Unit 1: Mathematics In Our World PDF 2020

West Visayas State University | 2020 Unit 1: MATHEMATICS IN OUR WORLD Learning Outcomes 1. Identified patterns in nature and regularities in the world. 2. Argued about the nature of mathematics, what it...

West Visayas State University | 2020 Unit 1: MATHEMATICS IN OUR WORLD Learning Outcomes 1. Identified patterns in nature and regularities in the world. 2. Argued about the nature of mathematics, what it is, how it is expressed, represented and used. 3. Articulated the importance of mathematics in one’s life. 4. Expressed appreciation for mathematics as a human endeavour. Introduction https://www.pinterest.ph/pin/407083253789715514/ https://pixabay.com/illustrations/covid-19-coronavirus-social-distance- 4975604/ https://www.facebook.com/earthshakerph/photos/a.19771280756746 53/3205812872806161 Progress and development of human society is swiftly realized through the usefulness of mathematical breakthroughs. The systematic applications of various mathematical concepts brought about comfort, ease, and order to humans and their activities. On the other hand, mathematics also plays an important role in the disintegration of human society. History is rich of various events that show millions of human life and countless of physical assets and properties wiped out. Regrettably this depressing reality also stands as witness to the application of mathematical concepts and ideas. 1  West Visayas State University | 2020 Clearly mathematics is the foundation of man’s daily endeavour and survival. Without its application a day to day grind would be useless and meaningless. Despite the enormous usage of mathematics, it does not fail to gain a scornful treatment from some. To them mathematics makes them anxious and gives them an undesirable difficulty. When they are confronted with numbers normally their reaction would be that of disdain and frustration. This is unfortunate for they must have failed to realize that mathematics is not only about numbers, symbols, rules, and formulas. They must also have failed to realize that without mathematics their life would be disorganized and chaotic. If they would only observe closely their surroundings and daily routine, they will realize that mathematics is embedded in whatever their senses can perceive and in whatever they can achieve every day. Lesson 1.1 will help you strengthen and broaden your understanding of the nature of mathematics. This is basic to your learning of this discipline so you will not hold on to your old notion that mathematics is just a set of rules and symbols. You will explore the patterns in nature and regularities in the world. On the other hand, it is true that mathematics is a powerful tool in man’s daily existence. Math is all around us, in everything we do. It is the building block for everything in our daily lives, including mobile devices, architecture (ancient and modern), art, money, engineering, and even sports (Hom, 2013). Mathematics was introduced to you informally by your parents by exposing you to basic counting and simple calculations. This was further given more emphasis when you started your formal education. This experience might have given you the notion that mathematics is just all about numbers and counting. You held on to this concept about mathematics for quite some time until probably on your own or for whatever reason, you were able to expand your knowledge about mathematics. In addition, you were able to hear from others, from people of various backgrounds, their own success stories about math at the same time you also had the chance to listen to stories that do not talk about success in doing mathematics. This experience perhaps has brought you at one instance of your life to ponder about what mathematics really is. Lesson 1.2 will allow you to know more about mathematics. The ideas you may learn from this lesson are aimed at supplementing or improving your views about mathematics. Mathematics is a huge thing, not just a body of knowledge. Thus, you are also encouraged to express your ideas about mathematics, how you think of it. 2  West Visayas State University | 2020 Finally, before the COVID-19 pandemic, we were in a world so fast-changing. Development seem to be limitless and human interaction boundless. In these scenarios mathematics is in active play. Mathematics had been vital in bringing about the world that we have today. Without it, our world would be missing a major component in its development. Even in its current situation, with the pandemic that we experience, our world still depends heavily in the application of mathematics whether it is obviously seen or kept behind the scenes. Without it, human society will face its downfall. In Lesson 1.3, you will learn the importance of mathematics and its various applications. 3  West Visayas State University | 2020 Pre - test Directions: Read each item very carefully then encircle your answer. 1. Patterns are regular, recurring, and repeating forms or designs that are commonly observed in natural objects. True False 2. Fibonacci patterns are often seen in nature. This is based on the Fibonacci sequence, which is a sequence of numbers in which _______________. a. a number is the sum of the two numbers that come after it b. a number is the sum of the two numbers that come before it c. each number gets infinitely smaller d. the numbers all add up to the same value 3. What are fractals? a. Patterns that are iterated on an infinitely smaller scale b. Cubed or tiled patterns c. Irregular stripes or spots d. When ripples are broken or 'fractured' 4. Many patterns and occurrences exist in nature, in our world, in our life. Mathematics helps makes sense of these patterns and occurrences. True False 5. Mathematics is a tool to quantify, organize, and control our world, predict phenomena, and make life easier for us. True False 6. Mathematics is just for the books, confined in the classroom. True False 7. Mathematics helps control nature and occurrences in the world for our own ends. True False 8. Mathematics has numerous applications in the world making it indispensable. True False 9. Mathematics helps predict the behavior of nature and phenomena in the world. True False 10. Everyone uses mathematics but different people use different mathematics at different times, for different purposes, using different tools, with different attitudes. True False 4  West Visayas State University | 2020 Lesson 1.1: PATTERNS AND NUMBERS IN NATURE AND THE WORLD Nature by Numbers You are going to watch the video Nature by Numbers by Cristobal Vila. If you do not have the copy, this can be watched through this link: https://vimeo.com/9953368. Share your Ideas What do you feel after watching the video? Can you write three (3) statements or ideas that you were able to deduce from it? (Note: You may write more than three statements or ideas if you so desire). 1.__________________________________________________________________ 2.__________________________________________________________________ 3.__________________________________________________________________ The statements or ideas that you wrote above contribute to the broadening of your understanding of the nature of mathematics. The video in particular shows the interconnectedness of mathematical ideas and how they are presented in nature. There are many patterns and occurrences that exist in nature, in our world, and in our life and that we make sense of these patterns and occurrences through the use of mathematics. Our Universe of Patterns There is a plethora of evidence that attest to our existence in a universe of patterns. Our wonderful universe entices us to observe and marvel on various phenomena and other works of nature. Patterns are just around us. They can even be the product of our day to day activities. Let us now take time to take a look at patterns. 