Mathematics In The Modern World PDF

Republic of the Philippines NUEVA VIZCAYA STATE UNIVERSITY Bayombong, Nueva Vizcaya INSTRUCTIONAL MODULE...

Republic of the Philippines NUEVA VIZCAYA STATE UNIVERSITY Bayombong, Nueva Vizcaya INSTRUCTIONAL MODULE IM No.1: IM-GEMATH-1STSEM-2024-2025 COLLEGE OF ARTS AND SCIENCES Bayombong Campus DEGREE GENERAL COURSE NO. GE MATH PROGRAM EDUCATION SPECIALIZATION MATHEMATICS COURSE TITLE MATHEMATICS IN THE MODERN WORLD YEAR LEVEL FIRST YEAR TIME FRAME 3 HRS WK NO. 1 IM NO. 1 I. CHAPTER 1 - The Nature of Mathematics II. LESSON TITLE A. Patterns and Numbers in Nature and the World B. The Fibonacci Sequence III. LESSON OVERVIEW Mathematics helps organize patterns and regularities in the world. The geometry of most patterns in nature can be associated, either directly or indirectly, to mathematical numbers. The limit and extent to which natural patterns adhere to mathematical series and numbers are amazing. Mathematics helps predict the behavior of nature and phenomena in the world. It helps control nature and occurrences in the world for the good of mankind. Because of its numerous applications, mathematics becomes indispensible IV. DESIRED LEARNING OUTCOMES The students should be able to: 1) Identify patterns in nature and regularities in the world; 2)Articulate the importance of mathematics in one’s life; 3) Argue about the nature of mathematics, what it is, how it is expressed, represented and used; and 4) Express appreciation of mathematics as a human behavior. V. LESSON CONTENT Mathematics is the study of the relationships among numbers, quantities, and shapes. It includes arithmetic, algebra, trigonometry, geometry, statistics and calculus. Mathematics nurtures human characteristics like power of creativity, reasoning, critical thinking, spatial thinking and others. A. Patterns and Numbers in Nature and the World Patterns in nature are visible regularities found in the natural world. These patterns persist in different context and can be modeled mathematically. Natural patterns may consists spirals, symmetries, mosaic, stripes, spots, etc. the world seems to make several distinct patterns, developing various complex steps of formation but a closer and deeper study reveals that these patterns have many similarities and resemblances. NVSU-FR-ICD-05-00 (081220) Page 1 of 9 Mathematics and Statistics Department “In accordance with Section 185. Fair use of a Copyrighted Work of Republic Act 8293, the copyrighted works included in this material may be reproduced for educational purposes only and not for commercial distribution”.  Republic of the Philippines NUEVA VIZCAYA STATE UNIVERSITY Bayombong, Nueva Vizcaya INSTRUCTIONAL MODULE IM No.1: IM-GEMATH-1STSEM-2024-2025 Plato, Pythagoras and Empedocles and other early Greek philosophers studied patterns and explain order in nature which lead to the modern understanding of visible patterns. In the 19th century, Belgian Physicist, Joseph Plateau examined soap films, leads him to formulate the concept of a minimal surface. German Biologist and Artist Ernst Haeckel painted hundreds of marine organisms to emphasize their symmetry. Scottish Biologists D’Arcy Thompson pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth. In the 20th century, British Mathematician Alan Turing predicted mechanisms of morphogenesis which give rise to patterns of spots and stripes. NVSU-FR-ICD-05-00 (081220) Page 2 of 9 Mathematics and Statistics Department “In accordance with Section 185. Fair use of a Copyrighted Work of Republic Act 8293, the copyrighted works included in this material may be reproduced for educational purposes only and not for commercial distribution”.  Republic of the Philippines NUEVA VIZCAYA STATE UNIVERSITY Bayombong, Nueva Vizcaya INSTRUCTIONAL MODULE IM No.1: IM-GEMATH-1STSEM-2024-2025 Hungarian Biologists Aristid Lindenmayer showed how the mathematics of fractals could create plant growth patterns. An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into some larger string of symbols, an initial "axiom" string from which to begin construction, and a mechanism for translating the generated strings into geometric structures. French American Benoit Mandelbrot showed how the mathematics of fractals could create plant growth patterns. Fractal is a curve or geometric figure, each part of which has the same statistical character as the whole. Fractals are useful in modeling structures (such as eroded coastlines or snowflakes) in which similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as crystal growth, fluid turbulence, and galaxy formation. W. Gary Smith adopts eight patterns in his landscape work, namely; scattered, fractured, mosaic, naturalistic drift, serpentine, spiral, radial and dendritic. These patterns occur in plants, animals, rock formations, river flow, stars or in human creations. (Goral, 2017) NVSU-FR-ICD-05-00 (081220) Page 3 of 9 Mathematics and Statistics Department “In accordance with Section 185. Fair use of a Copyrighted Work of Republic Act 8293, the copyrighted works included in this material may be reproduced for educational purposes only and not for commercial distribution”.  