MATHEMATICS IN THE MODERN

WORLD
GNED - 03
FIRST MODULE
LEARNING OBJECTIVES:

1. Identify and follow patterns, whether consciously or subconsciously.

2. Awareness of these patterns allowed humans to survive, similar many flora and fauna
also follow certain patterns such as the arrangement of leaves and stems in a plant, the
flower’s petal.
3. Analyze the Fibonacci sequence.
4. Predict the behavior of nature and phenomena in the world and helps humans exert
control over occurrences in the world in the world for the advancement of our
civilization.
A. Patterns and Numbers in Nature and the World

In the general sense of the word, patterns are regular, repeated, or

recurring forms or designs. We see pattern every day, from the layout
of floor tiles, designs of skyscrapers, to the way we tie our shoelaces.
Studying patterns help students in identifying relationships and finding
logical connections to form generalizations and make predictions.
Example 1. What Number Comes Next?
What number comes next in 1, 3, 5, 7, 9, ___ ?
Solution: Looking at the given numbers, the sequence is increasing, with each term
being two more than the previous term: 3 = 1 + 2; 5 = 3 + 2; 7 = 5 + 2; 9 = 7 + 2.
Therefore, the next term should be 11 = 9 + 2.
Example 2. What Comes Next?
What is the next figure in the pattern below?
Solution: Looking at the given figures, the lines seem to rotate a 90- degree intervals
in a counterclockwise direction, always parallel to one side of the square. Hence,
either A or B could be the answer. Checking the other patterns, the length of the
lines inside the square follow a decreasing trend. So again, either A or B could be
the answer. Checking the other patterns, the length of the lines inside the square
follow a decreasing trend. So again, either A or B could be the answer. Finally,
looking at the number of the lines inside the box, each succeeding figure has the
number of lines increase by 1. This means that the next figure should have five lines
inside. This means that the next figure should have five lines inside. This leads to
option A as the correct choice.
Snowflakes and Honeycombs

Recall that symmetry indicates that you can draw an imaginary line across an object and the
resulting parts are mirror images of each other.

The figure is symmetric about the axis indicated by

the dotted line. Note that the left and
right portions are exactly the same. This type of
symmetry, known as line or bilateral
symmetry is evident in most animals, including
humans. Look in a mirror and see how the left
and right sides of your face closely match.
Snowflakes and Honeycombs
There are other types of symmetry depending on the number of sides or faces that
are symmetrical. Note that if you rotate the spiderwort and starfish above by several
degrees, you can still achieve the same appearance as the original position. This is
known as rotational symmetry. The smallest angle that a figure can be rotated while
still preserving the original formation is called angle rotation. For the spiderwort, the
angle of rotation is 120° while the angle of rotation for the baby starfish is 72°.
A more common way of describing rotational symmetry is by order of rotation.
Order of Rotation

A figure has a rotational symmetry of order n (n-fold rotational symmetry) if 1/n of a complete
turn leaves the figure unchanged. To compute for the angle of rotation, we use the following
formula:
Angle of rotation = 360°/𝑛
Order of Rotation
Order of Rotation

It can be observed that the patterns on a snowflake repeat six times, indicating that there is a
six-fold symmetry. To determine the angle of rotation, we simply divide 360° by 6 to get 60°.
Many combinations and complex shapes of snowflakes may occur, which lead some people to
think that “no two are alike”. If you look closely, however, many snowflakes are not perfectly
symmetric due to the effects of humidity and temperature on the ice crystal as it forms.

Another marvel of nature’s design is the structure and shape of a honeycomb. People have
long wondered how bees, despite their very small side, are able to produce such arrangement
while human would generally need the use of a ruler and compass to accomplish the same feat.
It is observed that such formation enables the bee colony to maximize their storage of honey
using the smallest amount of wax.
You can try it out for yourself. Using several coins of the same size, try to cover as much area
od a piece of paper with coins. If you arrange the coins in a square formation, there are still
plenty of spots that are exposed. Following the hexagonal formation, however, with the
second row of coins snugly fitted between the first row of coins, you will notice that more
area will be covered.
Translating this idea to three-dimensional space, we can conclude that hexagonal that
hexagonal formations are more optimal in making use of the available space. These are referred
to as packing problem. Packing problems involve finding the optimum method of filling up a
given space such as a cubic or spherical container. The bees have instinctively found the best
solution, evident in the hexagonal construction of their hives. These geometric patterns are not
only simple and beautiful, but also optimally functional.