5  West Visayas State University | 2020 What patterns do you observe in the following pictures? https://www.memecenter.com/fun/45625/spring-summer-autumn-winter https://earthsky.org/brightest-stars/polaris-the-present-day-north- star (a) (b) https://www.accuweather.com/en/weather-news/photos- heavy-snow-slams-washington-and-oregon-triggering-major- travel-disruptions/70007391 https://www.youtube.com/watch?v=fUot7XSX8uA (c) https://torange.biz/tiger-45597 https://pixabay.com/photos/zebra-hartmann-s-mountain- 4034125 (d) 6  West Visayas State University | 2020 https://depositphotos.com/nl/stock- photos/cheetah.html https://www.pxfuel.com/en/search?q=spotted+hyena (e) https://www.lonelyplanet.com/articles/desert-of-dreams- experiencing-the-sahara https://www.transitionsabroad.com/listings/travel/articles/moroc co-sahara-desert-trekking.shtml https://cropwatch.unl.edu/2019/march-15-2019 (f) (g) https://www.reddit.com/r/mildlyint https://apod.nasa.gov/apod/ap121203 eresting/comments/5hw8ui/this_full.html _arc_rainbow_next_to_the_parking (h) _garage/ (i) 7  West Visayas State University | 2020 https://www.jagranjosh.com/articles/w hy-are-raindrops-spherical-in-shape- 1434950581-1 (j) You may have recognized some of the patterns that are presented in the pictures above. On the other hand you may have just realized that some of these patterns actually exist. These patterns that are shown to you are the following: a. stars move in circles across the sky b. seasons cycle at yearly intervals c. the six-fold symmetry of snowflakes -each snowflake is unique -it cannot have the same shape with another snowflake d. the tiger and the zebra are covered in patterns of stripes e. patterns of spots cover all over the body of the leopard and the hyena f. intricate trains of waves marching across the ocean g. trains of sand dunes marching across the desert h. colored arcs of light decorate the sky in the form of rainbows i. bright circular halo sometimes surrounds the moon on winter j. spherical drops of water fall from the clouds To recognize, classify, and exploit these patterns, the human mind and culture have developed a formal system of thought called mathematics (Stewart, 1995). Through the use of mathematics, we are able to organize and systematize our ideas about patterns and with that we are able to unfold a great secret i.e. nature’s patterns are not just there to be admired, they are vital clues to the rules that govern natural processes (Stewart, 1995). 8  West Visayas State University | 2020 Nature’s Patterns as Clues Johannes Kepler, a German astronomer, in his book Six-Cornered Snowflakes, argued that snowflakes must be made by packing tiny identical units together. The six-fold symmetry of snowflakes is a natural consequence of natural packing. If you pack a large number of identical coins by placing them as closely as possible, you get a honeycomb arrangement in which every coin -except those at the edges- is surrounded by six others, arranged in a perfect hexagon. A hexagonal pattern https://www.quora.com/Why-does-nature-prefer-hexagonal- shapes The regular nightly motion of the stars is a clue to the fact that the Earth rotates. You can watch a video of the Moving Stars of the Northern Hemisphere in youtube through this link: https://www.youtube.com/watch?v=HsJxGpDmJrQ. Waves and dunes are clues to the rules that govern the flow of water, sand, and air. https://www.shutterstock.com/tr/video/clip-3293060-sand- https://www.wallpaperup.com/13438/Water_landscapes_waves_sand_du dunes-desertsahara-desert-dunessand-wavesarabian- nes_africa_namib_d landscapedesert The tiger’s stripes and the hyena’s spots attest to mathematical regularities in biological growth and form. hexagonal https://www.latimes.com/world/la-fg-c1-china-siberian-tiger-20131001-dto-htmlstory.html https://www.amazon.com/Growth-Form-Complete-Revised/dp/0486671356 9 https://africa.cgtn.com/2016/11/27/man-in-zimbabwe-found-with-hyena-nose-tail-in-his-undergarments/  West Visayas State University | 2020 Rainbows tell about the scattering of light and indirectly confirm that raindrops are spheres. http://batak-bergaya.blogspot.com/2017/04/pelangi.html?m=1 Lunar halos are clues to the shape of ice crystals. miracle-and-wonderment-of-the-ice/ https://www.spokesman.com/stories/2019/feb/12/shawn-vestal-the- Libbrecht/6157c127da331b000311f2bbe9691aa4b1d9043a/figure/0 https://www.semanticscholar.org/paper/The-physics-of-snow-crystals- https://earthsky.org/todays-image/rare-lunar-halo-pyramidal-crystals Most ice halos around the sun or moon are made by plate- or column-like hexagonal crystals. The multiple halos in this image are from rare pyramidal ice crystals. https://slideplayer.com/slide/8528373/ Planets are clues to the https://quotes2remember.com/quotes/1703 rules behind gravity and motion. 10  West Visayas State University | 2020 You have just seen various patterns that occur naturally and these patterns not only provide clues to the rules that govern natural phenomena but also they possess beauty. There is beauty in nature’s clues and we can recognize it without any mathemtical training. There is also beauty in the mathematical stories that start from the clues and deduce the underlying rules and regularities. But the beauty is different as it applies to ideas rather than things that are observed. When a hexagonal snowflake is presented, a mathematician can figure out the geometry of ice crystals in the atomic level. Can you find any exceptions once you recognize any background pattern? Stewart (1995) enumerates and explains these exceptions. Nature’s Patterns are Numerical The simplest mathematical objects are numbers and the simplest of nature’s patterns are numerical. Here are some facts: a. In 28 days, the phases of the moon make a complete cycle from new moon to full moon and back again. b. Approximately, there are 365 days in a year. c. Number of legs of some creatures: people - 2 insects- 6 cats- 4 spiders- 8 d. Clover normally has 3 leaves. A four-leaf clover brings luck according a superstition. This reflects a deep-seated belief that exceptions to patterns are special. e. A pattern also occurs in the petals of flowers. The number of petals is one of the numbers that occurs in the sequence 3, 5, 8, 13, 21, 34, 55, 89. Below is a list of flowers with their corresponding number of petals. http://science.halleyhosting.com/nature https://www.minnesotawildflowers.info/ud https://www.applewoodseed.com/produc /basin/3petal/lily/calo/maculosus.htm ata/r9ndp23q/pd/ranunculus-acris-9.jpg t/french-marigold-dainty-marietta/ Lily-3 petals Buttercup – 5 petals Marigold-13 petals 11  West Visayas State University | 2020 https://www.slideshare.net/Gerda_Sidl/mathematics-in-nature- https://www.slideshare.net/Gerda_Sidl/mathematics-in- 26431479 nature-26431479 Chicory-21 petals shasta daisy-34 petals https://www.slideshare.net/Gerda_Sidl/mat https://pt.slideshare.net/Gerda_Sidl/mathematics- hematics-in-nature-26431479 in-nature-26431479/20 Michaelmas Daisy -55 petals Asteraceae family-89 petals The pattern of numbers given by the number of