answered by the
mathematical model of population growth.
The formula for exponential growth is, 𝐴 = 𝑃𝑒 𝑟𝑡
where:
𝐴 = size of the population after it grows;
𝑃 = initial number of people;
𝑟 = rate of growth;
𝑡 = time;
𝑒 ≈ 2.718 (This is the Euler’s constant with an approximate value of 2.718)
Example
2
Determine what is being asked in each problem.
1. Substitute the given values in the formula 𝐴 = 𝑃𝑒 𝑟𝑡 to find the missing quantity.
a. 𝑃 = 680,000; 𝑟 = 12% per year; 𝑡 = 8 years
b. 𝐴 = 731,093; 𝑃 = 525,600; 𝑟 = 3% per year
2. In the midyear of 2020, a country’s population is 109,581,078 with a growth rate of
approximately 1.35% per year. What will be the country’s population in 2050?
3. The exponential growth model 𝐴 = 25𝑒 0.02𝑡 describes the population of a town in
Northern Samar in thousands, 𝑡 years after 1998. What was the population of the
town in 1998?
Solution
1. a. To find the missing quantity, 𝐴, we substitute the given values to the formula:
𝐴 = 𝑃𝑒 𝑟𝑡 = (680,000) 𝑒 (0.12)(8)
we let 𝑟 = 12% = 0.12
0.96
𝐴 = (680,000) 𝑒
𝐴 = (680,000) (2.611696)
𝑒 0.96 ≈ 2.611696
𝑨 = 1,775,953
b. To find the missing quantity, 𝑡, we use the formula of 𝐴 to derive a formula for 𝑡.
𝐴
𝐴 = 𝑃𝑒 𝑟𝑡 ⇔ = 𝑒 𝑟𝑡
Dividing both of the formula by P to isolate 𝑒 𝑟𝑡 .
𝑃
731,093
525,600
= 𝑒 (0.03)𝑡
1.390968 = 𝑒 (0.03)𝑡
𝑙𝑛 1.390968 = 0.03𝑡
0.330000= 0.03𝑡
11 = 𝒕
Substitute the given values and let 𝑟 = 3% = 0.03
731,093
525,600
≈ 1.390968
Definition of logarithm
𝑙𝑛 1.390968 ≈ 0.330000
Divide both sides by 0.03
Checking of the solution by substitution of the given values and the obtained answer will serve as your
exercise.
2. Given are the following quantities: 𝑃 = 109,581,078; 𝑟 = 1.35% = 0.0135; 𝑡 = 30
years (Subtract: 2050 – 2020)
𝐴 = 𝑃𝑒 𝑟𝑡 = (109,581,078)𝑒 (0.0135)(30) = 164,295,239.
Thus, the population of the country in year 2050 is estimated to be 164,295,239 .
3. Since the exponential growth model describes the population 𝑡 years after 1998, we
consider 1998 as 𝑡 = 0 year and solve for the population size. 𝐴.
𝐴 = 25𝑒 0.02𝑡 = 25𝑒 0.02(0) = 25𝑒 0 = 25(1) = 25
Therefore, the population of the town in 1998 is 25,000.
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GE 1 – Mathematics in the Modern World
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1
Module
Section 1: The Nature of Mathematics
Self-Assessment Activity 2
Answer completely. (Use 6 significant digits in the approximated values)
1. Find the missing quantity in the formula 𝐴 = 𝑃𝑒 𝑟𝑡 by substitution of the given values:
a. 𝑃 = 505,050; 𝑟 = 5% per year; 𝑡 = 1 year
b. 𝑃 = 240,100; = 11% per year; 𝑡 = 10 years
c. Find 𝑟 correct to 4 significant digits.
𝐴 = 786,000; 𝑃 = 247,000; 𝑡 = 17 years
2. The exponential growth model 𝐴 = 45𝑒 0.19𝑡 describes the population of a city in the
Philippines in thousands, t years after 1995.
a. What is the population of the city in 1995?
b. What is the population after 25 years?
c. What is the population in 2045?