Let us illustrate this mathematically. Suppose you have circles of radius 1 cm, each of which
will then have an area of 𝜋𝑐𝑚²
. We are then going to fill a plane with these circles using square
packing and hexagonal packing.
For square packing, each square will have an area of 4𝑐𝑚2
.Note from the figure that for each square, it can fit only one circle (4
quarters). The percentage of the square’s area covered by circles will
be

𝒂𝒓𝒆𝒂 𝒐𝒇 𝒕𝒉𝒆 𝒄𝒊𝒓𝒄𝒍𝒆𝒔/𝒂𝒓𝒆𝒂 𝒐𝒇 𝒕𝒉𝒆 𝒔𝒒𝒖𝒂𝒓𝒆

× 𝟏𝟎𝟎% =𝝅𝒄𝒎²/𝟒𝒄𝒎² × 𝟏𝟎𝟎% = 𝟕𝟖. 𝟓𝟒%

For hexagonal packing, we can think of each hexagon as composed

of six equilateral triangles
with side equal to 2 cm.
The area each triangle is given by
𝐴 =𝑠𝑖𝑑𝑒²×√3/4=(2𝑐𝑚)²×√3/4=4𝑐𝑚²×√3/4= √3𝑐𝑚²

This gives the area of the hexagon as 6√3𝑐𝑚². Looking at the figure,
there are 3 circles that could fit inside one hexagon (the whole
circle in the middle, and 6 one thirds of a circle), which gives the
total area as 3𝜋𝑐𝑚². The percentage of the hexagon’s area
covered by circles will be

𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒𝑠/𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 ℎ𝑒𝑥𝑎𝑔𝑜𝑛× 100% =3𝜋𝑐𝑚²/6√3𝑐𝑚² ×100%

= 90.69%
Comparing the two percentages, we can clearly see that using
hexagons will cover a large area
than when using squares.
Tigers’ Stripes and Hyenas’ Spots

Patterns are exhibited in the external appearances of animals. We are familiar with
how a tiger looks-distinctive reddish-orange fur and dark stripes. Hyenas, another
predator from Africa, are also covered in patterns of spots. These seemingly random
designs are believed to be governed by mathematical equations. According to a
theory by Alan Turing, the man famous for breaking the Enigma code during World
War 11, chemical reactions and diffusion processes in cells determine these growth
patterns. More recent studies addressed the question of why some species grow
vertical stripes while others have horizontal ones. A new model by Harvard University
researchers predicts that there are three variables that could affect the orientation
of these stripes-the substance that amplifies the density of stripe pattern; the
substance that changes one of the parameters involved in stripe formation; and the
physical change in the direction of the origin of the stripe.
Tigers’ Stripes and Hyenas’ Spots
The Sunflower

Looking at a sunflower up close, you will

notice that there is a definite pattern of
clockwise and counterclockwise arcs or
spirals extending outward from the center
of the flower. This is another demonstration
of how nature works to optimize the
available space. This arrangement allows
the sunflower seeds to occupy the flower
head in a way that maximizes their access
to light and necessary nutrients.
The Snail’s Shell

We are also very familiar with spiral patterns. The most common
spiral patterns can be seen in whirlpools and in the shells of snails
and other similar mollusks. Snails are born with their shells, called
protoconch, which start out as fragile and colorless. Eventually,
these original shells harden as the snails consume calcium. As the
snails grow, their shells also expand proportionately so that they
can continue to live inside their shells. This process resulted in a
refined spiral structure that is even more visible when the shell is
sliced. This figure, called an equiangular spiral, follows the rule that
as the distance follows the rule that as the distance from the spiral
center increase (radius), the amplitudes of the angles formed by
the radii to the point and the tangent to the point remain
constant. this is another example of how nature seems follow a
certain set of rules governed by mathematics.
Flower Petals
Flower Petals

Flowers are considered as things of beauty. Their vibrant colors and fragrant colors and
fragrant odor make them vey appealing as gifts or decorations. If you look more closely, you
will note that different flowers have different number of petals. Take the iris and trillium, for
example. Both flowers have only 3 petals.
Flowers with five petals are said to be the most common. These include buttercup,
columbine, and hibiscus. Among those flowers with eight petals are clematis and delphinium,
while ragwort and marigold have thirteen. These numbers are all Fibonacci numbers, which
we will discuss in detail in the next section.
World Population

As of 2017, it is estimated that the world population is about 7.6 billion. World leaders,
sociologists, and anthropologists are interested in studying population, including its growth.
Mathematics can be used to model population growth. Recall that the formula for
exponential
growth 𝐴 = 𝑃𝑒𝑟𝑡, where A is the size of the population after it grows, P is the initial number
of people, r is the rate growth, and t is time. Recall further that e is Euler’s constant with an
approximate value of 2.718. plugging in values to this formula would result in the population
size after time t with a growth rate of r.
Example 3 Population Growth

The exponential growth model 𝐴 = 30𝑒0.02𝑡 describes the population of a city in the Philippines in thousands, t years after 1995.
a. What was the population of the city in 1995?
b. What will be the population in 2017?

