Mathematics in Our World: Patterns, Foundations, and Applications

The Beauty and Complexity of Mathematics

Mathematics presents itself as an essential filtering process for individuals navigating a world saturated with information. Filipino learners, in particular, face an explosion of knowledge that necessitates the filtration of enormous resources across various subject areas. These concerns are vital to human satisfaction and the uplifting of minds in reality.

To many outsiders, the discipline appears as an unknown territory. Its borders are perceived as protected by dense thickets of technical terms, and its landscapes are seen as a mass of indecipherable equations characterized by incomprehensible concepts. However, despite this perception, only a few people realize that numbers are rich with vivid images and provocative ideas. These materials are presented by Chardhidman Coaching/ $2020$ .

The Fundamental Role of Mathematics in the Modern World

Mathematics serves as the major underpinning of the world, providing the logic and critical thinking skills required in any endeavor. It stands as the pillar of organized life for the present day. In the absence of numbers and mathematical evidence, resolving daily issues would be impossible, leading to confusion and chaos. Mathematical evidence is used to handle:

Measurements and rates.
Wages, tenders, and discounts.
Claims, supplies, and jobs.
Stocks, contracts, and taxes.
Money exchange and consumption.
Sports data.

Numbers are employed across various fields to calculate results effectively, forecast the behavior of one variable into another known variable, and identify the specific requirements for medicine dosages to cure illnesses. Mathematics is further used to verify if specific solutions are applicable to general setups, ascertain the chronology of past events, and determine situational patterns.

Historical Origins and the Evolutionary Necessity of Math

Mathematics has been the companion and helper of humanity since the beginning of existence. It was invented when humans first sought to answer the question, "How many?"

Algebra: This branch was invented to facilitate calculations, measurements, analysis, and engineering.
Trigonometry: This science emerged from the human desire to locate the positions of high mountains and stars.

Mathematics developed because humans felt a need for it; it is necessary for both the long-term planning of life and the daily planning of any individual. This "mathematical rapprochement" is required for any process, and successfully reaching the heights of life requires believing in the role of mathematics, from the ordinary citizen to experts.

Mathematics as a Key to Nature and Mastery

Mathematics is deeply related to natural phenomena and provides the way to solve many secrets of nature. It is a necessary prerequisite for understanding other branches of knowledge, as no science, art, or specialty exists for which mathematics was not the key. The discipline and mastery of any other science or art are directly related to the "size" or extent of mathematics involved.

We utilize mathematics as a vital way to think about nature and the world in general. By examining patterns and numbers, such as the Fibonacci sequence and other arrays, humans can forecast, estimate, or control the behavior of nature and world phenomena. Ultimately, the application of numbers acts as a significant aid in decision-making processes.

Core Learning Outcomes for Mathematical Study

As part of Module No. $1$ (Chardhidman Coaching/ $2020$ ), students are expected to:

Identify patterns in nature and global regularities.
Articulate the importance of mathematics in daily life.
Argue about the nature of mathematics, including what it is and how it is expressed, represented, and used.
Express appreciation for mathematics as a significant human endeavor.

Patterns and Regularities in the Natural World

Patterns are inherently parallel to counting; they are correlated in such a way that counting exists where there is a pattern, and counting exists where there is logic. Natural patterns follow a logical setup. Mathematics acts as a "unifying mechanism" that exists everywhere patterns do.

According to Collins ( $2018$ ), a Regularity is the fact that the same thing always happens in same circumstances. A Pattern is a discernible world regularity or a man-made design where elements repeat in a predictable manner.

Nature Patterns are visible forms of regularities found in the natural world. Ian Stewart ( $1995$ ) cited that we live in a universe of patterns. These can be:

Sequential.
Spatial.
Temporal.
Linguistic.

Regularity is evidenced by the repetition of names or events, such as date sequences in a calendar (months, days, annum, and holidays celebrated in the same sequence every year). Patterns are also found in construction materials (brick walls, walkways), architectural elements (rooftops, tiled floors, stairways), and even in the sand through the action of the wind.