petals of the flowers found on the preceding page is obtained by adding the previous two numbers together. For example, 3+5=8, 5+8=13, 8+13=21 and so on. These numbers can also be found in the spiral patterns of seeds in the head of a sunflower and these are the very numbers existing in a sequence called Fibonacci sequence. You are higly encouraged to read further about this interesting and widely applied mathematicl concept. http://www.inquirebotany.org/en/discussions/pl ant-maths-692.html 12  West Visayas State University | 2020 Geometric Patterns Triangles, squares, pentagons, hexagons, circles, ellipses, spirals, cubes, spheres, cones, and others can be found in nature. Some of these shapes are far more common or more evident. For example, the rainbow is a collection of circles, one for each color. The entire circle is not normally seen but just an arc; but when seen from the air, rainbows can be complete circles. The ripples on a pond are circles; the human eye and the design on butterfly’s wings are also circles. https://www.canadiancontractor.ca/voices/the-unexpected-ripple- http://www.ceravoloimages.com/gallery3/atmos effect-of-mandatory-wsib-in-ontario/attachment/ripples-in-a-pond/ pheric_optics/rainbows/chasing_rainbows.html A rainbow seen from the air Ripples in a pond https://www.fertility-docs.com/programs-and- services/pgd-screening/choose-your-babys-eye-color.php https://www.youtube.com/watch?v=1BcRs9OCaTs Human eyes Circles on butterfly’s wings Different wave patterns can also be observed in nature. Some types are the following: b. Waves that spread in a V-shape a. Waves that surge toward behind a moving boat a beach in parallel ranks. https://mirjamglessmer.com/tag/s Human eyes https://www.khou.com/article/news/local/the- hip-generated-waves/ beaches-are-ripe-for-rips/314551505 13  West Visayas State University | 2020 c. Waves radiating outward from d. Tidal Bore an underwater earthquake https://www.indiatoday.in/education-today/gk- https://www.jagranjosh.com/general-knowledge/what-is-a-tidal- current-affairs/story/tsunami-india-359485-2016- bore-1556004787-1 12-26 e. Swirling spiral whirlpools and f. Structureless random frothing tiny vortices in rivers of turbulent flow https://www.awajishima-kanko.jp/en/uzushio/ https://pixabay.com/photos/waterfall-rapids-river-water- 2420715/ g. Wave patterns seen in the atmosphere-spiral of a hurricane https://www.pinterest.com/pin/286119382546282937/ 14  West Visayas State University | 2020 Wave Patterns on Land The most visible mathematical landscape on Earth are found in the great ergs, or sand oceans, of the Arabian and Sahara deserts. Sand dunes form when the wind blows steadily in a fixed direction. Some types of sand dunes: a. Transverse dunes- similar to the ocean waves, they line up in parallel straight rows at right angles to the prevailing wind direction. The picture at the right shows a transverse dunes in the Caraveli Province of Peru. https://www.researchgate.net/figure/Transverse-dunes-in- the-Caraveli-Province-of-Peru-Note-the-almost-perfect- alignment_fig3_309719801 b. Barchanoid ridges- sand dunes that line up in wavy rows https://www.flickr.com/photos/rwolf/4 988415995/in/photostream/ c. Shield-shaped barchan dunes https://www.worldatlas.com/articles/types-of- sand-dunes.html 15  West Visayas State University | 2020 d. Parabolic dunes This dune is formed when the sand is slightly moist and there is little vegetation to bind it together. They are shaped like a U with the rounded end pointing in the https://www.worldatlas.com/articles/types-of-sand-dunes.html direction of the wind. e. Star-shaped dunes- have arms that radiate out from a center pyramid-shaped mound. They grow upward instead of outward and are a result of multidirectional winds. They are also arranged in a random pattern of spots. https://www.haikudeck.com/desert-visual- dictionary-education-presentation- Qqo9gOEsPS#slide2 Stripes and Spots in the Animal Kingdom Tigers and zebras are covered in patterns of stripes, leopards, hyenas, and giraffes are covered in patterns of spots. https://www.123rf.com/photo_91311887_ spotted-hyena-crocuta-crocuta-also- known-as-the-laughing-hyena-.html https://qz.com/1556012/why-zebras- have-black-and-white-stripes/ http://allears.net/wp- content/uploads/archive/tp/ak/ak115.jpg https://phys.org/news/20 12-02-alan-turing-1950s- tiger-stripe.html 16  West Visayas State University | 2020 The shapes and patterns of plants and animals provide a rich source for the mathematically minded who may look for answers to the following questions: a. Why do so many shells form spirals? b. Why are starfish equipped with a symmetric set of arms? c. Why do so many viruses assume regular geometric shapes, the most striking being that of an icosahedron- a regular solid formed from twenty equilateral triangles? d. What is the shape of the virus that caused the COVID-19 pandemic? e. Why are so many animals bilaterally symmetric? f. Why is that symmetry so often imperfect, disappearing when you look at the detail, such as the position of the human heart or the differences between the two hemispheres of the human brain? g. Why are most of us right-handed, but not all of us? https://hemantmore.org. in/science/biology/criteri a-animal- classification/1149/ http://4- designer.com/2013/09/Shells-and- starfish-02-HD- https://www.britanni picture/#.XUgIB2YRXIU ca.com/science/icosa hedral-virus https://www.pinterest.ph/pin/560768591077808681/?autologin=tru https://socratic.org/questions/in-what-anatomical-position-of- e the-body-is-the-heart-located The human brain has two almost symmetrical halves called cerebral hemispheres. 17  West Visayas State University | 2020 Nature’s symmetries can be found on every scale, from the subatomic particles to that of the entire universe. Many chemical molecules are symmetric. Some examples are shown below. Methane Benzene https://www.dreamstime.com/stock-illustration- http://chemistry.elmhurst.edu/vchembook/204tetrahedral.html benzene-structure-consists-carbon-hydrogen-vector- image49549968 The methane molecule is Benzene has the six-fold tetrahedron- a triangular-sided symmetry of a regular pyramid- with one carbon atom at hexagon. its center and four hydrogen atom at its corners. Buckminsterfullerene The Buckminsterfullerene molecule is a truncated icosahedral cage of sixty carbon atoms. https://bcachemistry.wordpress.com/ tag/buckminsterfullerene/ Icosahedron Truncated Icosahedron An icosahedron is a regular solid with twenty triangular faces. “truncated”- means the corners of https://commons.wikimedia.or the icosahedron are https://mathworld.wolfram.com/T runcatedIcosahedron.html g/wiki/File:Zeroth_stellation_of _icosahedron.png cut off. 18  West Visayas State University | 2020 Patterns of Movement a. Human walk- the feet strike the ground in a regular rhythm. (See video on this link: https://www.youtube.com/watch?v=bl3H-zkNB68) b. Four-legged creature like a horse- when it walks, there is a more complex but equally rhythmic pattern. (See video on this link: https://www.youtube.com/watch?v=jjaQ6qr2jp0) c. Scuttling insects (See video on this link: https://www.youtube.com/watch?v=oJ3JLBjdj30) d. Flight of birds (See video on this link: https://www.youtube.com/watch?v=owiwCIhc0I0) e. Pulsations of jellyfish (See video on this link: https://www.youtube.com/watch?v=jpm4rQ8v_oQ) f. Wavelike movements of fish, worm, and snake (See video on this link:https://www.youtube.com/watch?v=sQ3LNWHNbHc) g. Sidewinder, a