Welcome to the first module in Mathematics in the Modern World!
What is mathematics? Where is mathematics? What role does mathematics play
in your world?
MATHEMATICS
It is a science that deals with the logic of shape, quantity and arrangement.
Math is all around us, in everything we do.
Math is the building block for everything in our daily lives, including mobile devices,
architecture (ancient and modern), art, money, engineering, and even sports.
MATHEMATICS IN THE MODERN WORLD
It is about mathematics as a system of knowing or understanding our surroundings
- deals with nature of mathematics, appreciation of its practical, intellectual, and aesthetic
dimensions, and application of mathematical tools in daily life.
COURSE DESCRIPTION
This course deals with nature of mathematics, appreciation of its practical,
intellectual, and aesthetic dimensions, and application of mathematical tools in daily life.
The course begins with an introduction to the nature of mathematics as an
exploration of patterns (in nature and the environment) and as an application of inductive
and deductive reasoning. By exploring these topics, students are encouraged to go
beyond the typical understanding of mathematics as merely a set of formulas but as a
source of aesthetics in patterns of nature, for example and a rich language in itself (and
of science) governed by logic and reasoning.
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The course then proceeds to survey ways in which mathematics provides a tool
for understanding and dealing with various aspects of present-day living, such as
managing personal finances, making social choices, appreciating geometric designs,
understanding codes used in data transmission and security, and dividing limited
resources fairly. These aspects will provide opportunities for actually doing mathematics
in a broad range of exercises that bring out the various dimensions of mathematics as a
way of knowing, and test the students’ understanding and capacity. (CMO No. 20, series
of 2013)
COURSE OUTLINE
1. Nature of Mathematics
1.1 Patterns and Numbers in Nature and the World
1.2 The Fibonacci Sequence
1.3 Mathematics for our World
2. Speaking Mathematically
2.1 Variables
2.2 The Language of Sets
2.3 The Language of Relations and Functions
3. Problem Solving
3.1 Inductive and Deductive Reasoning
3.2 Problem Solving with Patterns
3.3 Problem-Solving Strategies
4. Statistics
4.1 Measures of Central Tendency
4.2 Measures of Dispersion
4.3 Measures of Relative Position
4.4 Normal Distributions
4.5 Linear Regression and Correlation
After completing this module, develop a piece of art connecting to patterns in
nature and mathematical sequences. It may be a mural, magazine cover, website, etc. In
a separate sheet of paper, you are going to explain the mathematical connection between
your work and sequence.
It is hope that this module has achieved its aim of producing a concise self-learning
kit which nevertheless considers all the significant topics comprehensively and coherently
enough for you.
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LESSON 1
THE NATURE OF MATHEMATICS
“Mathematics is the language with which God wrote the universe.”
Galileo Galilei
The emergence of digital technology has sparked a monumental rise in the rate at
which we consume and produce data. Before the internet, it could take hours to get
several volumes of resources from the library for a research paper. Today, a few minutes
(or seconds, depending on the speed of your connection) using your mobile device’s
browser could get you the same information, or even more. A few decades ago, it took
hours for photographs to be printed and shared, while now, it only takes a matter of
seconds for your perfect selfie to be uploaded and viewed by your relatives and friends
on the other side of the world.
In this fast-paced society, how often have you stopped to appreciate the beauty of
the things around you? Have you ever paused and pondered about the underlying
principles that govern the universe? How about contemplating about the processes and
mechanisms that make our lives easier, if not more comfortable? Most people do the
same routine tasks every single day and the fundamental concepts that make these
activities possible are often overlooked.
As rational creatures, we also tend to identify and follow patterns, whether
consciously or subconsciously, because it feels natural, like our brain is hardwired to
recognize them. Early humans recognized the repeating interval of day and night, the
cycles of the moon, the rising and falling of tides, and the changing of the seasons.
Awareness of these patterns essentially aided humans with survival. In a similar fashion,
many flora and fauna also follow certain patterns, i.e the arrangement of leaves and stems
in a plant, the shape of a snowflake, the flowers’ petals, or even the shape of a snail’s
shell.
What do you think? Do you also notice these patterns around you? What other
examples could you think of?
In this discussion, we will be looking at patterns and regularities in the world, and
how MATHEMATICS comes into play, both in nature and in human endeavor.
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1.1 : PATTERNS AND NUMBERS IN NATURE AND THE WORLD
DEFINITION
PATTERNS are regular, repeated, or recurring forms or designs.