 STOP 
Break Time (10 – 15 minutes)
Look around you. Try to observe for patterns in your bedroom, house, or backyard.
Do they have geometric patterns or numbers patterns? What makes these patterns appealing?
The Fibonacci Sequence
To start the new discussion, try to answer the puzzle below:
Form a rectangle using all of the squares. The measures of the sides are indicated in each square.
Department of Mathematics, College of Science, University of Eastern Philippines
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Answers to SAA 2:
1. (a) 530,945, (b). 721,300, (c). 6.809%,
2. (a) 45,000, (b). 5,201,293, (c). 601,187,707
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GE 1 – Mathematics in the Modern World
Author: Mary Jane B. Calpa
1
Module
Section 1: The Nature of Mathematics
One of the answers to the puzzle is shown below:
Notice that the number pattern formed when the squares are placed side by side.
The sequence gives as 1, 1, 2, 3, 5, 8. Does this sequence looks familiar?
In the previous discussion, it is said that sunflower seed arrangement displays a very
interesting numerical sequences called the Fibonacci sequence.
The sequence is 1, 1, 2, 3, 5, 8, 13, 21, … was discovered by an
Italian named Leonard Pisano Bigollo who is known in mathematical
history by several names: Leonardo of Pisa and Fibonacci.
Fibonacci’s 1202 book “Liber Abaci” introduced the sequence
to Western European mathematics, although there are some claims
that the sequence has been discovered earlier in Indian mathematics.
It is said that Fibonacci discovered the number sequence
through a practical problem involving the growth of a hypothetical
population of rabbits based on idealized assumptions.
This problem has an assumption that a pair of rabbits will be
born each month and will reproduce a new pair of rabbits two months
after they were born.
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Section 1: The Nature of Mathematics
The assumption is illustrated below:
https://www.storyofmathematics.com/medieval_fibonacci.html
Clearly, each number on the sequence is determined by adding the two preceding numbers.
Notice that if we divide two consecutive terms in the
Fibonacci sequence, the quotient approaches a particular
number. It is the number phi, 𝜙 = 1.6180339887… or the
1+ √5
irrational number 2 . This number is called the golden
ratio.
In mathematics and in arts, two quantities are in
golden ratio if their ratio is the same as their sum to the
larger of the two quantities.
In symbols, 𝑎 and 𝑏, where 𝑎 > 𝑏 > 0, are in a golden ration
𝑎
𝑎+𝑏
if 𝑏 =
.
𝑎
1
𝑛
Fibonacci
Number 𝐹𝑖𝑏(𝑛)
1
2
1
3
2
4
3
5
5
6
8
7
13
8
21
9
34
10
55
11
89
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𝐹𝑖𝑏(𝑛 + 1)
𝐹𝑖𝑏(𝑛)
1
= 1.00000
1
3
=1.50000
1
2
2
5
3
8
=2.00000
≈1.66667
=1.60000
5
13
8
21
13
34
21
55
34
89
=1.62500
≈1.61538
≈1.61905
≈1.61765
≈1.61818
55
144
89
≈1.61798
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Module
Section 1: The Nature of Mathematics
Here are some of man’s greatest works that would reminds us of the Fibonacci sequence and
the golden ratio.
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Example
3
1
Module
Section 1: The Nature of Mathematics
Answer completely.
1. Find the next three terms of the sequence: 1, 1, 2, 3, 5, 8, …
2. Let 𝐹𝑖𝑏(𝑛) be the nth term of the Fibonacci sequence, with 𝐹𝑖𝑏(1) = 1,
𝐹𝑖𝑏(2) = 1, 𝐹𝑖𝑏(3) = 2, and so on.
a. Find 𝐹𝑖𝑏(10)
b. Find 𝐹𝑖𝑏(17)
3. If 𝐹𝑖𝑏(22) = 17,711 and 𝐹𝑖𝑏(24) = 46,368, what is 𝐹𝑖𝑏(23)
Solution
1. To find the next three terms, we add the two preceding numbers:
5 + 8 = 13;
8 + 13 = 21;
13 + 21 = 34
Thus, the next three terms of the sequence are 13, 21, 34.
