AVO_Module1-GE4-MMW Mathematics in Modern World PDF

“MATHEMATICS IN MODERN WORLD” 1 “MATHEMATICS IN OUR WORLD” MS. ANGEL ERIKA V. OLAES  NO. 1 Mathematics in Modern World Patterns and Numbers in Nature and in the World Fibonacci Sequence Patterns and Regularities in...

“MATHEMATICS IN MODERN WORLD” 1 “MATHEMATICS IN OUR WORLD” MS. ANGEL ERIKA V. OLAES  NO. 1 Mathematics in Modern World Patterns and Numbers in Nature and in the World Fibonacci Sequence Patterns and Regularities in the World Behavior of Nature Applications of Mathematics in the World Argue about the nature of mathematics: what it is, how it is expressed, represented, and used. Discuss the concept Fibonacci and its applications. Identify patterns in nature and regularities in the world. Appreciate the nature and uses of mathematics in everyday life. Establish the relationship between the Fibonacci sequence with the golden ratio. investigate the relationship of the golden ratio and Fibonacci number, in natural world. Determine the application of the Golden ratio in arts architecture.  NO. 1 Mathematics in Modern World This module discusses about …” what is physical is subject to the laws of mathematics, and what is spiritual to the laws of God, and the laws of Mathematics are but the expression of the thoughts of God.” – Thomas Hill.  NO. 1 Mathematics in Modern World https://www.youtube.com/watch?v=64643Op6WJo NO. 1 Mathematics in Modern World Mathematics is a tool. Play with it any way you want and see if you can make something. Don’t worry if you break the tool, we’ll rebuild it, together. Today, we’ll be talking about the essence of mathematics and how it shapes the world around us. The intention behind this lesson is to show the beauty of math to people, how it governs nature without most of us even noticing it. One of the things about Mathematics that we love the most is its weird ability to reveal hidden beautiful patterns in our everyday life, the nature around us. These patterns can be sequential, three-dimensional, temporal, and even linguistic. There are connections between things that don’t seem connected, but can be observed with the intellect of math. NO. 1 Mathematics in Modern World One beautiful example is — fireflies flashing in unison and a pattern that can be solved mathematically. NO. 1 Mathematics in Modern World In short, we can say mathematics is the science of patterns. Talking more about patterns, let’s have a glimpse of “Chaos Theory, which is a hot topic among many mathematicians. ‘Chaos’ is an interdisciplinary theory stating that within the apparent randomness of complex systems, there are underlying patterns, constant feedback loops, repetition, self-similarity, fractals and self-organization. NO. 1 Mathematics in Modern World One of the simplest examples to understand is the ‘Butterfly Effect’ that describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state, e.g. a butterfly flapping its wings in Brazil can cause a tornado in Texas. Such kind of phenomena are often described by fractal mathematics. Chaos theory and chaotic models have applications in many areas including geology, economics, biology, meteorology etc., and can help demystify the huge dynamic complex systems.  NO. 1 Mathematics in Modern World What is a fractal? A fractal is a never-ending pattern. They are the images of dynamic systems — the pictures of ‘Chaos’. Geometrically, they exist in between our familiar dimensions, nature is full of fractals, for instance: trees, mountains, seashells, clouds, ferns, even human body! These things look very complex and non-mathematical. Now, think what it took to produce what you see. You’ll realize it takes endless repetition and that gives rise to one of the defining characteristics of a fractal, a self-similarity. Fractals have huge applications in astronomy, fluid mechanics, image compression etc. as they hold the key to describe the real world better than traditional science.  NO. 1 Mathematics in Modern World Let’s have a closer look at some of the real- world fractal examples around us. 1. Fern — As you look deeper and deeper, you see a never- ending repetitive pattern. 1. Koch Snowflake — A beautiful example of a fractal with infinite perimeter but finite area. The idea is, make an equilateral triangle. Now make another equilateral triangle above the previous one, but in opposite direction. You’ll see small equilateral triangles on the boundary. Keep doing the same for them, and keep doing, keep doing…When you keep doing it, soon after some depth, you’ll start seeing the resemblance of pattern with a snowflake. 1. Fractal Antenna — Above example of ‘Koch snowflake’ shows a fractal of perimeter increasing infinitely while it’s area can be bounded. Using such property, fractal antenna was invented, using a self-similar deign to maximize the length of material that can receive much weaker signals and transmit signals over long distance without compromising the area and volume taken by the antenna due to its fractal nature. This is very compact and have useful applications in cellular telephone and microwave communications.  NO. 1 Mathematics in Modern World Now let’s have a look at one of the famous mathematical number sequences, the ‘Fibonacci Sequence’. The Fibonacci sequence is a recursive sequence, generated by adding the two previous numbers, the first two numbers of the sequence being 0 and 1. So, Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 … An interesting fact is that the number of petals on a flower always turns out to be a Fibonacci number. Statistically, this sequence appears a lot in botany. Another example is if you look at the bottom of pine cone, and count clockwise and anti-clockwise number of spirals, they turn out to be adjacent fibonacci numbers.  