desert snake- moves like a single coil or a helical spring, thrusting its body forward in a series of S-shaped curves, in an attempt to minimize its contact with the hot sand. (See video on this link: https://www.youtube.com/watch?v=B3NbPUTD5qA) h. Tiny bacteria-propel themselves using microscopic helical tails, which rotate rigidly, like a ship’s screw. (See video on this link: https://www.youtube.com/watch?v=18ZY91xSe78) New Kinds of Pattern a. Fractals-these are geometric shapes that repeat their structure on ever-finer scales. Although fractals are very complex, they are made by repeating a simple process. Fractals are found in nature, for instance, trees, rivers, coastlines, mountains, clouds, sea shells, hurricanes. Fractals are also found in geometry and algebra referred to as mathematical fractals. Natural Fractals-Branching Neurons from the human cortex. The branching of our brain cells creates the incredibly complex network that is responsible for all https://fractalfoundation.org/fractivities/FractalPacks- EducatorsGuide.pdf 19  West Visayas State University | 2020 Our lungs are branching fractals with a sur-face area ~100 m2. The similarity to a tree is significant, as lungs and trees both use their large surface areas to exchange oxygen and CO2. https://fractalfoundation.org/fractivities/FractalPacks- EducatorsGuide.pdf https://fractalfoundation.org/fractivities/FractalPacks-EducatorsGuide.pdf Lichtenberg “lightning”, formed Oak tree, formed by a sprout by rapidly discharging electrons branching, and then each of the in lucite. branches branching again, etc. River network in China, formed by erosion from repeated rainfall flowing downhill for millions of years. https://fractalfound ation.org/fractivitie s/FractalPacks- EducatorsGuide.pdf 20  West Visayas State University | 2020 Natural Fractals-Spirals https://fractalfoundation.org/fractivities/FractalPacks-EducatorsGuide.pdf A fossilized A hurricane is a self-organizing A spiral galaxy is the largest ammonite from spiral in the atmosphere, natural spiral comprising 300 million years driven by the evaporation and hundreds of billions of stars. ago. A simple, condensation of sea water. primitive organism, it built its spiral shell by adding pieces that grow and twist at a constant rate. https://fractalfoundation.org/fractivities/FractalPacks-EducatorsGuide.pdf The turbulent motion of fluids A fiddlehead fern is a The plant kingdom is full self-similar plant that of spirals. An agave cactus creates spirals in systems ranging from a soap film to forms as a spiral of forms its spiral by growing spirals of spirals. new pieces rotated by a the oceans, atmosphere and fixed angle. Many other the surface of jupiter. plants form spirals in this way, including sunflowers, pinecones, etc Geometric Fractals Sierpinski Triangle https://fractalfoundation.org/fractivities/FractalPacks-EducatorsGuide.pdf The Sierpinski Triangle is made by repeatedly removing the middle triangle from the prior generation. The number of colored triangles increases by a factor of 3 each step, 1,3,9,27,81,243,729, etc. 21  West Visayas State University | 2020 Koch Curve The Koch Curve is made by repeatedly replacing each segment of a generator shape with a smaller copy of the generator. At each step, or iteration, the total length of the curve gets longer, eventually approaching infinity. Much like a coastline, the length of the curve increases the more closely you measure it. https://fractalfoundation.org/fractivities/FractalPacks-EducatorsGuide.pdf Algebraic Fractals-Mandelbrot Set Algebraic fractals are created by repeatedly calculating a simple equation over and over. Because the equations must be calculated thousands or millions of times, we need computers to explore them. Not coincidentally, the Mandelbrot Set was discovered in 1980, shortly after the invention of the personal https://fractalfoundation.org/fractivities/FractalPacks- EducatorsGuide.pdf computer. b. Chaos- a kind of apparent randomness whose origins are entirely deterministic, that tis, the future behavior of a system is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos or simply chaos. Chaotic behavior is observed in many natural systems such as weather and climate. It is also present in some systems with artificial components such as road traffic. 22  West Visayas State University | 2020 Uses of Understanding Nature’s Patterns The development of new mathematical theories helps us to unravel the secrets of the more elusive of nature’s patterns. This discovery has both practical and intellectual impact. Our understanding of nature’s secret irregularities has vast uses including the following: a. to steer artificial satellites to new destinations with far less fuel b. to help avoid wear on the wheels of locomotives and other vehicles that run on a railroad. c. to improve the effectiveness of heart pacemakers d. to manage forests and fisheries e. to make more efficient dishwashers f. it gives us the opportunity to deepen our views about the universe in which we live, and of our own place in it. Activity 1 Go out and observe your surroundings and look for patterns that exist in objects, plants, trees, buildings, etc. Take a photo of the pattern that catches your attention, describe the pattern that you perceive and write a brief explanation how mathematics is embedded in that arrangement. (Note: The teacher and the students may decide on the date of submission of this activity). Activity 2 Read about Fibonacci sequence. Then come up with a 2-page discussion on its applications in various fields. Self- Assessment Question 1 You have learned from this lesson that patterns are nature’s clues to rules that govern natural processes and that important discoveries were made by studying the underlying patterns observed. It is however, noteworthy to pay attention to some exceptions, that is, those phenomena that seem to deviate from the observed patterns. What do you think is the significance of paying attention to and studying about these exceptions? 23  West Visayas State University | 2020 Lesson 1.2: GETTING TO KNOW MATHEMATICS At the right is a cartoon strip of the six blind men and the elephant. In that story the six blind men were asked what is an elephant. What do you think does the picture suggest? What does it imply in the way we perceive mathematics? https://linkiya.wordpress.com/2015/07/13/the-six-blind- men-and-the-elephant-differing-perspectives-on-grades/ What is Mathematics? https://career.sa.ucsb.edu/students/career-planning/choosing- https://denisegaskins.com/2015/10/26/what-is-mathematics/ major/mathematical-sciences Mathematics is a formal system of thought for recognizing, classifying, and exploiting patterns (Stewart, p. 1). Mathematics is a study of patterns and relationships Mathematical ideas are interwoven with each other. Patterns and relationships arise in all parts of mathematics--in numbers and chance, in geometry and data--and not just in the realm of formulas and functions. "Patterns can be either real or imagined, visual or mental, static or dynamic, qualitative or quantitative.... They can arise from the world around us... or from the inner workings of the human mind." (Devlin, 1994) 24  West Visayas State University | 2020 Mathematics is a way of thinking Mathematics is a way of thinking and sense making. It is creative, beautiful, individual, and dynamic. It is being curious, asking questions, figuring out why things work, breaking problems