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1. A science of measures – Measurement is an integral part of modern science as well
as of engineering, commerce, and daily life. It is an activity that involves with a
concrete system with the aim of representing aspects of that system in abstract terms
if “concrete” implies “real”.
Intellectual game – Intellectual games are games of skill that require significant
intelligence and cognitive effort. These games may be largely based on a wide or
deep knowledge. These games also require strong dose of patience, restraint, and
concentration.

2. The art of drawing conclusions – Making sense of mathematics using logical thinking
is the foundation of reasoning and proof standard. Reasoning is a way to use
mathematical knowledge and to generate and solidify new mathematical ideas.
A tool subject – Mathematics is applied in the fields of engineering, theoretical and
applied physics, astronomy, aeronautics, architecture, geology, and geodetic survey.
Mathematics is also applied even in life sciences and in industry and business.

3. A system of logical procedure – Problem-solving is an important component of
mathematics. It is a vehicle for teaching and reinforcing mathematical knowledge and
helping to meet everyday challenges.
An intuitive method – Mathematicians have traditionally regarded intuition as a way
of understanding proofs and conceptualizing problems.

4. CHARACTERISTICS OF MATHEMATICS
Characteristic
s of
Mathematics
Logical
Sequenc
e
Structur
e
Classificatio
n
Abstractnes
s
Mathematical
language
and
symbolism
Applicabilit
y
Generalization
Precision and
accuracy

5. UNIT 1: PATTERNS AND NUMBERS IN
NATURE AND THE WORLD
Let us analyze the given pattern:
A pattern is a series or sequence that repeats.
Patterns indicate a sense of structure and organization.

6. PATTERNS IN MATHEMATICS

8. What have you noticed in the pictures?

9. UNIT 1: PATTERNS AND NUMBERS IN
NATURE AND THE WORLD
Symmetry indicates that you can draw an imaginary line across an object and
resulting parts are mirror images of each other.
It is known as line or bilateral symmetry.
if you rotate an object 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 of rotation.
Angle of rotation = .

10. Bilateral Symmetry

11. Rotational Symmetry

12. UNIT 1: PATTERNS AND NUMBERS IN
NATURE AND THE WORLD
Packing problems
involve finding the optimum
method of filling up a given
space such as cubic or
spherical container.
Square Packing
Each square will have an area of 4cm2
and
suppose each circles has a radius of 1cm. note that
each square, it can fit only one circle. The
percentage of one square’s area covered by circle
will be
𝐴𝑟𝑒𝑎𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒𝑠
𝐴𝑟𝑒𝑎𝑜𝑓 h
𝑡 𝑒𝑠𝑞𝑢𝑎𝑟𝑒𝑠
×100%=
𝜋𝑐𝑚2
4𝑐𝑚
2
×100% ≈78.54 %

13. UNIT 1: PATTERNS AND NUMBERS IN
NATURE AND THE WORLD
We can think of each hexagon as composed of six equilateral triangles with side equal to
2cm. the area of each triangle is given by
This gives are of the hexagon as. Looking at the figure, there are 3 circles could
fit inside one hexagon which gives the total area as. The percentage of the hexagon’s
area covered by circles will be
Comparing the two percentages, we can clearly see that using hexagons will cover a
larger than using squares.

14. UNIT 1: PATTERNS AND NUMBERS IN
NATURE AND THE WORLD
Sunflower
Sunflower has a definite pattern of clockwise
and counterclockwise arcs or spirals extending outward
from the center of the flower. This arrangement allows
sunflower seeds to occupy the flower head in a way that
maximizes their access to light and necessary nutrients.

15. UNIT 1: PATTERNS AND NUMBERS IN
NATURE AND THE WORLD
Snail’s Shell
Snails are born with their shells called
protoconch. As the snails grow, their shell also expand
proportionately so that they can continue to live inside
their shells. This process results in a refined spiral
structure that is even more visible when shell is sliced.
Equiangular spiral follows the rule that as the distance
from the spiral center increases, the amplitudes of the
angle formed by the radii to the point and the tangent to
the point remain constant.

16. UNIT 1: PATTERNS AND NUMBERS IN
NATURE AND THE WORLD
World Population
Mathematics can be used to model population growth.
Using the formula for exponential growth:, where A is the size of
the population after it grows, P is the initial number of people, r is
the rate of growth, and t is time. Note that e is Euler’s constant
with approximate value of 2.718.
Example. The exponential growth model describe 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 2024?
Exponential growth – is a specific way that a
quantity may increase over time.

