LEARNING MODULE
SURIGAO STATE COLLEGE OF TECHNOLOGY
GE MATH
MODULE 1
The Nature of Mathematics
Organized by:
JIMBO JUANITO B. VILLAMOR, MA
MARIA FE C. GUERRA, MAEd
FLORIA C. DIALDE, MA
RICKY T. OSORIO, MA
RHEA MAE C. BABAS, MA
GINA C. BOTOY, MA
EDILMAR P. MASUHAY, MS
JONATHAN C. MACABODBOD, EdD
JULIUS G. CAADAN, PhD
ROSEMARIE C. ARCAYA, PhD
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1. Title: The Nature of Mathematics
2. Topic: The Nature of Mathematics
3. Time Frame: 8 hrs.
4. Introduction
Why Study Mathematics? Mathematics has become one of the most
powerful tools in understanding the world around us. How nature including the flora
and fauna behave and react can be explained in simple terms if not up to its
complexities.
Quite often, people confine mathematics to numbers and arithmetic. Such
partial and superficial understanding of the nature of this body of knowledge has
restrained a liberal culture of learning in our educational system. This learning
module on patterns in nature can exhibit the potential of learners more about
reasoning, making logical inferences and generalizations, and seeing relationships
in both the visible and invisible patterns in the natural world.
5. Objectives
At the end of the lesson, the students are expected to:
 Discuss the importance of mathematics in one’s life;
 Identify patterns in nature and regularities in the world;
 Explain the nature of mathematics, what it is, how it is expressed,
represented and used; and
 Solve basic problem solving in Fibonacci sequence and other number
patterns, including population growth; and
 Express appreciation for mathematics as a human endeavor.
6. Learning Activities
Activity 1: Think and Share
Mathematics is all around us. As we discover more and more about our
environment and our surroundings we see that nature can be described
mathematically. The beauty of a flower, the majesty of a tree and mountain, even
the rocks upon which we walk can exhibit nature’s sense of symmetry and
patterns.
Below are pictures of some common entities in nature. Describe orally the
following pictures below according to how you perceived it.
A.
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B.
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C.
D.
Have you ever stopped to look around and notice all the amazing shapes
and patterns we see in the world around us? Mathematics forms the building
blocks of the natural world and can be seen in stunning ways. Here are some other
examples of math in nature, but there are many other examples as well.
Activity 2: Video Watching
To mathematically explain patterns in the world, watch a video which contains
discussions on the following math concepts:
1. Fibonacci Sequence: https://youtu.be/SjSHVDfXHQ4
2. Golden Ratio: https://youtu.be/9mozmHgg9Sk
3. Tesselations: https://www.youtube.com/watch?v=7GiKeeWSf4s
Guide Questions:
1. What math concept is fully discussed on each video?
2. What is about these concepts? Cite few more real-world cases or examples
that would show the concepts.
3. What are your insights about these mathematical truths or certainties?
4. What did Galileo mean when he said, “Mathematics is the alphabet by
which God has written the universe”? Do you agree on this adage? Why?
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Points to Ponder: Patterns and Numbers in Nature and the World
Ian Stewart, a British mathematician, once quoted that “Mathematics is the
science of patterns, and nature exploits just about every pattern there is.”
Philosophers and mathematicians have, for long, dedicated themselves to the
cause of explaining nature, beginning from the very early ventures of ancient
Greeks. After all, mathematics is, in its very essence, a search for patterns of all
kinds – and what better place to find such irregularities than nature itself? A closer
look into nature leads to some very interesting implications about the underlying
beauty of our universe.
Consider the example of a crystal. A ‘perfect’ crystal is one that is fully
symmetrical, without any structural defects. Of course, perfect crystals do not really
exist; the physical world is rarely perfect. Mathematics is an abstract language,
and the laws of physics serve to apply these abstractions to the real world.