apart, seeking regularity, making predictions, and creating logical arguments (Meyer, 2017). http://academicsfreedom.blogspot.com/20 13/07/teaching-mathematics-as-way-of- thinking.html https://www.edubloxtutor.com/logical- thinking/ Mathematics is an art Mathematics is characteized by order and consistency. Numerous patterns can be found in numbers and geometric figures. Tessellations, weaving and tiling are a few explicit examples of mathematics in art. By exploring the orderliness and consistency of mathematics, we learn to appreciate its beauty. https://www.esquiremag.ph/culture/design/philippine-indigenous- fabrics-are-making-a-comeback-a00225-20171017-lfrm Mathematics is a language Mathematics is used to communicate complex processes and thoughts efficiently using symbols and specific and precise terms. Mathematics has its own register, or https://medium.com/predict/mathematics-the-language-of-the- universe-47325ade387 special vocabulary which needs to be learned to be able one to communicate well about mathematics and to speak and think like mathematicians. Mathematicians would not use “equal”, “congruent”, and “similar” https://www.universetoday.com/120681/mathematics-the- interchangeably as these terms mean beautiful-language-of-the-universe/ different things. 25  West Visayas State University | 2020 Mathematics is a tool Many occupations require the knowledge of mathematics. Scientists, engineers, businessmen, and many other professions use a great deal of mathematics to do their best. https://ieshasmall.com/what-is-the-point-of-maths-a- maths-teachers-unexpected-response/ Where is mathematics? We see hints or clues of it in nature http://eumindmat hs3.weebly.com/p awar-public- maths-in-nature- 1.html In our daily routine https://www.slideshare.net/tcarey10/math-in-daily-life https://sciencestruck.com/math-in-everyday-life 26  West Visayas State University | 2020 In our work https://commons.wikimedia.org/wiki/File:Figure_01_02_19.jpg https://ecmiindmath.org/2016/02/16/a-short-history-of- industrial-mathematics-in-linz-austria/ In people and communities https://www.bulatlat.com/tag/trifpss/ In events https://www.australiancurriculumlessons.com.au/2014/05/28/predictin g-using-data-sets-mathematics-lesson-plan-sporting-events/ https://passyworldofmathematics.com/olympic-games-mathematics/ https://www.manilatimes.net/the-best-of-festivals-at-aliwan-fiesta- 2018/392691/ 27  West Visayas State University | 2020 Please watch this video on youtube: https://www.youtube.com/watch?v =lyGe_NDJjC0 What Mathematics is About? https://www.youtube.com/watch?v=Kp2bYWRQylk http://hkc361blog.blogspot.com/2014/01/what-is-mathematics.html When you hear the word “mathematics”, the first thing that comes to your mind is numbers. Numbers are the heart of mathematics, an all-pervading influence, the raw materials out of which a great deal of mathematics is forged. The absence of numbers is a major handicap. It creates the overwhelming impression that mathematics is mostly a matter of numbers-which isn’t really true. Numbers are just one type of object that mathematicians think about. A large part of the early pre-history of mathematics can be summed up as the discovery. The earliest mathematical ideas/concepts, objects discovered/invented includes the following: 1) counting numbers- e.g. 1, 2, 3… 2) fractions- e.g. , 3) the concept of zero was invented and accepted as denoting numbers between 400 and 1200 AD. 28  West Visayas State University | 2020 There was a period in history when “one” was not considered a number because it was thought that a number of things ought to be several of them. 4) negative numbers- can be thought of as representing a debt, e.g. -1, -5, - 19. Negative numbers can also be represented as follows: a) negative temperature- is one that is colder than freezing b) negative velocity- an object with negative velocity is one that moves backward. 5) Irrational numbers- e.g. √ 6) Real numbers- an enlargement of the number system that includes the irrationals. 7) Imaginary and complex numbers- the next enlargement of the number system e.g. √ Currently, these numbers are known as the ff: 1. Natural Numbers- involving 0, 1, 2, 3,… 2. Integers- the whole numbers including the negative whole numbers. 3. Rational Numbers- involving positive and negative fractions 4. Real numbers- more general set of numbers 5. Complex Numbers- still more general set of numbers These five number systems are inclusive than the previous: natural numbers, integers, rationals, real numbers, and complex numbers. Mathematics is also about Operations. Examples include addition, subtraction, multiplication, and division. An operation is something that is applied to two mathematical objects to get a third object. https://www.assignmentpoint.com /science/mathematic/four-basic-operations- mathematics.html 29  West Visayas State University | 2020 Another type of mathematical object is Function. It is thought of as a rule that starts with a mathematical object-usually a number- and associates to it another object in a specific manner. Functions are often defined using algebraic formulas, which are just shorthand https://wumbo.net/notation/function/ ways to explain what the rule is, but they can be defined by any convenient method. Mathematics is also about Transformation Another type of mathematical object with the same meaning as “function”. Transformation is the rule that transforms the first object into the second. Transformation is used when the rules are geometric. Operations and functions are said to be processes rather than things. Thingification processes is one of the most powerful general weapons in the mathematicians’ armory. Mathematical “things” have no existence in the real world: they are abstractions. But mathematical processes are also abstractions, so processes are no less “things” than the “things” to which they are applied. Example 1 Two is not actually a thing but a process-the process one carries out when two camels or two sheeps are associated with the symbols “1,2” chanted in turn. A number is a process that has long ago been thingified so thoroughly that everybody thinks of it as a thing. It is just as feasible to think of an operation or a function as a thing. Example 2 We might think of “square root” as if it were a thing- and it means here not the square root of any particular number, but the function itself. The square root function is a kind of sausage machine. You stuff a number in at one end and its square roots pop out at the other. 30  West Visayas State University | 2020 Mathematics has a Landscape. Mathematics is not just like a collection of isolated facts: it is more like a landscape; it has an inherent geography that its users and creators employ to navigate through what would otherwise be an impenetrable jungle. There is what we call a metaphorical feeling of distance. Example 3 The circumference of a circle, is very close to the fact that the circumference of a circle is. There is an immediate connection between these two facts- the diameter is twice the radius. On the other hand, unrelated ideas are more distant from each other; for example, the fact that there are exactly six different ways to arrange three objects in order is a long way away from facts about circle. The mathematical landscape has a metaphorical feeling of prominence. This character can be sensed on mathematical ideas that are used widely and seen far away such as Pythagoras’s theorem about right triangles or the basic techniques of calculus. The ingredient that binds the landscape together is Proof, which determines the route from one fact to another. Professional mathematicians consider a statement valid if it is proved beyond any possibility of logical error. A proof is a sequence of statements, each of which either follows from previous statements in the sequence or from agreed axioms- unproved but explicitly stated assumptions that in effect define the area of mathematics being studied. Proof tells a story- a story about mathematics that works. It has to tell a convincing story which has a beginning and an end, and a story line that gets you from one to the other without any logical holes appearing. It must be convincing to be accepted by mathematicians. Proofs knit the fabric of mathematics together, and if a single thread is weak, the entire fabric may unravel. 