Republic of the Philippines NUEVA VIZCAYA STATE UNIVERSITY Bayombong, Nueva Vizcaya INSTRUCTIONAL MODULE IM No.1: IM-GEMATH-1STSEM-2024-2025 Spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Scattered A dispersed settlement, also known as scattered settlement, is one of the main types of settlement patterns used by landscape historians to classify the rural settlements found in England and other parts of the world. Radial Classic radial design is symmetrical, with elements situated equally around the center. Think of something like a clock face: there is a central point where the clock hands meet, digits encircling this central point, and an equal number of digits on either side of the center. This same type of radial design is used in visual art. Take the stained glass rose window from the famous Notre Dame Cathedral, for instance. Radial designs are generated outward from a center point creating a circular pattern or design. Mosaic A mosaic is a piece of art or image made from the assembling of small pieces of colored glass, stone, or other materials. It is often used in decorative art or as interior decoration. Most mosaics are made of small, flat, roughly square, pieces of stone or glass of different colors, known as tesserae. Fractured Fractured patterns arise abundantly in natural and engineered systems, and their geometries depend on material properties and on the ways in which the material is deformed or forces act on it. Two-dimensional fracture patterns can be characterized by their network topology (how fractures connect to each other) and their heterogeneity (whether fractures appear clustered or uniformly distributed in space). NVSU-FR-ICD-05-00 (081220) Page 4 of 9 Mathematics and Statistics Department “In accordance with Section 185. Fair use of a Copyrighted Work of Republic Act 8293, the copyrighted works included in this material may be reproduced for educational purposes only and not for commercial distribution”.  Republic of the Philippines NUEVA VIZCAYA STATE UNIVERSITY Bayombong, Nueva Vizcaya INSTRUCTIONAL MODULE IM No.1: IM-GEMATH-1STSEM-2024-2025 Dendritic Dendritic-having a branched form resembling a tree Serpentine Something that is serpentine is curving and winding in shape, like a snake when it moves. Naturalistic Drift Natural lines of drift are those paths across terrain that are the most likely to be used when going from one place to another. These paths are paths of least resistance: those that offer the greatest ease while taking into account obstacles (e.g. rivers, cliffs, dense unbroken woodland, etc.) and modes of transit (e.g. pedestrian, automobile, horses.). Common endpoints or fixed points may include water sources, food sources, and obstacle passages such as fords or bridges. Numbers are everywhere in nature. Mathematicians noticed that numbers appear in many different patterns in nature: bird’s to wings, clovers’ three leaflets, deer’s four hooves, buttercup’s five petals, insect’s six legs, rainbow’s seven colors, octopus’ eight arms and many others. As man of science studied numbers, they also realized their significance in everyday life. NVSU-FR-ICD-05-00 (081220) Page 5 of 9 Mathematics and Statistics Department “In accordance with Section 185. Fair use of a Copyrighted Work of Republic Act 8293, the copyrighted works included in this material may be reproduced for educational purposes only and not for commercial distribution”.  Republic of the Philippines NUEVA VIZCAYA STATE UNIVERSITY Bayombong, Nueva Vizcaya INSTRUCTIONAL MODULE IM No.1: IM-GEMATH-1STSEM-2024-2025 Sequencing A sequence, in mathematics, is a string of objects, like numbers, that follow a particular pattern. Ex. 3,6,9,12,15….. Determine the next term in the given sequences of numbers. Arithmetic Sequence 1: Find the next term in the sequence 7,15,23,31,___. 2: Find the next term in the sequence 31,24,17,10,__ 3. Find the next three terms in -14,-10,-6,-2,____ Geometric Sequence 4: Find the next term in the sequence 2,6,18,54,__ 5: Find the next two terms in 8,4,2,1,0.5,___ Triangle Numbers Sequence 6: Find the next terms in the sequence 1,3,6,10,15 Square Numbers Sequence 7: Find the next terms in the sequence 1,4,9,16,25,__ Fibonacci Numbers Sequence 8: Find the next terms in the sequence 1,1,2,3,5,8,13,__ B. The Fibonacci Sequence In mathematics, the Fibonacci numbers, commonly denoted Fn, form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is, F0 = 0 , F1 = 1 and Fn = Fn – 1 + Fn – 2 for n > 1. The beginning of the sequence is thus: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377… Each number in th sequence is the sum of the two number which precede it. In some older books, the value F0 = 0 is omitted, so that the sequence starts with F1 = F2 = 1 and the recurrence Fn = Fn – 1 + Fn – 2 is valid for n > 2. The Fibonacci spiral: an approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling; NVSU-FR-ICD-05-00 (081220) Page 6 of 9 Mathematics and Statistics Department “In accordance with Section 185. Fair use of a Copyrighted Work of Republic Act 8293, the copyrighted works included in this material may be reproduced for educational purposes only and not for commercial distribution”.  Republic of the Philippines NUEVA VIZCAYA STATE UNIVERSITY Bayombong, Nueva Vizcaya INSTRUCTIONAL MODULE IM No.1: IM-GEMATH-1STSEM-2024-2025 Fibonacci numbers are strongly related to the golden ratio: Binet's formula expresses the nth Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are named after Italian mathematician Leonardo of Pisa, later known as Fibonacci. In his 1202 book Liber Abaci, Fibonacci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, an uncurling fern, and the arrangement of a pine cone's bracts. NVSU-FR-ICD-05-00 (081220) Page 7 of 9 Mathematics and Statistics Department “In accordance with Section 185. Fair use of a Copyrighted Work of Republic Act 8293, the copyrighted works included in this material may be reproduced for educational purposes only and not for commercial distribution”.  