Solution
a. Since our exponential growth model describes the population t years after 1995, we
considered 1995 as 𝑡 = 0 and then solve for A, our population size.
A = 30𝑒0.02𝑡
A = 30𝑒(0.02)(0)
Replace t with t = 0
A = 30𝑒0
A = 30(1) 𝑒0 = 1
A = 30
Therefore, the city population in 1995 was 30,000
Example 3 Population Growth

b. We need find A for the year 2017. To find t, we subtract 2017 and 1995 to get t = 22,
which we then plug in to our exponential growth model.
A = 30𝑒0.02𝑡
A = 30𝑒(0.02)(22)
Replace t with t = 22
A = 30𝑒0.44
A = 30(1.55271) 𝑒0.44 =1.55271
A = 46.5813
EXERCISE SET
EXERCISE SET:

Determine what comes next in the given pattern.

1. A, C, E, G, I, _____
2. 27 30 33 36 39______
• Substitute the given values in the formula A = P𝑒
𝑟𝑡 to find the messing quantity.
3. P = 680,000; r = 12% per year; t = 8 year
The Fibonacci Sequence

as we have seen in the previous section, the human mind is hardwired to recognize
patterns. In mathematics, we can generate patterns by perming one or several
mathematical operations repeatedly. Suppose we choose the number 3 as the first
number in our pattern. We then choose to add 5 to our first number, resulting in 8,
which is our second number. Repeating this process, we obtain 13, 18, 23, 28, . . . as
the succeeding numbers that form our pattern. In mathematics, we call these ordered
lists of numbers a sequence.

Sequence
A sequence is an ordered list of numbers, called terms, that may have repeated values.
The arrangement of these terms is set by a definite rule.
Example 1 Generating Sequence

Analyze the given sequence for its rule and identify the next three terms
a. 1, 10, 100, 1000
b. 2, 5, 9, 14, 20

Solution
a. Looking at the set of numbers., it can be observed that each term is a power of 10:
1=100, 10 =101, 100 = 102, and 1000 = 103. Following this rule, the next three terms are
104 = 10,000, 105 = 100,000, and 106 = 1,000,000.
b. The difference between the first and second terms (2 and5) is 3. The difference between
the second and third terms (5 and 9) is 4. The difference between the third and fourth
terms (9 and 14) is 5. The difference between the fourth and fifth terms is 6. Following
this rule, it can be deduced that to obtain the next three terms, we should add 7, 8, 9,
respectively, to the current term. Hence, the next three terms are 20 + 7 = 27, 27 + 8 =
35, 35 + 9 = 44.
The Fibonacci Sequence

It is named after the Italian mathematician Leonardo of Pisa, who was better known by
his
nickname Fibonacci. He is said to have discovered this sequence as he looked at how a
hypothesized group of rabbits bred and reproduced. The problem involved having a
single pair
of rabbits and then finding out how many pairs of rabbits will be born in a year, with the
assumption that a new pair of rabbits is born each month and this new pair, in turn, give
birth
to additional pairs of rabbits beginning at two months after they were born. He noted
that the
set of numbers generated from this problem could be extended by getting the sum of
the two
previous terms.
The Fibonacci Sequence

Starting with 0 and 1, the succeeding terms in the

sequence can be generated by adding the two
numbers that came before the term:
The Fibonacci Sequence

While the sequence is widely known as Fibonacci sequence, this

pattern is said to have been discovered much earlier in India.
According to some scholarly articles, Fibonacci sequence is
evident in the number of variations of a particular category of
Sanskrit and Prakrit poetry meters. In poetry, meter refers to the
rhythmic pattern of syllables.Fibonacci sequence has many
interesting properties. Among these is that this pattern is very
visible in nature. Some of nature’s most beautiful patterns, like the
spiral arrangement of sunflower seeds, the number of petals in a
flower, and the shape of a snail’s shell-things that we looked at
earlier in the chapter-all contain Fibonacci numbers. It is also
interesting to note that the ratios of successive Fibonacci numbers
approach the number ∅ (Phi), also known as
the Golden Ratio. This is approximately equal to 1.618.
The Fibonacci Sequence