Basic Samples of Patterns in Nature

$1$ . Symmetry: Defined as an agreement in dimensions, due proportion, and arrangement. An object is invariant to various transformations, including reflections.
$2$ . Spiral: A curve emanating from a point, moving farther away as it revolves around that point. A cutaway of a nautilus shell displays chambers arranged in an approximately logarithmic spiral.
$3$ . Meander: A series of regular sinuous curves, bends, loops, turns, or windings in a river channel, stream, or watercourse. It is produced by a river swinging from side to side, eroding sediments on the outer concave bank and depositing them at a point bar.
$4$ . Wave: A disturbance transferring energy through matter or space with little to no associated mass transport. It consists of oscillations or vibrations of a physical medium or field around fixed locations.
$5$ . Foam: A substance formed by trapping pockets of gas in a liquid or solid. Examples include a bath sponge or the head on a glass of beer. In most foams, the gas volume is large, separated by thin films of liquid or solid. Soap foams are specifically known as suds.
$6$ . Tessellation: The tiling of a plane using one or more geometric shapes (tiles) with no overlaps and no gaps. This can generalize to higher dimensions and various geometries.
$7$ . Crack Fracture: The separation of an object or material into two or more pieces under the action of stress.
- Normal Tensile Crack: Displacement develops perpendicular to the surface.
- Shear Crack (Dislocation): Tangential displacement.
$8$ . Stripes: A series of bands or strips, often of the same width and color, occurring along a length.
$9$ . Fractal: A never-ending pattern that is infinitely complex and self-similar across different scales. Fractals are images of dynamic systems and the "pictures of chaos," created by repeating a simple process in an ongoing feedback loop. Examples include trees, rivers, coastlines, mountains, clouds, seashells, and hurricanes.
$10$ . Affine Transformations: Processes including rotation, reflection, and scaling. Plant forms like Broccoli and Cauliflower heads utilize these processes to generate their structure and spiraling features.

The Fibonacci Sequence

The Fibonacci numbers, denoted as $F_n$ , form an array of numbers where, given two terms, the next term is determined by adding the two preceding terms. Starting from $0$ and $1$ , the recurrence relation is defined as:

${F_{n}=F_{n-1}+F_{n-2}}$

In some formulations (such as in some older books), $F_{0}=0$ is omitted, and the sequence starts with $F_{1}=F_{2}=1$ , with the recurrence ${F_{n}=F_{n-1}+F_{n-2}}$ valid for $n > 2$ .

The beginning of the sequence is:

$0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \ldots$

Fibonacci numbers are strongly related to the Golden Ratio. Binet's formula expresses the $n$ th Fibonacci number in terms of $n$ and the golden ratio. As $n$ increases, the ratio of two consecutive Fibonacci numbers tends toward the golden ratio.

Procedural Steps to Solve for Fibonacci Terms

To calculate terms in the sequence, follow these steps using the general formula ${F_{n}=F_{n-1}+F_{n-2}}$ (Note: The transcript uses the notation ${F_{n}=F_{n+1}+F_{n-2}}$ on pages $12$ and $13$ ):

Step $1$ : Create a table with two columns. The number of rows depends on how many terms you wish to calculate.
Step $2$ : In the left column, enter the sequence of terms using sequential ordinal numbers (e.g., $1$ st, $2$ nd, $3$ rd, $4$ th, $5$ th). This indicates the position in the sequence.
Step $3$ : Enter $1$ in the first row of the right-hand column as the starting point.
Step $4$ : Add the first term ( $1$ ) and $0$ to get the second term. Conceptually, $0$ precedes the first term: $1 + 0 = 1$ .
Step $5$ : Add the first term ( $1$ ) and the second term ( $1$ ) to get the third term: $2$ .
Step $6$ : Add the second term ( $1$ ) and the third term ( $2$ ) to get the fourth term: $3$ .
Step $7$ : Add the third term ( $2$ ) and the fourth term ( $3$ ) to get the fifth term: $5$ .

Additional Patterns and Applications in the Modern World

Geometry: Also known as "Earth measurement," it deals with the measurement, properties, and relationships of shapes, points, lines, and angles. It is essential for building houses and buildings.
Measurement: The process of associating numbers with physical phenomena and determining ratios of quantities.
Money: The assets, property, and resources owned by an entity. It is a medium of exchange (coins and banknotes) used by market participants for goods and services.
Time: The indefinite continued progress of existence and events in the past, present, and future viewed as a whole.
Estimations: Also known as forecasting or regression. It is a rough calculation used to predict the value, quantity, extent, or how many people/things may occur.
Fractions: Numerical quantities representing equal parts of a whole or a collection. Used to determine what portion of a whole is needed.
Decimals: A system based on the number $10$ , $10$ th parts, and powers of $10$ . Decimals are used $24/7$ in money, weight, and length, providing more precision than whole numbers.
Percents: Defined as "per hundred." Percents allow for easier comparison than fractions and are used in shops for discounts and financial institutions for interest rates on loans or investments.
Probability: The extent to which an event is likely to occur, measured by the ratio of favorable cases to the total number of possible cases. It is used in decision-making when the outcome is uncertain (e.g., drawing an ace from a deck or picking a specific colored candy).
Problem Solving: A philosophy of teaching and learning where students work together to solve scenarios significant to their community. It requires access to current knowledge and input from experts.