Completing the sequence, we have1, 1, 2, 3, 5, 8, 13, 21, 34.
2. a. 𝐹𝑖𝑏(1) = 1, 𝐹𝑖𝑏(2) = 1, 𝐹𝑖𝑏(3) = 2, 𝐹𝑖𝑏(4) = 3, 𝐹𝑖𝑏(5) = 5, 𝐹𝑖𝑏(6) = 8,
𝐹𝑖𝑏(7) = 13 , 𝐹𝑖𝑏(8) = 21, 𝐹𝑖𝑏(9) = 34
𝑭𝒊𝒃(𝟏𝟎) = 𝑭𝒊𝒃(𝟖) + 𝑭𝒊𝒃(𝟗) = 𝟐𝟏 + 𝟑𝟒 = 𝟓𝟓
𝑏. 𝐹𝑖𝑏(10) = 55, 𝐹𝑖𝑏(11) = 89, 𝐹𝑖𝑏(12) = 144, 𝐹𝑖𝑏(13) = 233, 𝐹𝑖𝑏(14) = 377,
𝐹𝑖𝑏(15) = 610, 𝐹𝑖𝑏(16) = 987
𝑭𝒊𝒃(𝟏𝟕) = 𝑭𝒊𝒃(𝟏𝟓) + 𝑭𝒊𝒃(𝟏𝟔) = 𝟔𝟏𝟎 + 𝟗𝟖𝟕 = 𝟏𝟓𝟗𝟕
3. 𝐹𝑖𝑏(23) = 𝐹𝑖𝑏(24) − 𝐹𝑖𝑏(22) = 46,368 – 17,711 = 28,657
Self-Assessment Activity 3
Answer completely.
1. Let 𝐹𝑖𝑏(𝑛) be the nth term of the Fibonacci sequence, with 𝐹𝑖𝑏(1) = 1,
𝐹𝑖𝑏(2) = 1, 𝐹𝑖𝑏(3) = 2, and so on.
a. Find 𝐹𝑖𝑏(20)
b. Find 𝐹𝑖𝑏(25)
2. Evaluate the following sum.
a. 𝐹𝑖𝑏(1) + 𝐹𝑖𝑏(2)
b. 𝐹𝑖𝑏(1) + 𝐹𝑖𝑏(2) + 𝐹𝑖𝑏(3)
c. 𝐹𝑖𝑏(1) + 𝐹𝑖𝑏(2) + 𝐹𝑖𝑏(3) + 𝐹𝑖𝑏(4)
3. What will be the sum of 𝐹𝑖𝑏(1) + 𝐹𝑖𝑏(2) + ⋯ + 𝐹𝑖𝑏(10)?
4. If we construct a number sequence using the following:
𝐹𝑖𝑏(1) + 𝐹𝑖𝑏(2), 𝐹𝑖𝑏(1) + 𝐹𝑖𝑏(2) + 𝐹𝑖𝑏(3) +….+ 𝐹𝑖𝑏(1) + 𝐹𝑖𝑏(2) + ⋯ + 𝐹𝑖𝑏(10),
what pattern can be observed?
Math for our World
The world and the universe are full of complexities and uncertainties. These are very evident
in our day-to-day living. From the simplest household chores to challenging and laborious tasks,
from observation and prediction of weather conditions and natural phenomena to its survival from
the aftermath, from legislation to implementation of governing laws, from theater shows to high
definition videos and films; from ancient navigational methods to global national satellite systems,
these and the like are some of the intricacies of the universe. With this, people of the ancient times
and the modern world have learned to live and to cope.
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4. The differences of two preceding numbers form a
Fibonacci sequence.