31  West Visayas State University | 2020 How is mathematics done? Mathematics is done with curiosity. The desire to learn or know something. Mathematical curiosity includes more than simply a desire to learn or know mathematics. It also includes a desire to explore mathematical ideas through posing mathematically interesting problems after one has “finished” a problem (Knuth 2002). https://www.portfoliopartnership.com/does- curiosity-give-you-a-competitive-edge-2/ Mathematics is done with a penchant for seeking patterns and generalities. If you have a penchant for seeking patterns and generalities, you have a special liking for it or a tendency to do it. You devote a special time for doing what you are interested about. Your attention may be in finding something special about some patterns such as the ones in the pictures below. https://thamesandhudson.com/patterns-of-india-a- colouring-book-9780500420744 http://theconversation.com/personalities-are-like- traditions-unique-patterns-of-behaviour-that-build-over-a- lifetime-of-improvisation-108096 https://www.emeryetcie.com/en/portfolio/patchwork/ 32  West Visayas State University | 2020 Mathematics is done with the desire to know the truth. You may want to unravel the truth about the facts and mysteries of the mathematical universe. https://www.worldsciencefestival.com/videos/mysteries-of- the-mathematical-universe/ Mathematics is done with trial and error. Trial and error refers to the process of verifying that a certain choice is right (or wrong). You simply substitute that choice into the problem and check. Some questions can only be solved by trial and error; for others you must first decide if there isn't a faster way to arrive at the answer. In the problems that you solve you test all choices for your benefit. Once you have the right answer, there is no need to check the rest of the choices. Mathematics is done without fear of facing more questions and problems to solve. https://myjobadvice.wordpress.com/2017/ 03/09/problem-show-your-skills-solve-it/ 33  West Visayas State University | 2020 Who uses Mathematics? Mathematicians (pure and applied) https://www.pinterest.ph/pin/560135272387356375/?lp= true https://www.flickr.com/photos/24660089@N04/7429694 772/ Scientists (natural and social) https://www.flickr.com/photos/hhc hu/2511842518 Practically everyone But, different people use different mathematics at different times, for different purposes, using different tools, with different attitudes. https://www.volunteerforever.co m/article_post/visit-indigenous- aboriginal-communities-around- the-world https://www.pinterest.ph/pin/34 1992165432959048/?lp=true You can watch this video to conclude this lesson: https://www.youtube.com/watch?v=WFE-17fLCd4 34  West Visayas State University | 2020 Activity 3 1. Aside from the related mathematical concepts discussed in this lesson, can you cite other mathematical concepts that are also related? Then briefly explain/illustrate the relationship between/among these mathematical concepts. 2. Aside from Pythagorean Theorem and Calculus what do you think are other prominent mathematical ideas/concepts that were invented/discovered? Activity 4 Write a two-to three-page synthesis paper focusing on one of the following aspects of mathematics: (a) Mathematics helps organize patterns and regularities in the world. (b) Mathematics helps predict the behavior of nature and phenomena in theworld. (c) Mathematics helps control nature and occurrences in the world for our own ends Self - Assessment Questions 2 Answer the following: 1. The world is currently experiencing the COVID-19 pandemic. In what aspect or aspects of this event that the role of mathematics is clearly evident? Explain briefly. 2. What new ideas about mathematics did you learn so far? 35  West Visayas State University | 2020 Lesson 1.3: IMPORTANCE OF MATHEMATICS What Mathematics is for? https://www.youtube.com/watch?v=uMO95_YlEKM https://theconversation.com/mathematics-forget- simplicity-the-abstract-is-beautiful-and-important- 91757 Mathematics helps us to solve puzzles Each of nature’s patterns is a puzzle. Mathematics is a more or less systematic way of digging out the rules and structures that lie some observed pattern or regularity, and then these rules and structures are used to explain what’s going on. Example 1 Kepler performed a mathematical analysis of astronomical observations made by Danish Astronomer Tycho Brahe. He came to conclude that planets move in ellipses. Example 2 Isaac Newton discovered that the motion of an object is described by a mathematical relation between the forces that act on the body and the acceleration it experiences. Example 3 Isaac Newton and Gottfried Leibniz invented a new brand of mathematics called the calculus to answer questions about rates of change. Calculus brings out two of the main things that mathematics is for: a) provide tools that enable scientists calculate what nature is doing (external aspect of mathematics or applied mathematics) b) provide new questions for mathematicians to sort out to their satisfaction (internal aspect of mathematics or pure mathematics) 36  West Visayas State University | 2020 Example 4 Physicists were able to find other laws of nature that could explain natural phenomena in terms of rates of change namely, heat, sound, light, fluid, dynamics, elasticity, electricity, and magnetism. Mathematics is a useful way to think about nature What do we want mathematics to tell us about the patterns we observe? a) to understand how and why they happen To understand the spiral form of a snail shell, mathematics describes the atomic structures of the molecules used in shell; it describes the strength and rigidity of shell material as compared to the weakness and pliability of the snail’s body. https://www.pinterest.ph/pin/311874342920034101/?lp =true In the discovery of genes – the molecular structure of the DNA, the genetic material – relied mostly on the existence of mathematical clues. Many pieces of mathematics were involved in the discovery that DNA has the double- helical structure. Recently, scientists have confirmed a new DNA Structure inside human cells. Read the article in this link: https://www.sciencealert.com/scientists https://www.thoughtco.com/double-helix-373302 -have-confirmed-a-new-dna-structure- inside-living-cells-i-motif-intercalated. 