Assuming the object as perfect helps our cause. So, it is just that identifying a
crystal as a symmetrical, uniform structure helps us in making approximations
about its aspects. Similarly, meanders or bends in rivers find explanation in the
branch of fluid dynamics pertaining to physics. On a more cosmic scale, the
characteristic spiral of galaxies that we are all too familiar with is a result of the
laws of gravitation and can be modeled as such. Finding such patterns and
abstractions facilitates our understanding of the world around us.
Patterns in nature are visible regularities of form found in
the natural world. These patterns recur in different contexts and can sometimes
be modelled mathematically. Natural patterns include symmetries, trees, spirals,
meanders, waves, foams, tessellations, cracks and stripes.
The following are the basic ideas about patterns in nature and the world:
1.1. Fibonacci Sequence
Named for the famous mathematician, Leonardo Fibonacci, this number
sequence is simple, yet profound pattern.
Often called ‘Nature’s Universal Rule’, the Fibonacci sequence is perhaps one
of the most famous mathematical sequences. The origin of this sequence is
much contested, although it is commonly attributed to the Italian
mathematician, Leonardo Fibonacci. In his famous work ‘Liber Abaci’, he
introduced a hypothetical problem involving rabbits and employed the
sequence to find the number of rabbits after a certain period of time.
In this sequence, each number is the sum of the two numbers that precede it.
Take a look:
0….1….1….2….3….5….8….13….21….34
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Illustration:
This sequence begins with the numbers 1 and 1 or 0 and 1, and then each
subsequent number is found by adding the two previous numbers.
After 1 and 1, the next number is 2, that is, 1+1. The next number is 3, taken
from 1+2, and then 5, taken from 2+3 and so on.
Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …
Fibonacci Sequence in Nature
The Fibonacci sequence can be observed in a stunning variety of phenomena
in nature.
Nautilus shells, one of the most iconic examples of the Fibonacci sequence,
follow the proportional increase of 1.61.
The total number of petals of a flower is often a number present in the Fibonacci
sequence, as with irises and lilies. Most pineapples have either five, eight,
thirteen or twenty-one spirals; these are also Fibonacci numbers.
Something strange happens when the sequence approaches infinity. The ratio
between two consecutive numbers converges to 1.61803… : ‘phi’, or as you
might call it, the ‘golden ratio’.
1.2. Golden Ratio
The "golden ratio" is a unique mathematical relationship. Two numbers are in
the golden ratio if the ratio of the sum of the numbers (a + b) divided by the
larger number (a) is equal to the ratio of the larger number divided by the
smaller number (a/b).
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The golden ratio is about 1.618, and represented by the Greek letter phi, Φ.
The ratios of sequential Fibonacci numbers (2/1, 3/2, 5/3, etc.) approach the
golden ratio. In fact, the higher the Fibonacci numbers, the closer their
relationship is to 1.618.
The golden ratio is sometimes called the "divine proportion," because of its
frequency in the natural world. The number of petals on a flower, for instance,
will often be a Fibonacci number.
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The seeds of sunflowers and pine cones twist in opposing
spirals of Fibonacci numbers.
Even the sides of an unpeeled banana will usually
be a Fibonacci number—and the number of ridges
on a peeled banana will usually be a larger
Fibonacci number.
1.3. Tesselations
Tessellation (Tiling) is a shape that repeats to form a pattern.
Tessellation Definition
A tessellation is created when a shape is repeated
over and over again covering a plane without any
gaps or overlaps.
Another word for a tessellation is a tiling.
Tiling Definition
When you fit individual tiles together with no gaps
or overlaps to fill a flat space like a ceiling, wall, or
floor, you have a tiling.
What are Tessellations?
The word 'tessera' in latin means a small stone cube. They were used to make
up 'tessellata' - the mosaic pictures forming floors and “tilings” in Roman
buildings. The term has become more specialized and is often used to refer to
pictures or tiles, mostly in the form of animals and other life forms, which cover
the surface of a plane in a symmetrical way without overlapping or leaving
gaps.