The Golden Ratio can also be expressed as the ratio

between two numbers, if the latter is also
the ratio between the sum and the larger of the two
numbers. Geometrically, it can also be
visualized as a rectangle perfectly formed by a
square and another rectangle, which can be
repeated infinitely inside each section.
EXERCISE SET
Exercise Set:

1. Find Fib (19)

2. Evaluate the sum of Fib (1) + Fib (2) + Fib (3) + Fib (4)
3. Determine the pattern in the successive sum from the previous
question. What will the
sum of Fib (1) + Fib (2) + . . . + Fib (10)
Mathematics for Organization

A lot of events happen around us. In the blink of an eye, several

children have already
been born, liters of water have been consumed, or thousands of
tweets have been posted. For us to make sense of all available
information, we need mathematical tools to help us make sound
analysis and better decisions. For instance, a particular store can
gather data on the shopping habits of its customers and make
necessary adjustments to help drive sales. Scientists can plot bird
migration routes to help conserve endangered animal populations.
Social media analysts can crunch all online postings using software
togauge the netizen’s sentiments on particular issues or personalities.
 Mathematics for Prediction

It is sometimes said that history repeats itself. As much as we can use mathematical models
using existing data to generate analysis and interpretations, we can also use them to make
predictions. Applying the concepts of probability,experts can calculate the chance of an
event occurring. The weather is prime example.Based on historical patterns,meteorologists
can make forecasts to help us prepare for our day-to-day activities. They can also warn us
of weather disturbances that can affect our activities for weeks or months. Astronomers also
use patterns to predict the occurrence of meteor showers or eclipses. In 2017
announcements were made about heavenly phenomena such as the Draconid Meteor
Shower and “The Great American Eclipse”. They were able to tell when these phenomena
would occur and where would be the best places to view them.
Mathematics for Prediction
Mathematics for Control

We have demonstrated by means of examples around us the patterns are definitely

present in universe. There seems to be an underlying mathematical structure in the way
that natural object and phenomenon behave. While photographers could capture a
Single moment through a snapshot, videographers could record events as they unfold.
Painters and sculptors could create masterpieces in interpreting their surroundings,
poets could use beautiful words to describe an object, and musicians could capture
and reproduce sounds that they hear. These observations of nature, as well as their
interactions and relationships, could be more elegantly described by means of
mathematical equations. As stated by astrophysicist Brian Greene. “With a few
symbols on a page, you can describe a wealth of physical phenomena”
 Mathematic for Control

It is interesting then to ponder on how mathematics, an invention of the human mind,

seems to permeate the natural laws that hold the universe together. There have been
instances when a natural phenomenon has been speculated to exist because mathematics
says so but no hard evidence had been found to support its existence. Such phenomena
were proven to exist only when advancements in technology have allowed us to expand
our horizons. For example, in 1916, Albert Einstein hypothesized the existence of
gravitational waves based on his theory of general relativity. This is when “ripples”
are formed in the fabric of space-time due to large and violent cosmic events, very much
like when a pebble is thrown on a stagnant pond. About a hundred years later, the Laser
Interferometer Gravitational Wave Observatory (LIGO) announced that it found
evidence of this phenomena.
Mathematic for Control

A large cosmic disturbance could cause ripples in space-

time, like a pebble thrown in pond.Though the use of
mathematics, man is also able to exert control over
himself and the effect of nature. The threat of climate
change and global warming has been the subject
of much debate over the years. It is believed that unless
man changes his behavior, patterns are said to indicate
that sea levels could rise to catastrophic levels as the
polar caps melt due to the increase in global
temperatures. To ensure that greenhouse gas
concentrations in the atmosphere are kept at levels that
would not interfere with the climate system, the United
Nations Framework Convention on Climate Change
(UNFCCC) was signed in 1992 and has 197 parties as of
December 2015.
Mathematics is Indispensable

In this chapter, it was highlighted how mathematics plays a huge

role in the underpinning of our world. We have seen it in living
creatures and natural phenomena. We have also looked at
examples of how mathematical concepts could be applied.
Whether you are on your way to becoming a doctor, an engineer,
an entrepreneur, or a chef, a knowledge of mathematics will be
helpful. At the most basic level, logical reasoning and critical
thinking are crucial skills that are needed in any endeavor. As
such, the study of mathematics should be embraced as it paves
the way for more educated decisions and in a way, brings us
closer to understanding the natural world.
End of Module One
GNED 03 MATHEMATICS IN THE MODERN WORLD