EXAMPLES
▪ layout of floor tiles
▪ designs of buildings
▪ the way we tie our shoelaces
Patterns indicate a sense of structure or organization such that it would seem that
only humans are capable of producing these intricate, creative and amazing formations.
It is from this perspective that some people see an “intelligent design” in the way the
nature is created.
TYPES OF PATTERNS
SYMMETRY
DEFINITION
SYMMETRY indicates that you can draw an imaginary line across
an object and the resulting parts are mirror images of each other.
EXAMPLES
▪ butterfly
▪ Leonardo da Vinci’s Vitruvian Man
▪ starfish
The butterfly is symmetric about the axis indicated by the
black line. Note that the left and right portions
are exactly the same. This type of symmetry is
called bilateral symmetry.
Leonardo da Vinci’s Vitruvian Man
shows the proportion and symmetry of the human body.
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There are other types of symmetry depending on the number of
sides or faces that are symmetrical. Note that if you rotate the starfish
in Figure 3 by 72° , you can still achieve the same appearance as the
original position. This is known as the rotational symmetry.
SPIRAL
DEFINITION
SPIRAL is a curved pattern that focuses on a center point and a
series of circular shapes that revolve around it.
EXAMPLES
▪ pineapple
▪ pine cones
▪ hurricanes
The reason for why plants use a spiral form because they are constantly trying to
grow but stay secure. A spiral shape causes plants to condense themselves and not take
up as much space, causing it to be stronger and more durable against the elements.
FRACTAL
DEFINITION
FRACTAL is a detailed pattern that looks similar at any scale and
repeats itself over time. A fractal's pattern gets more complex as
you observe it at larger scales.
EXAMPLES
▪ fern
▪ lightning
▪ trees branching
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TESSELLATION
DEFINITION
TESSELLATION is a repeating patterns of polygons that covers a
flat surface with no gaps or overlaps.
EXAMPLES
▪ turtle
▪ honeycomb
ACTIVITY 1.1
Give at least three examples in each types of patterns that you can see in nature.
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SYMMETRY
SPIRAL
FRACTAL
TESSELLATION
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NATURAL PATTERNS
▪
SNOWFLAKES AND HONEYCOMB
Snowflakes are single ice crystals that often exhibit a six-fold
symmetry. This symmetry, known as line or bilateral symmetry is
very evident in most animals, including humans. Take a look in
mirror and see how the left and right sides of your face closely
match.
The pattern on the snowflake repeat six times, indicating that
there is a 6-fold symmetry. Using the formula. The angle of rotation
is 60°. Many combinations and shapes may occur, which lead some
people to think that “no two are alike”. Snowflakes aren’t perfect symmetric, however,
due to the effects of humidity and temperature on the ice crystals 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 size are able to produce
such configurations while humans would generally need
the use of a ruler and a compass to accomplish the same
feat. It is considered that such a 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 denomination, try to cover as much area of a piece of paper with coins. If
you arrange the coins in a square configuration, there are still plenty of spots that are
exposed. Following the hexagonal formation, however, with the second row of coins
snugly fitted in between the first row of coins, you will notice that more are will be
covered.
Translating this idea to three-dimensions, we can deduce that hexagonal
formations are more optimal in making use of the available space. These packing
problems, which, in the simplest sense, are those that involve in finding the optimum
method of filling up a given space, such as 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.
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TIGER STRIPES AND HYENA’S SPOT
Patterns exhibited in the external appearances of
animals. We are all familiar with how a tiger looks, with its
distinctive reddish-orange fur and dark stripes. Hyenas,
another predator from Africa are also covered in patterns of
spots. These seemingly unorganized or random designs are
believed to be governed by mathematical equation.
According to the theory of Alan
Turing, a British mathematician, the man famous for breaking
the Enigma code during World War II, chemical reactions and
diffusion in cell determine these growth patterns, as well as
influence other factors. More recent studies addressed the
question of why some species grow vertical stripes, while
others have horizontal ones. This new model follow the logic
that the local patterns produced by the various chemical processes will repeat given
a larger space such as an animal’s fur.
▪
▪
THE SUNFLOWER
Looking at the 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 of which could be seen in whirlpools, or in
the shells of snails or other similar mollusks. Snails are
born with their shells, called protoconch, and these start
out as very fragile and colorless. Eventually, these original
shells harden as the snails consume a calcium-rich diet. As the snails grow, their shells
also expand in the same proportion for them to be able to continue to live inside it.
This process results in a refined spiral structure that is even more visible when shell
is sliced. This figure, called an equiangular
spiral, follows the rule 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. This is yet another example of how
nature seems to follow a certain set of rules
governed by mathematics.