Answers to SAA 3:
1. (a) 6,765, (b) 75,025, 2. (a) 2, (b) 4, (c) 7, 3. 143,
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Section 1: The Nature of Mathematics
Technologies were invented and areas of knowledge were cultivated for better
understanding of the underlying aspects that would lead to the world’s development. Furthermore,
contexts of the real world and the universe were revealed for varied reasons through science and
mathematics.
Man’s exploration of the world began with its curiosity of the objects within and outside
Earth’s surface. The Babylon were successful in predicting eclipses. Aristarchus of Samos,
determined that the sun is the center of the universe through mathematical computation.
Hipparchus calculated the size of the moon and its distance from Earth, and determined specific
locations of geographical points through the use of longitude and latitude measurements. The
earliest civilizations of Babylon, China, and Egypt devised calendars and completed public works,
such as irrigational canals for agricultural purposes. Tax collections were made possibly easy when
Blaise Pascal invented the world’s first digital calculator when the first mechanical computer was
invented by Charles Babbage. (Todd, 2003)
Computations done by scientists and mathematicians paved the way to greater discoveries
and creations of expedient methods, devices, and tools. Galileo Galilei who invented the first
thermometer was able to measure temperatures of water and air. Similarly, Gabriel Daniel
Fahrenheit introduced the boiling and freezing points of liquid when he invented the first mercury
thermometer. James Gregory’s invention of reflecting telescope, and Ole Roemer’s calculations of
measuring the speed of light made great contributions to physics and other related fields. (Todd,
2003)
Isaac Newton’s explanation of the Universal Law of Gravitation and Laws of Motion, and
Albert’s Einstein Theory of Relativity, the motion particles when suspended within a liquid, and the
mathematical formula e = mc2 are some of the notable contributions in the field of science and
mathematics.
In its purest sense, mathematics is an abstraction. It is extremely useful in describing and
predicting events of the world. It has the ability to model effectively numerous aspects of the world
by creating abstract structures that have properties or attributes to its real-world counterparts.
Models, if they behave in a manner that truly parallel with originals, are used to make conclusions
and/or predictions about the real world. (Post, 1981)
Mathematics finds wide applications in arts, nature, music, medicine, chemistry, biology,
astronomy, and in other disciplines. The artist of antiquity and of the modern times described their
works using the Golden Ratio and Fibonacci Numbers. They are very evident in paintings,
architecture, sculpture, dance, and even in music. Throughout the ages, music and mathematics go
hand-in-hand when Pythagoras was able to establish a fundamental relationship between vibrating
strings and harmony. Today, music is stored and played digitally. The sound production, recording,
and engineering use advanced technology equipped with mathematics and science.
Nicolas Copernicus, by using uniform circular motion, modelled that the planets revolved
around the sun. The Heliocentrism paved the way to greater discoveries and explorations of the
solar system. Mathematics calculations are used to launch satellites, rockets and other space robes,
and used to describe the natural order and occurrences of the universe.
In medicine, mathematics is use to: 1) predict complex medical situations; 2) model
biological processes that underlie a disease, and 3) develop formulas from chemistry and physics,
and medical technologies. (Lerner and Lerner, 2006).
In this section, we have just seen how mathematics help our lives better and make the world
a better place to live in. As we learn mathematical concepts, we also apply them in our day-to-day
interactions and in solving societal problems. Moreover, mathematics is useful in making
conclusions and/or predictions of the events of the world. It is used to organize patterns and
regularities as well as irregularities. It is, therefore, mathematics is considered an integral part of
our world.
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Section 1: The Nature of Mathematics
Summary
The lessons in Section 1 (1.1 Mathematics in our World) allow us to “get-to-know”
mathematics. Far from the idea that it is full of difficulties and complexities, mathematics is a
study of patterns, an art, a language, a process of thinking, and a set of problem-solving tools.
We were able to see through the beauty of the world through observed patterns that are
mathematical in nature. Patterns that are in the natural objects and man-made creations. The
ratios of the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, …) converge to the Golden Ratio that made
these creations more aesthetically pleasing.