37  West Visayas State University | 2020 How Do Snails Get Their Shells? In the context of biological development, the geometry of the snail itself explains why it develops a spiral shell. The shell grows an extra ring of material around its rim, consequently, the snail grows https://animals.howstuffworks.com/marine-life/do-snails-get- shells.htm bigger, so as the size of the rim. Mathematics is used to relate the resulting geometry to various variables – such as growth rate and eccentricity of growth- that are involved. On the other hand, if an evolutionary explanation is sought, the strength of the shell can be looked into and calculate whether a long thin cone is stronger or weaker than a tightly coiled spiral. http://enhg.org/Home/Resources/UnitedArabE mirates/Shellfish.aspx b) to organize the underlying patterns and regularities in a most satisfying way Three-dimensional geometry is used as an organizing principle to place the three-in-a- row pattern of stars in Orion’s belt in the current positions in the heavens. With this principle, the stars are found to be at very different distances from the earth. https://earthsky.org/?p=13996 38  West Visayas State University | 2020 Resonance Resonance is a relationship between a periodically moving bodies in which their cycles are located together, so that they take up the same relative positions at regular intervals. This common cycle time is called the period of the system. This concept is used to organize the pattern in the periods of revolution of Jupiter’s satellites, Ganymede, Europa, and Io. When these three satellites are back in exactly the same relative position as before, this phenomenon is called as resonance. Jupiter’s satellites are said to be in 4:2:1 resonance. http://www.seasky.org/solar-system/jupiter-menu.html http://www.seasky.org/solar-system/jupiter-menu.html The moon’s rotational period is the same as its period of revolution around the earth- a 1:1 resonance of its orbital and its rotational period. Mercury’s rotational and orbital periods are in a 2:3 resonance. The Hilda group of asteroids (an asteroid beltlet) is in 2:3 resonance with Jupiter. All of the Hilda asteroids circle the sun three times for every two revolutions of Jupiter. The Hilda asteroids are a group that have orbits in a 3:2 resonance with Jupiter, i.e. a Hilda asteroid orbits the Sun three times for every two orbits of Jupiter. Although individual Hilda asteroid has an ordinary elliptical orbit, collectively the group has a triangular form with apexes that remain fixed with respect to Jupiter. In this video clip, the Hildas are highlighted in red with the rest of the asteroids shown in brown. The over 500,000 asteroid positions are calculated at interactive rates on the GPU using the VESTA engine for astronomical visualization. (https://www.youtube.com/watch?v=DibIiHcy-xM) c) to predict how nature will behave Lunar and solar eclipses and the return of comets are predicted by astronomers with their understanding the motion of heavenly bodies. Tides are predicted many years ahead since tides are controlled by the position of the sun and moon to the earth. 39  West Visayas State University | 2020 d) to control nature for our own ends A thorough understanding of how a system works can make us attempt to control the system to make it do what we want. Some examples of control systems are the following: 1. thermostat on a boiler, which keeps it a fixed temperature https://www.ebay.com/itm/868MHz-Wireless-Digital-LCD-RF-Thermostat- For-Boiler-Gas-Electric-Heating-System-/263960538650 Wireless Digital LCD RF Thermostat For Boiler/Gas/Electric Heating System 2. coppicing woodland http://www.lsrtreesurgery.co.uk/coppicing/ 3. launching of space shuttle https://en.wikipedia.org/wiki/File: Space_Shuttle_Columbia_launching.jpg 40  West Visayas State University | 2020 4. use of electronic pacemakers to help people with heart disease http://caifl.com/cardiac-devices/pacemakers/ e) to make practical use about what we have learned about our world The most down-to-earth aspect of mathematics is its practical applications this is how mathematics earns its value. Our world rests on mathematical foundations and mathematics is unavoidably embedded in our global culture. Our lives are strongly affected by mathematics it is kept as far as possible behind the scenes. Mathematics underlie the booking of vacation. It involves the use of computers and telephone lines which are designed through the use of mathematical and physical theories. It also involves the scheduling of as many flights as possible around any particular which is done using the concept of optimization. Moreover, provision of accurate radar images for the pilots made is made possible through the use of signal-processing methods is also involved. Three-dimensional geometry is used to produce special effects on the screen when someone watches a television. Coding methods are used to transmit TV signals by satellite. Thousands of different applications of mathematics underlie every step of the manufacture of every component of the spacecraft that launches the satellite into position Statistical theories of genetics provide the foundation in identifying which genes would make a particular type of plant resistant to disease. 41  West Visayas State University | 2020 Provides tools for calculation Mathematics provides an appropriate tool for every task at hand. They usually differ in how good they serve particular purposes associated with a task. Mathematical tools vary in respect to conditions like suitability, efficacy, optimality, and others and they can be changed, made better, or lose their adequacy. Provides new questions to think about Mathematics allows us to generalize concepts at higher levels of abstraction. For example, consider a pattern of shapes: line, square, cube. This is a pattern: a one-dimensional object, two-dimensional object, and three-dimensional object. We can ask then (continuing the pattern): what would a four-dimension object look like? Can we make sense of a fractional dimension? What about a negative dimension? Can we make sense of these concepts? There are gaps in patterns we encounter that are ambiguous or seem to have no solvable answers. And when you study patterns, you see more patterns in the study itself (Seifert, 1999). Why is it important to learn or know Mathematics? The mathematics that underlie https://www.tea human work must be understood in the past, cherswithapps.c om/important- learn-math- otherwise airliners, television, spacecraft, someday-might- accidently/ and disease resistant plants wouldn’t have been invented. It must also be understood in the present, otherwise these inventions won’t continue to function. New mathematics has to be invented in the future to be able to solve problems that either have not arisen before or have not been given solutions otherwise our society will fall apart when change requires solutions to new problems or new solutions to old problems. If mathematics were to be frozen, so that it never went a single step farther, our civilization would start to go backward. The really important breakthroughs in mathematics are always unpredictable. It is their very unpredictability that makes them important: they change our world in ways we didn’t see coming. Mathematics is important for it puts order in our life; it puts order in disorder; it helps us become better persons; and it helps the world a better place to live in. 42  West Visayas State University | 2020 Various Applications of Mathematics Mathematics is used in a large variety of real worl applications. The general public applies arithmetic in grocery shopping, financial mathematics is applied to commerce and economics, statistics