▪
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▪
FLOWER PETALS
Flowers are easily considered as things of beauty. Their
vibrant colors and fragrant odors make them very appealing as
gifts or decorations. Looking at the flowers closely, you will note
that each species has a different number of petals. Take the lily
and iris, for example, with both of them having only 3 petals in
their flowers. Flowers with five petals are said to be the most
common. These include the buttercup, the columbine, and the 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’ll discuss in a little bit
more detail in the next section.
▪
WORLD POPULATION
It is estimated that as of 2017, the world population is about 7.6 billion. All these
people are spread across various continents and countries. World leaders,
sociologists, and anthropologists are all interested in studying the population,
including its growth. Mathematics could be used to model the total world population
growth.
FORMULA FOR EXPONENTIAL GROWTH
𝑨 = 𝑷𝒆𝒓𝒕
A
-
size of the population after it grows
P
-
intial amount of people
r
-
rate of growth
t
-
time
e
-
Euler’s constant with an approximate value of 2.718
Example:
The exponential growth model 𝐴 = 30𝑒 0.02𝑡 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?
Since our exponential growth model describes the population t years after 1995,
we consider 1995 as t = 0 and then solve for A, our population size.
𝐴 = 30𝑒 0.02𝑡
𝐴 = 30𝑒 (0.02)(0)
𝐴 = 30𝑒 0
𝐴 = 30(1)
𝐴 = 30
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𝑒0 = 1
Therefore, the city population in 1995 was 30, 000.
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b. What will be the population in 2017?
We need to find A for the year 2017. To find t, we subtract 2017and 1995 to get
t=22, which we then plug into our exponential growth model.
𝐴 = 30𝑒 0.02𝑡
𝐴 = 30𝑒 (0.02)(22)
𝐴 = 30𝑒 0.44
𝑒 0.44 is approximately 1.55271
𝐴 = 30(1.55271)
𝐴 = 46.5812
Therefore, the city population would be about 46, 581 in 2017.
CHECK YOUR PROGRESS
The exponential growth model 𝐴 = 50𝑒 0.07𝑡 describes the population of a city in
the Philippines in thousands, t years after 1997.
a. What is the population after 20 years?
b. What is the population in 2037?
Take Note: The formula for exponential growth could also be used for exponential decay,
with the rate “r” being a negative value. The time “t” should coincide with the given rate
i.e if the time is in years, then the growth rate should also be a yearly growth rate.
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ACTIVITY 1.2
Substitute the given information in the formula 𝐴 = 𝑃𝑒 𝑟𝑡 to find the missing
quantity. Show your solution.
1. P = 680, 000
;
r = 12% per year
;
t = 8 years
2. A = 1, 240, 000 ;
r = 8% per year
;
t = 30 years
3. A = 786, 000
;
P = 247, 000
;
t = 17 years
4. A = 731, 093
;
P = 525, 600
;
r = 3% per year
5. Suppose the population of a certain bacteria in a laboratory ample is 100. If it
doubles in population every 6 hours, what is the growth rate? How many
bacteria will there be in two days?
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1.2 : FIBONACCI SEQUENCE
The human mind is hard-wired to recognize patterns, as we’ve seen when we
noted som eof nature’s wonders. In mathematics, we could generate patterns by
performing one or several mathematical operations over and over. Suppose we start with
the number 3 as the first number in our pattern. We then choose to ass the number 5 to
our first number, resulting in 8, which is our secod 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.
DEFINITION
SEQUENCE is an ordered list of numbers, called terms, that may
have repeated values.
EXAMPLES
▪ 2, 4, 6, 8, 10
▪ 18, 14, 10, 6, 2
The arrangements of these terms is set by some definite rule.
Analyze the given sequence for its rule and identify the next three terms.
a. 1, 10, 100, 1000
Looking at the set of numbers, it can be observed that each term is a power of 10:
0
1 = 10 , 10 = 101 , 100 = 102 , and 1000 = 103 . Following this rule, the next three terms
are 104 = 10000, 105 = 100000, and 106 = 1000000.
b. 2, 5, 9, 14, 20
The difference between the first and second terms (2 and 5) is 3. The difference
between the second and third terms (5 and 9) is 4. The diffrence between the third and
fourth terms (9 and 14) is 5. Following this rule, it can be deduced that to obtain the next
termthe current term should be incresed by 2+n, where n is the position of the current
term. Hence the followingthree terms are 20+2+5 = 27, 27+2+6 = 35, and 35+2+7 = 44.