With a careful understanding, we were able to see the beauty and the significance of
mathematics in our day-to-day living. We are reminded how evident mathematics in the world.
The module has bought us to realization the usefulness of mathematics that makes our lives
better and our world a better place to live.
Responses To Consider
Congratulations! You have finished Module 1.
After working with this module, what are the new ideas about mathematics did you
learn? What is it about mathematics that might have changed your thoughts about it? What
is most useful about mathematics for humankind?
You are encouraged to provide a lecture notebook where you can write all your
responses and solutions to the activities and SAAs. Answers to SAAs are provided at the
bottom part of the page. If you have difficulty in obtaining the correct answer, you can go
over again with the examples. To be successful in mathematics, you have to do mathematics.
Do it without the fear of facing more problems and questions to solve. For further
understanding, use the references, suggested readings, and other materials indicated in the
module.
References
Aufmann, R., Lockwood, J., et.al, Mathematics in the Modern World, Rex Bookstore, Inc., 2018.
Lerner, K.L., Lerner, B.W., Real-life Math, Vol. 2, Thomson Gale, 2006.
Nocon, R., Nocon, E., Essential Mathematics for the Modern World, C & E Publishing, Inc. 2018.
Post, T.R., The Role of Manipulative Materials in the Learning Mathematical Concepts.
Retrieved from: http://www.cehd.umm.edu/ci/rationalnumberproject/81_4.html
Images Sources:
hhtp://www.jobtestprep.co.uk
www.psychometric-success.com
https://www.library.illinois.edu/mtx/2018/10/09/mathematics-in-nature/
https://www.weareteachers.com/teacher-dresses-ms-frizzle/
https://www.smithsonianmag.com/science-nature/science-behind-natures-patterns180959033/
http://mustafacil-online.blogspot.com/2015/08/manmade-patterns.html
https://newsinfo.inquirer.net/941295/batok-tattooing-tattooing-mambabatok
https://www.our7107islands.com/basey-samar-the-new-banig-capital-of-the-philippines/
http://alvicsbatik.weebly.com/mindanao-accessories---page2.html
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https://www.benefits-of-honey.com/honeycombpattern.html#:~:text=Studies%20on%20the%20geometry%20of,and%20square%20makes
%20smaller%20area.
https://www.bigwalls.net/climb/ca
https://listverse.com/2013/04/21/10-beautiful-examples-of-symmetry-in-nature/
mf/index.html
https://www.storyofmathematics.com/medieval_fibonacci.html
Other Materials
https://vimeo.com/9953368
htpps://youtu.be/pb0MSMGSley (BBC’s Documentary “The Language of the Universe”)
Suggested Readings
Stewart, Ian, Nature’s Numbers
Adam, John A., Mathematics in Nature: Modeling Patterns in the natural World
Adam, John A., A Mathematical Nature Walk
Akiyama & Ruis, A Day’s Adventure in Math Wonderland
Enzensberger, The Number Devil
Note To Students
You can discuss the lessons with your GE 1 instructor/professor through the different
modes of communication (email, Messenger, Moodle, Google Meet, Zoom, Google classroom,
etc.). Your GE 1 instructor/professor will contact you using the email address and/or mobile
number who have provided the University upon your registration.
If you have not received a message from your assigned faculty at least two (2) weeks
from the resumption of classes (October 5, 2020), please send your concerns to the
Department of Mathematics Chair using the following address: mjcalpauepcs@gmail.com, or
through your respective municipal links. Please include your FULL NAME, STUDENT
NUMBER, COURSE – YEAR, and GE 1 CLASS ID NUMBER.
Deadline of submission of Worksheet and Reflection Paper to the Municipal Link:
October 23, 2020
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Section 1: The Nature of Mathematics
GE 1 – Mathematics in the Modern World
Author: Mary Jane B. Calpa
Note to the Municipal Link:
WORKSHEET 1
1. The worksheet and reflection should
be forwarded to:
MARY JANE B. CALPA
Chair, Department of Mathematics
College of Science,
University of Eastern Philippines
Catarman, Northern Samar
2. Please check the date of submission
indicated on the student’s information
below.