17. EXPONENTIAL GROWTH

18. CUMULATIVE NUMBERS OF CONFIRMED DEATHS
DUE TO COVID-19

19. ORIGINS OF EXPONENTIAL GROWTH

20. DAILY COVID CASES IN PH

22. UNIT 2 : FIBONACCI SEQUENCE
Learning Objectives:
After successful completion of this lesson, you should be able to:
1. identify patterns in nature and regularities in the world.
2. Understand the Fibonacci sequence

23. UNIT 2 : FIBONACCI 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: Analyze the given sequence for its rule and identify the next three
terms.
a. 1, 10, 100, 1000
b. 2, 5, 9, 14, 20
c. 1, 1, 2, 3, 5, 8

24. UNIT 2 : FIBONACCI SEQUENCE
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,… is
called the Fibonacci sequence.

25. UNIT 2 : FIBONACCI SEQUENCE
Around 1200 AD when Pizano or Leonardo of Pisa (1170-1250)
published the Liber Abbaci, or “Book of Calculation,” an arithmetic
text which concentrated mainly on financial computation and
promoted the use of Hindu-Arabic numerals.
One of the exercises in his book is about the rabbit habit.

26. UNIT 2 : FIBONACCI SEQUENCE

27. UNIT 2 : FIBONACCI SEQUENCE
Fibonacci observed numbers in nature.
A calla lily flower has only 1 petal, an Asiatic dayflower has 2, a
trillium has 3, hibiscus has 5, cosmos flower has 8, corn marigold
has 13, some asters have 21, and a daisy can have 34, 55, or 89
petals.

29. GOLDEN RECTANGLE
The Fibonacci sequence forms a golden rectangle; a
perfect rectangle
A golden rectangle can be broken into squares the size of
the next Fibonacci number down and below.
To get the golden rectangle, break it down into smaller
squares based from Fibonacci sequence and divide each
with an arc, the pattern begins to take shapes.

31. GOLDEN RATIO
The Golden Ratio can be expressed as the ratio between two numbers.
1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.6667
8/5 = 1.6
13/8 = 1.6250
21/13 = 1.6154
34/21 = 1.6190
55/34 = 1.6177
89/55 = 1.6182
= = 1.61803

32. GOLDEN RECTANGLE IN ARCHITECTURE

33. GOLDEN RATIO AND BEAUTY

34. “WHENEVE
R THERE IS
NUMBER,
THERE IS
BEAUTY” –
PROCIUS,
GREEK
PHILOSOPH
ER

35. GOLDEN
RATIO
AND
PHOTOG
RAPHY

39. UNIT 3 : MATHEMATICS FOR OUR WORLD
Learning Objectives:
After successful completion of this lesson, you should be able to:
1. identify patterns in nature and regularities in the world.
2. identify the characteristics of mathematics.
3. argue about the nature of mathematics, what it is, how it is expressed,
represented, and used.

40. UNIT 3 : MATHEMATICS FOR OUR WORLD
Mathematics for Organization
A lot of events happen around us. In just a blink of an eyes, several babies has been born,
liters of water have been consume or thousands of tweets have been posted. This implies that we need
mathematics to organize things, making analysis and better decision. Some of the use of mathematics in
terms of organization are:
A store owner can gather data of the shopping habits of costumers and make necessary adjustments
to increase sales.
Scientist can plot bird migration routes to conserved engaged animals.
Social media can crunch all online posting using software to gauge the netizen sentiment or a
particular issue.
Managers can determine the famous artist to for the audience to buy their concert tickets.
Store manager can survey the cake flavors usually ordered by the costumers.

41. UNIT 3 : MATHEMATICS FOR OUR WORLD
Mathematics for Control
We have demonstrated by means of examples around us that patterns are
definitely present in the universe. With this we can able to control things or event using
mathematics. Also, mathematics in terms of control can be used by:
Photographers can capture single moments through snap shot.
Videographers could record events as they unfold.
Painters and sculptors can create master piece using their surroundings.
Poets can use beautiful words to describe an object
Musician can capture and produce sounds as they hear.

42. UNIT 3 : MATHEMATICS FOR OUR WORLD
Mathematics is Indispensable
In this chapter it is highlighted how mathematics plays a huge role in
understanding of our world. Whether you are becoming a doctor, engineer, or lawyer,
mathematics can be helpful. At the most basic level logical reasoning and critical
thinking are crucial things needed in your endeavor. Mathematics is everywhere and
mathematics is always part or our everyday life.

43. UNIT 3 : MATHEMATICS FOR OUR WORLD
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. Some of the use of mathematics in terms of prediction are:
Applying concepts of probability to calculate the chance of an event occurring.
We can also predict weather to plan an event outside.
Astronomers also use patterns to predict the occurrence of meteor showers or eclipse.
Doctors can determine the gender of the baby.
Sports analysis can determine who will win in today’s game base of their previous scores.

44. ASSIGNMENT # 1:
Research the different patterns in nature, its name & description.
Include at least 3 different sample images for each type.
Print your work on a short bond paper.
Write your name at the top, starting with your surname.
Answer the question: What role do patterns in nature play in ecosystems, and how
might they affect biodiversity or the balance of life?
Submission is due March 10 &12, 2025.