CHECK YOUR PROGRESS
Analyze the given sequence for its rule and identify the next three terms.
a. 16, 32, 64, 128
b. 1, 1, 2, 3, 5, 8
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The sequence in item B is a very special sequence called 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 rabbit and the n 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, gives birth to additional pairs of rabbits beginning at two months after they were born
he noted that the set of number generated from this problem could be extended by getting
the sum of the two previous terms.
Starting with 0 and 1, the succeeding terms in the sequence could be generated
by adding the two numbers that came before:
0+1=1
0, 1, 1
1+1=2
0, 1, 1, 2
1+2=3
0, 1, 1, 2, 3
2+3=5
0, 1, 1, 2, 3, 5
3+5=8
0, 1, 1, 2, 3, 5, 8
5 + 8 = 13
0, 1, 1, 2, 3, 5, 8, 13
…
0, 1, 1, 2, 3, 5, 8, 13, …
While the sequence carries Fibonacci’s name, this particular pattern is said to have
been discovered much earlier in India. According to some scholarly articles, this 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. Counting these as they appear
in poetry leads to the general rule for the formation of the Fibonacci sequence, which is
This special sequence has many interesting properties. Among these is that this
pattern is very visible in nature. Some of nature’s most beautiful patterns, like spiral
arrangement of sunflower seeds, the number of petals in a flower, and the shape of a
snail’s shell – things that we looked earlier in this chapter – all contain Fibonacci numbers.
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It is also interesting to note that the ratios of successive Fibonacci numbers approach the
number Φ (Phi), also known as the Golden Ratio. This is approximately equal to 1.618.
1/1 = 1.0000
2/1 = 2.0000
3/2 = 1.5000
5/3 = 1.6667
8/5 = 1.6000
13/8 = 1.6250
21/13 = 1.6154
34/21 = 1.6191
55/34 = 1.6177
89/55 = 1.6182
This 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. Shapes and figures that
bear this proportion are generally considered to be aesthetically pleasing. As such, this
ratio is visible in many works of art and architecture such as in the Mona Lisa, the Notre
Dame Cathedral, and the Parthenon. In fact, the human DNA molecule also contains
Fibonacci numbers, being 34 angstroms long by 21 angstroms wide for each full cycle of
the double helix spiral. As shown in the list above, this approximates the Golden Ratio at
a value of about 1.619 (1 angstrom=10−10 meter or 0.1 nanometer)
Golden Rectangle with the Golden Spiral
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ACTIVITY 1.3
I.
Let Fib (n) be the nth term of the Fibonacci sequence, with Fib (1) = 1, Fib (2) = 1,
Fib (3) = 2 and so on.
1. Fib (8).
2. Fib (19).
3. If Fib (22) = 17, 771 and Fib (24) = 46, 368, what is Fib (23)?
4. Fib (13)
5. Fib (16)
II.
III.
Evaluate the following sums.
6. Fib (1) + Fib (2)
=
______________
7. Fib (1) + Fib (2) + Fib (3)
=
______________
8. Fib (1) + Fib (2) + Fib (3) + Fib (4) =
______________
9. Fib (6) + Fib (10)
______________
=
10. Fib (12) + Fib (23)
=
Compare and contrast patterns and sequence.
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1.3 : MATHEMATICS FOR OUR WORLD
We have witnessed in the preceedings ections how mathematics is evident in the
natural world – specifically in how the patterns that we observe in nature follow logical
and mathematical structures. It is therefore of great importance to lean mathematical
concepts and procedures and apply them in solving our problems, from our individual
concerns to matters that affect the rest of the world.
MATHEMATICS FOR ORGANIZATION
So many events happen all around us. In the blink of an eye, several children have
already been born, so many liters of water have been consumed, or several thousands
of tweets have been posted. For us to make sense of all these data, we need
mathematical tools to help us make sound analysis and better decisions.
A particular store could gather data on the shopping patterns of their customers
and adjust their pricesor product placements to help drive sales. Scientists could not plot
bird migration routes to help endangered animal population. Social media analysts could
crunch all the online postings using software to gauge the netizens’sentiments on
particular issues or personalities. Software could generate a similar map of words that
are most talked about in social media. The bigger the font, the more netizens are talking
about the concept or topic.