Republic of the Philippines NUEVA VIZCAYA STATE UNIVERSITY Bayombong, Nueva Vizcaya INSTRUCTIONAL MODULE IM No.1: IM-GEMATH-1STSEM-2024-2025 The ratio of any two successive Fibonacci Numbers is very close to the Golden Ratio, referred to and represented as phi () which is approximately equal to 1.618034… The bigger the pair of Fibonacci Numbers is considered, the closer is the approximation. NVSU-FR-ICD-05-00 (081220) Page 8 of 9 Mathematics and Statistics Department “In accordance with Section 185. Fair use of a Copyrighted Work of Republic Act 8293, the copyrighted works included in this material may be reproduced for educational purposes only and not for commercial distribution”.  Republic of the Philippines NUEVA VIZCAYA STATE UNIVERSITY Bayombong, Nueva Vizcaya INSTRUCTIONAL MODULE IM No.1: IM-GEMATH-1STSEM-2024-2025 Mathematics solves puzzles in nature, describes changing quantities via calculus, modelling change, and predicts and controls physical systems. Mathematics also about operations (also known as functions and transformations), about logical relationships between facts, and about proof. Drops, dynamics and daisies are three examples of “simplicity emerging from complexity”. Mathematics is an ordinary exercise of the human mind in abstracting the results of observation to find similarities and differences between phenomena. Nature, as an object of mathematical study, bridges the gap between the concreteness if the everyday environment and abstraction of mathematics. Mathematics, in turn, allow us to summarize, formalize, and extrapolate, from observations that have been recorded. NVSU-FR-ICD-05-00 (081220) Page 9 of 9 Mathematics and Statistics Department “In accordance with Section 185. Fair use of a Copyrighted Work of Republic Act 8293, the copyrighted works included in this material may be reproduced for educational purposes only and not for commercial distribution”.  Republic of the Philippines NUEVA VIZCAYA STATE UNIVERSITY Bayombong, Nueva Vizcaya INSTRUCTIONAL MODULE IM No.1: IM-GEMATH-1STSEM-2024-2025 E. Nature and Occurrence in the world as controlled by mathematics for human ends Mathematics relies on both logic and creativity, and it is pursued both for variety of practical purposes and for its intrinsic interest. The essence of mathematics lies in its beauty and its intellectual challenge. Because mathematics plays such a central role in modern culture, some basic understanding of the nature of mathematics is requisite for scientific literacy. The application of mathematics to medicine is an exciting and novel area of research within the discipline of applied mathematics. Nurses routinely use addition, fractions, ratios and algebraic equations each workday to deliver the right amount of medication to their patients or monitor changes in their health. A component in which mathematics contributes significantly to health and medicine concerns life expectancy. This varies across the world and even among different ethnic and gender groups within the same country and is due to the numerous factors that can positively or negatively affect the life people lead. Political scientist use math and statistics to predict the behavior of group of people. They study the population using many different applications of math, including computer science, database management, statistics and economics. Analysis and study in economics help explain the interdependent relation between different variables. As student advance their study of economics, they realize that there is more to it than just theories. There is no better way of explaining the concepts of prices, quantity of goods sold and costs without the use of mathematics. F. Applications of Mathematics in the World Farming and gardening also provide rich mathematical opportunities. Mathematics has enable farming to be more economically efficient and has increase productivity. Basic geometry, proportions, multiplication and measurement skills are used every day by farmers. Planning a market list and grocery shopping requires math knowledge, starting from the fundamental operations to estimation and percentages. Today’s trends like using credit card to pay, or atm debit or electronic banking are all applications of mathematics. Anywhere in the house, there is mathematics and working in the kitchen requires mathematical knowledge. Long and short travels involves math in various ways. A contractor or construction worker, knows that building anything and creating something requires a broad range of mathematics. The art of applying mathematics to complex real-world problems is called engineering mathematics. It combines mathematical theory, practical engineering and scientific computing to address the fast- changing technology. Many experts agree that without strong math skills, people tend to invest, save, or spend money based on their emotions. Without good planning of time, the day can slip idly and tasks and duties accrue. In a swift changing world, creating and following schedule prove beneficial, but it takes more mathematical skills than simply using a clock and calendar to manage time well and be on top of others. Generalization Many patterns and occurrence exist in nature, in our world and in our life. Mathematics help make sense of the patterns and occurrence. Mathematics is a tool to quantify, organized and control the world, predict phenomena and make life easier for us. NVSU-FR-ICD-05-00 (081220) Page 10 of 9 Mathematics and Statistics Department “In accordance with Section 185. Fair use of a Copyrighted Work of Republic Act 8293, the copyrighted works included in this material may be reproduced for educational purposes only and not for commercial distribution”.