Examples:
Rectangles
Octagons and Squares
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Tessellations in Nature
Tessellations form a class of patterns found in nature. The arrays of
hexagonal cells in a honeycomb or the diamond-shaped scales that pattern
snake skin are natural examples of tessellation patterns. Distinct shapes are
formed from several geometric units (tiles) that all fit together with no gaps or
overlaps to form an interesting and united pattern. Tessellating patterns are
abstract and non-representational which makes their interpretation open to the
imagination of all people.
Honeycomb
Snake skin
Flower
Leaves
Drought
Sunflower
1.4. Fractals
A fractal is a never-ending pattern. Fractals are infinitely complex patterns
that are self-similar across different scales. They are created by repeating a
simple process over and over in an ongoing feedback loop. Driven by recursion,
fractals are images of dynamic systems – the pictures of Chaos. Geometrically,
they exist in between our familiar dimensions. Fractal patterns are extremely
familiar, since nature is full of fractals.
For instance: trees, rivers, coastlines, mountains, clouds, seashells,
hurricanes, etc. Abstract fractals – such as the Mandelbrot Set – can be
generated by a computer calculating a simple equation over and over.
Examples:
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1.5. Other Patterns in Nature
Hexagons in Nature
Another of nature’s geometric wonders is the hexagon. A regular
hexagon has 6 sides of equal length, and this shape is seen again and again
in the world around us.
The most common example of nature using hexagons is in a bee hive.
Bees build their hive using a tessellation of hexagons. But did you know that
every snowflake is also in the shape of a hexagon?
Honeycomb
Bubbles
We also see hexagons in the bubbles that make up a raft bubble.
Although we usually think of bubbles as round, when many bubbles get
pushed together on the surface of water, they take the shape of hexagons.
Concentric Circles in Nature
Another common shape in nature is a set of concentric circles. Concentric
means the circles all share the same center, but have different radii. This
means the circles are all different sizes, one inside the other.
A common example is in the ripples of a pond when something hits the
surface of the water. But we also see concentric circles in the layers of an onion
and the rings of trees that form as it grows and ages.
If you live near woods, you might go looking for a fallen tree to count the
rings, or look for an orb spider web, which is built with nearly perfect concentric
circles.
Patterns in Outer Space
Moving away from planet earth, we can also
see many of these same mathematical features
in outer space.
For instance, the shape of our galaxy is a
Fibonacci spiral. The planets orbit the sun on
paths that are concentric. We also see
concentric circles in the rings of Saturn. But we also see a unique symmetry in
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outer space that is unique (as far as scientists can tell) and that is the symmetry
between the earth, moon and sun that makes a solar eclipse possible.
Every two years, the moon passes between the sun and the earth in such
a way that it appears to completely cover the sun. But how is this possible when
the moon is so much smaller than the sun?
1.6. Number Patterns
Number pattern is a pattern or sequence in a series of numbers. This pattern
generally establishes a common relationship between all numbers.
For example:
0, 5, 10, 15, 20, 25, ...
A sequence is a pattern of numbers that are formed in accordance with a
definite rule.
We can often describe number patterns in more than one way. To illustrate
this, consider the following sequence of numbers {1, 3, 5, 7, 9, …}.
Clearly, the first term of this number pattern is 1; and the terms after the first
term are obtained by adding 2 to the previous term. We can also describe this
number pattern as a set of odd numbers.
Activity 3: Patterns of Numbers
Determine the next 3 terms in the following sequences.
1. 13, 21, 34, 55, __, __, …
2. 55, 89, 144, 233, __, __, …
3. 2, 3, 5, 8, __, __, …
4. 21, 34, 55, 89, __, __, …
5. 89, 144, 233, 377, __, __, …
6. 34, 55, 89, 144, __, __, …
7. 8, 13, 21, 34, __, __, …
8. 3, 5, 8, 13, __, __, …
Activity 4: Generating a Sequence
A. Analyze the given sequence for its rule and identify the next three terms
1. 1, 10, 100, 1000
2. 2, 5, 9, 14, 20
3. 16, 32, 64, 128
4. 1, 1, 2, 3, 5, 8
B. Let Fib(n) be the nth term of a Fibonacci sequence, with Fib(1) = 1,
Fib(2) = 1, Fib(3) = 2. Find:
1. Fib(8)
2. Fib(19)
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1.7. Abstract Reasoning Patterns
An abstract reasoning test uses shapes and patterns to assess your logic,
fluid intelligence and problem-solving skills. You’ll be expected to quickly
interpret a series of images to deduce the rule or pattern that connects them
(for example, a repetition of colour, shape or size). Abstract tests are common
for research, software development and engineering roles.