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 us ethem to make predictions. Applying the concepts of probability, mathematicians
could calculate the chances of an event occurring. The wather is a prime example. Based
on historical patterns, meteorologists could make relatively accurate forecasts to help
prepare us for our day-to-day activities, or warn us of weather systems that could affect
entire populations for weeks or months. Astronomers could also use these regular
patterns to predict the occurrence of meteor showers or eclipses. In 2017,
announcemnets were made regarding when heavenlyphenomena such as the Draconid
Meteor Shower and “The Great American Eclipse” would occur and where would be the
best places to view them.
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MATHEMATICS FOR CONTROL
We have demonstrated by means of examples around us that patterns ae definitely
present in the universe. There seems to be underlyingmathematical structure in the way
that natural objects and phenomenon behave. While photographers could capture a
single moment through a snapshot, fimmakers could record events live as they unfold,
painter and sculptors could create masterpieces in interpreting their surroundings, poet
could use beautiful words to describe an object, and songwriters 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.”
It is interesting then to ponder on how mathematics, an invention of the human
mind, seems to perpermeate the innate laws that hold the universe togethe. There have
been instances when natural phenomenon have been predicted to exist because the
mathematics says it, but have only been proven after many years when advancements in
technology have allowed us to expand our horizions and view a little more into the abyss.
Through the use of mathematics, man is also able to exert control over himself and
the effects of nature. The ongoing threat of climatechange 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 the sea levels could rise to catastrophic levels a s
the polar caps melt due to the increase in global temperatures, fueled by deforestation
and the release of poluutants into the atmosphere.
MATHEMATICS IS INDISPENSABLE
Mathematics play a huge role in the underpinnings of our world. We have seen it in living
creatures and natural phenomena. We have also lookedat examplesof how mathematical
concepts could be applied. Whethet you are on your way to becoming a doctor, an
engineer, an entrepreneur, or a chef, a knowledge of mathematics could onlybe helpful.
In 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.
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CHAPTER EXERCISES
Patterns and Sequences
1. Which of the figures can be used to continue the series given below?
2. Which of the figures, you think best fits the series below?
3. Which of the figures can be used to continue the series given below?
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4. Which number should come next in this series?
18, 26, 34, 42
a. 46
c. 50
b. 52
d. 56
5. Which number should replace the question mark (?)?
17
8
5
5
13
7
5
4
6
12
6
3
10
6
4
?
a. 4
c. 6
b. 5
d. 7
6. What completes the following pattern?
_____________
CSD, ETF, GUH, ___, KWL
7. What number should come next in the sequence?
_____________
22, 21, 25, 24, 28, 27, …
8. What letter comes next in this pattern?
_____________
O, T, T, F, F, S, S, E, …
9. What number comes next?
_____________
1, 8, 27, 64, 125, …
10. What is the next shape?
_____________
EXPONENTIAL GROWTH AND DECAY
11-13. A house is purchased for P 1, 000, 000 in 2002. The value of the house is
given by the exponential growth model 𝐴 = 1000000𝑒 0.645𝑡 . Find when the house would
be worth P 5, 000, 000.
14-16. An artifact originally ha 12 grams of carbon-14 present. The decay model 𝐴 =
12𝑒
describes the amount of carbon-14 present after t years. How many grams
of carbon-14 will be present in this artifact after 10, 000 years?
0.000121𝑡
FIBONACCI SEQUENCE
17. Starting at the first Fibonacci number, Fib (1) = 1 and the second Fibonacci
number, Fib (2) = 1, what is the 15th Fibonacci number, Fib (15)?
18. What is Fib (20)?
19. Given Fib (30) = 832, 040 and Fib (28) = 317, 811, what is Fib (29)?
20. What is Fib (11)?
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JOURNAL WRITING
What ideas about Mathematics did you learn?
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What is it Mathematics that might have changed your thoughts about it?
______________________________________________________________________
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What is most useful about mathematics to human kind?
______________________________________________________________________
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FINAL TASK
Develop a piece of art connecting to patterns in nature and mathematical
sequences. It may be a mural, magazine cover, website, etc. In a separate sheet of paper,
you are going to explain the mathematical connection between your work and sequence.
REFERENCES
www.scribd.com/document/Math-in-the-Modern-World-Chapter-1-docx
Mathematics in the World book from RBSI
https://www.iqtestexperts.com/pattern-recognition- sample.php
https://www.iqtestexperts.com/maths-sample.php
http://www.mathscareers.org.uk/article/how-the-tiger-got-its- stripes/ Anna Clarice M.
Yanday Pangasinan State University Chapter 1: Nature of Mathematics
https://www.slideshare.net/AnnaClariceYanday/mathematics-in-the-modern-worldlecture-1
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