is used in many fields (e.g. marketing and experimental sciences), number theory is used in information technology and cryptography, surveyors apply trigonometry, operations research techniques are applied to logistics across diverse industries, and the list of applications of mathematics is endless (Wilson, 2008). Applications of Mathematics in Technology https://mathigon.org/applicatio Predicting the Weather ns Internet and Phones Computers https://mathigon.org/applications Reading CDs and DVDs Public Key Cryptography Satellite Navigation https://mathigon.org/applications Digital Music Computer Circuits Public Transportation 43  West Visayas State University | 2020 Applications of Mathematics in Engineering https://mathigon.org/applications Construction Automotive Design Robotics https://mathigon.org/applications Rockets and Satellites Microwaves Surveying Applications of Mathematics in Media https://mathigon.org/applications Making Music Movie Graphics Music Shuffling 44  West Visayas State University | 2020 Applications of Mathematics in Medicine and Health https://mathigon.org/applications MRI and Tomography Neurology Epidemics Analysis https://mathigon.org/applications Pharmacy and Medicine Plastic Surgery Counting Calories Applications of Mathematics in Finance and Business https://mathigon.org/applications Finance and Banking Gambling and Insurance Betting Loans, Interest, Fraud Detection https://mathigon.org/applications Pricing Strategies Mortgages 45  West Visayas State University | 2020 Applications of Mathematics in Science and Nature https://mathigon.org/applications Cosmology Glacier Melting Carbon Dating https://mathigon.org/applications Search for Alien Life Wildfire Modelling Applications of Mathematics in Sports https://mathigon.org/applications Football Scoring Skate Park Design Swimsuit Design 46  West Visayas State University | 2020 Self - Assessment Questions 3 Answer the following questions: 1. What is most useful about mathematics for humankind? 2. What is it about mathematics that might have changed your thoughts about it? Summary In Lesson 1.1, you have learned that various patterns and occurrences exist in nature, in our world, and in our life. What do you want to do with these patterns? Perhaps you have admired and marveled at them and communed with them which reminded you of what you are. What is significant about these patterns is that mathematics is embedded in them. If you closely observe them you might be able to come up with some generalizations that can serve as an explanation to some of life’s unsolved puzzles. Mathematics is your tool to exploit and make sense of these patterns. You can take full advantage of this powerful tool for you to understand why these patterns exist and how you can make use of these patterns to improve and find meaning in your own existence. In Lesson 1.2, you had the opportunity to get to know mathematics better and deeper. Perhaps, you were able to widen your existing notion of mathematics and made a realization that mathematics is such an indispensable tool for man’s existence and therefore, you need to improve your own mathematical knowledge and skills. Mathematics is something that you cannot do away with. Your activities from simple to complex bear its presence. Whether consciously or unconsciously you do mathematics in various ways and purposes. It’s basically part of your own endeavor whatever your role in society is. In Lesson 1.3, you navigated through the importance and various applications of mathematics. The world as you knew it achieved the peak of its development with the use of mathematics. In different periods of human civilization mathematics is studied to enable to produce inventions that could bring about major changes in the society. Across the various fields of human endeavor, the very tool which is in active play is mathematics. 47  West Visayas State University | 2020 Post - test Directions: Read each item very carefully then encircle your answer. 1. Patterns are regular, recurring, and repeating forms or designs that are commonly observed in natural objects. True False 2. Fibonacci patterns are often seen in nature. This is based on the Fibonacci sequence, which is a sequence of numbers in which _______________. e. a number is the sum of the two numbers that come after it f. a number is the sum of the two numbers that come before it g. each number gets infinitely smaller h. the numbers all add up to the same value 3. What are fractals? e. Patterns that are iterated on an infinitely smaller scale f. Cubed or tiled patterns g. Irregular stripes or spots h. When ripples are broken or 'fractured' 4. Many patterns and occurrences exist in nature, in our world, in our life. Mathematics helps makes sense of these patterns and occurrences. True False 5. Mathematics is a tool to quantify, organize, and control our world, predict phenomena, and make life easier for us. True False 6. Mathematics is just for the books, confined in the classroom. True False 7. Mathematics helps control nature and occurrences in the world for our own ends. True False 8. Mathematics has numerous applications in the world making it indispensable. True False 9. Mathematics helps predict the behavior of nature and phenomena in the world. True False 10. Everyone uses mathematics but different people use different mathematics at different times, for different purposes, using different tools, with different attitudes. True False 48  West Visayas State University | 2020 References Gutierrez, A. (2017). Indigenous Filipinos Fabrics Are Making a Comeback. https://www.esquiremag.ph/culture/design/philippine-indigenous-fabrics-are-making- a-comeback-a00225-20171017-lfrm Hom, E. J. (2013). What is Mathematics? https://www.livescience.com/38936- mathematics.html Knuth, E. (2002). Fostering Mathematical Curiosity. The Mathematics Teacher, 95(2), 126-130. Retrieved from http://www.jstor.org/stable/20870953 Nocon, R. and Nocon, E. Essential Mathematics for the Modern World Stewart, I. A. N. (1995). Nature's Numbers: The Unreal Reality of Mathematics Vistru-Yu, C. PowerPoint Presentation. CHED-GET AdMU Training Nature by Numbers. https://vimeo.com/9953368. https://reader.elsevier.com/reader/sd/pii/S089396590800253X?token=3E122F7FFDB 64A1D1EDF2928D7EA132691A46665AE7C9EBAEAA29CECAB2996F73D7CE809486BC 32AB4349BB451B5F195 https://mathigon.org/application https://www.slideshare.net/knowellton/module-63-mathematics https://brilliant.org/wiki/sat-trial-and- error/#:~:text=Trial%20and%20error%20refers%20to,is%20right%20(or%20wron g).&text=Some%20questions%20can%20only%20be,all%20choices%20for%20your %20benefit. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.231.9177&rep=rep1&type =pdf https://www.quora.com/Why-does-mathematics-provide-new-questions-to-think- about https://www.sciencealert.com/scientists Video Links: Nature by Numbers- https://vimeo.com/9953368. Human walk- https://www.youtube.com/watch?v=bl3H-zkNB68 Four-legged creature like a horse-https://www.youtube.com/watch?v=jjaQ6qr2jp0 Scuttling insects-https://www.youtube.com/watch?v=oJ3JLBjdj30 Flight of birds-https://www.youtube.com/watch?v=owiwCIhc0I0 Pulsations of jellyfish-https://www.youtube.com/watch?v=jpm4rQ8v_oQ Wavelike movements of fish, worm, and snake- https://www.youtube.com/watch?v=sQ3LNWHNbHc Sidewinder, a desert snake- https://www.youtube.com/watch?v=B3NbPUTD5qA Tiny bacteria-https://www.youtube.com/watch?v=18ZY91xSe78 https://www.youtube.com/watch?v=DibIiHcy-xM 49