NO. 1 Mathematics in Modern World Let’s have a look at a property of Fibonacci numbers. I’m going to write continuous sums of squared Fibonacci numbers. Squared Fibonacci Sequence: 0, 1, 1, 4, 9, 25, 64, … Continuous sums: 0=0x1 0+1=1x1 0+1+1=1x2 0+1+1+4=2x3 0+1+1+4+9=3x5 0 + 1 + 1 + 4 + 9 + 25 = 5 x 8 … and so on. (You see every time product of the sum is two consecutive Fibonacci numbers) NO. 1 Mathematics in Modern World Learn more about the Fibonacci sequence and natural spirals in this fascinating video series by mathematician Vi Hart, who talks fast, but she's interesting and will remind you of the way your brain once hopped from subject to subject: https://youtu.be/ahXIMUk SXX0  NO. 1 Mathematics in Modern World The reason for why plants use a spiral form like the leaf picture above is 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.  NO. 1 Mathematics in Modern World GOLDER RATIO MANIFEST IN NATURE Chameleon tails A chameleon tail is famous for its tight spiraling shape. (Photo: Ryan M. Bolton/Shutterstock) Seashells A seashell is one of the most well-known examples of the golden ratio spiral in nature. (Photo: Tramont_ana/Shutterstock) Fern fiddleheads The curled-up fronds of a young fern are known as fiddleheads. (Photo: Zamada/Shutterstock) Ocean waves Despite their tumultuous nature, ocean waves are another example of the golden ratio manifesting in nature. (Photo: irabel8/Shutterstock)  NO. 1 Mathematics in Modern World Golder Ratio manifest in Arts Leonardo da Vinci The Golden Section was used extensively by Leonardo Da Vinci.  NO. 1 Mathematics in Modern World Michelangelo In Michelangelo’s painting of “The Creation of Adam” on the ceiling of the Sistine Chapel, look at the section of the painting bounded by God and Adam.  NO. 1 Mathematics in Modern World GOLDER RATIO MANIFEST IN ARCHITECTURE Ancient Greek architecture used the Golden Ratio to determine pleasing dimensional relationships between the width of a building and its height, the size of the portico and even the position of the columns supporting the structure. The final result is a building that feels entirely in proportion. Ancient Greek architecture uses the Golden Ratio to determine pleasing dimensions Ration in animals: NO. 1 Mathematics in Modern World RATION IN ANIMALS: All the key The eyes, beak, facial features of wing and key the tiger fall at body markings of golden sections of the penguin all fall the lines defining at golden sections the length and of its height. width of its face. Ration in animals: NO. 1 Mathematics in Modern World APPLICATION OF MATHEMATICS IN THE WORLD The body of knowledge and practice known as mathematics is derived from the contributions of thinkers throughout the ages and across the globe. It gives us a way to understand patterns, to quantify relationships, and to predict the future. Math helps us understand the world — and we use the world to understand math. The world is interconnected. Everyday math shows these connections and possibilities. The earlier young learners can put these skills to practice, the more likely we will remain an innovation society and economy.  NO. 1 Mathematics in Modern World For students to function in a global context, math content needs to help them get to global competence, which is understanding different perspectives and world conditions, recognizing that issues are interconnected across the globe, as well as communicating and acting in appropriate ways. In math, this means reconsidering the typical content in atypical ways, and showing students how the world consists of situations, events and phenomena that can be sorted out using the right math tools. NO. 1 Mathematics in Modern World GE4-MMW-1.3.1 “Data Gathering” Pen & Paper none  NO. 1 Mathematics in Modern World GE4-MMW-1.3.1 Look around your place see if there’s Fibonacci sequence and natural spirals around you. List and explain at least 5 application below (with explanation): 1.. 2.. 3.. 4.. 5..  NO. 1 Mathematics in Modern World GE4-MMW-1.3.1 For Flexible Distance Learning: Screenshot of hand written answer on bondpaper and uploaded at Edmodo Apps For Modular Distance Learning: Handwritten bondpaper and submitted at AISAT Campus Five days after the discussion. September 30, 2024 NO. 1 Mathematics in Modern World GE4-MMW-1.4.1 “Data Gathering” Pen & Paper none  NO. 1 Mathematics in Modern World GE4-MMW-1.4.1 Give 2 best examples of what is the importance of mathematics and how it can help us in our daily lives. Write your answer below: 1. ______________________: __________________________________________________________________ __________________________________________________________________ __________ 2.______________________: _________________________________________________________________ __________________________________________________________________ __________  NO. 1 Mathematics in Modern World GE4-MMW-1.4.1 For Flexible Distance Learning: Screenshot of hand written answer on bondpaper and uploaded at Edmodo Apps For Modular Distance Learning: Handwritten bondpaper and submitted at AISAT Campus Five days after the discussion. September 30, 2024  NO. 1 Mathematics in Modern World “Don’t let what you cannot do interfere with what you can do.” – John Wooden