Thank you.
1.1 Mathematics in Our World
To the Students:
1. Fill out the “Student’s Information” completely.
2. Write all your solutions/answers on the space
provided below each item.
3. Write legibly. Use blue- or black-ink ball pen only.
4. Submit on or before the indicated deadline.
5. For queries, please contact your respective GE 1
instructor/professor.
Student’s Information:
Student Number:
Last Name, First Name M.I.:
Class ID Number:
Professor/Instructor’s Name:
Course – Year:
Date of Submission:
A. Answer completely.
1. a. Draw the next three shapes in the pattern.
Solution & answer here:
b. What is the next figure? Draw your answer on the empty block.
2.
3.
Fill in the missing numbers.
a. 2, 4, , 16, 32, …
b. 100, 81, 64, , 36, …
c. 0, 1.5, 4, , 12, …
d. 1, 7, 17, , 49, …
e. , 6, 11, 16, 21, …
What letter comes next in this sequence: O T T F F S S E? Justify your answer.
Solution & answer here:
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4. Given 𝐹𝑖𝑏(30) = 832,040 and 𝐹𝑖𝑏(28) = 317,811, what is 𝐹𝑖𝑏(29)?
GE 1 – Mathematics in the Modern World
Solution & answer here:
5. The exponential growth model A = 1.5 𝑒 0.015𝑡 describes the number of tourists of a beach
resort in thousands, 𝑡 years after 2000.
a. How many tourists visited the beach resort in 2020?
b. How many tourists visited the beach resort after 15 years?
Solution & answer here:
6. A house was purchased for ₱1,000,000 in 2002. The value of the house is given by the
exponential growth model A = 1,000,000𝑒 0.645𝑡 . Find 𝑡 when the house would be worth
₱5,000,000.
Solution & answer here:
𝐹𝑖𝑏(𝑛+1)
as 𝑛 gets larger is said to approach the Golden Ratio, which is
𝐹𝑖𝑏 (𝑛)
𝐹𝑖𝑏(𝑛)
approximately equal to 1.618. What happens to the inverse of this ratio,
? What
𝐹𝑖𝑏(𝑛+1)
number does this quantity approach? How does this compare to the original ratio?
7. The ratio
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Solution & answer here: (number 7)
8. A wood 120 meters in length is to be cut into two parts such that the ratio of the parts
constitutes the Golden Ratio. What must be the lengths of the woods?
Solution & answer here:
9. Ask for a family member to help you on this activity.
Use a ruler or measuring tape to measure the following:
a. distances from A, B, C as indicated in the figure on the right;
b. length of your hand;
c. distance from your wrist to your elbow.
Calculate the following ratios using the indicated units of measure.
1. ratio 1 =
2. ratio 2 =
3. ratio 3 =
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝐶
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝐵
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝐵
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝐴
(in centimeters)
(in inches)
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑦𝑜𝑢𝑟 𝑤𝑟𝑖𝑠𝑡 𝑡𝑜 𝑦𝑜𝑢𝑟 𝑒𝑙𝑏𝑜𝑤
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑦𝑜𝑢𝑟 ℎ𝑎𝑛𝑑
(in inches and in centimeters)
Determine which of the measure/s give an approximate value of 𝜙.
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Solution & answer here: (number 9)
10. Answer completely: What is the importance of mathematics in my chosen field of
specialization?
Examples: “What is the importance of mathematics in community development? in
agriculture? in biology? in political science? in forestry? in environmental science?
Cite references and use extra A4-size bond paper if necessary.
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Section 1: The Nature of Mathematics
B. Choose only one (1) activity.
(You may provide separate A4 -size bond paper/s or you may also use this page.)
•
Look for patterns in nature. Write a 1- to 2-page report about the observed patterns. Include
a picture of these patterns.
•
Create your family tree up to the third generation using lines, shapes, or any geometrical
figures.
•
Draw your “mathematics” face mask. Use patterns and colors.
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