For example, you might be asked to select, out of a number of possible
options, which image completes a sequence or statement of fact, which image
is missing from the overall picture or which image doesn’t correlate to the others
shown.
Activity 5: Abstract Reasoning
Determine what completes the series.
1.8.
Population Growth Patterns
Mathematics can be used to model population growth. Using the
exponential growth formula, 𝑨 = 𝑷𝒆𝒓𝒕 , where,
A is the size of population after it grows,
P is the initial number of people,
r is the rate of growth,
t is time, and
e is Euler’s constant, ≈ 2.718
Illustration:
The exponential growth model formula, 𝑨 = 𝟑𝟎𝒆𝟎.𝟎𝟐𝒕 , describes the population of
a city in the Philippines in thousands, t years after 1995.
1. What was the population of the city in 1995?
2. What will be the population by the end of 2021?
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Solution:
1. 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
= 30𝑒 0.02(0)
= 30𝑒 0
= 30(1)
𝑨 = 𝟑𝟎
Therefore, the city population in 1995 was 30,000.
2. We need to find A by the end of 2021. To find t, we subtract 2021 and 1995 to
get t = 26. Hence,
𝑨 = 30𝑒 0.02𝑡
= 30𝑒 0.02(26)
= 30𝑒 0.52
= 30(2.718)0.52
= 30(1.68194)
𝑨 = 𝟓𝟎. 𝟒𝟓𝟖𝟐
Therefore, the city population would be about 50,458 by the end of 2021.
7. Self-evaluation
Fill in the blanks with word to make the statement true.
1. A Fibonacci sequence is often called ______________ .
2. A ______________ is a pattern of numbers that are formed in accordance with
a definite rule.
3. The arrays of hexagonal cells in a honeycomb is an example of
______________ .
4. The rings of Saturn illustrates ____________.
5. Fractals are infinitely ____________ that are self-similar across different
scales.
8. Learning Assessment Task
Answer the items in Worksheet No. 1
9. References:
Mathematics in the Modern World by Rex Bookstore, 2018
https://youtu.be/SjSHVDfXHQ4
https://youtu.be/9mozmHgg9Sk
https://www.youtube.com/watch?v=7GiKeeWSf4s
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https://www.mathsisfun.com/numbers/fibonacci-sequence.html
https://spectramagazine.org/mathematics/the-fibonacci-sequence-in-nature/
https://www.nationalgeographic.org/media/golden-ratio
http://www.csun.edu/~lmp99402/Math_Art/Tesselations/tesselations.html
https://fractalfoundation.org/resources/what-arefractals/#:~:text=Fractals%20are%20infinitely%20complex%20patterns,systems
%20%E2%80%93%20the%20pictures%20of%20Chaos.
https://www.mathsisfun.com/geometry/tessellation.html#:~:text=A%20pattern%2
0of%20shapes%20that,are%20no%20overlaps%20or%20gaps.
http://www.spacemakeplace.com/tessellationpatterns/#:~:text=Tessellations%20form%20a%20class%20of%20patterns%20fo
und%20in%20nature.&text=Distinct%20shapes%20are%20formed%20from,an%
20interesting%20and%20united%20pattern.
https://www.mathsteacher.com.au/year8/ch15_graphs/03_number/patterns.htm
https://www.splashlearn.com/math-vocabulary/number-sense/numberpatterns#:~:text=Number%20pattern%20is%20a%20pattern,by%20skip%20coun
ting%20by%205.
https://www.practiceaptitudetests.com/abstract-reasoningtests/#:~:text=An%20abstract%20reasoning%20test%20uses,colour%2C%20sha
pe%20or%20size).
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