Mathematics In Our World Module 1 PDF

MATHEMATICS IN OUR MODULE 1 WORLD …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 e...

MATHEMATICS IN OUR MODULE 1 WORLD …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 Mathematics is defined as the study of numbers and arithmetic operations. Others describe mathematics as a set of tools or a collection of skills that can be applied to questions of “how many” or “how much”. Still, others view it as a science which involves logical reasoning, drawing conclusions from assumed premises, and strategic reasoning based on accepted rules, laws or probabilities. Mathematics is also considered an art which studies patterns for predictive purposes or a specialized language which deals with form, size, and quantity. MODULE Nevertheless, mathematics has been given different definitions. People OVERVIEW have viewed it as a set of problem-solving tools, as a language, as a study of patterns, as an art, or as a process of thinking, among others. MODULE OBJECTIVES LESSONS IN THE MODULE At the end of this module, challenge yourself to: LESSON 1: Patterns and Numbers in 1. argue about the nature of mathematics, Nature and in the World what it is, how it is expressed, represented LESSON 2: Fibonacci Sequence and used; 2. discuss the concept of Fibonacci and its LESSON 3: Patterns and Regularities in application; the World 3. identify patterns in nature and regularities in the world; 4. appreciate the nature and uses of mathematics in everyday life; 5. establish the relationship between the Fibonacci sequence with the golden ratio; 6. investigate the relationship of the golden ratio and Fibonacci number in natural world; and 7. determine the application of the Golden ratio in arts and architecture. MODULE 1: MATHEMATICS IN OUR WORLD 1  MODULE 1 Patterns and Numbers in Nature LESSON 1 and in the World At the end of this lesson, you should be able to: a. argue about the nature of mathematics, what it is, how it is expressed, represented and used; b. appreciate the nature and uses of mathematics in everyday life; and c. determine the patterns and numbers in nature of the world. LEARNING OUTCOMES Patterns in nature are visible regularities of form found in the natural world and can also be seen in the universe. These patterns recurring in different context can sometimes be modelled mathematically. Man has developed a formal system of thought for recognizing, classifying, and exploiting patterns which we called mathematics. By applying mathematics to organize and systematize ideas about patterns, we have discovered a pattern in nature. Nature patterns which are not just to be admired, they are vital clues to the rules that govern natural processes. Patterns possess utility as well as beauty and once we have learned to recognize a background pattern, we can immediately appreciate it. INTRODUCTION Patterns in nature are visible regularities of form found in the natural world and can also be seen in the universe. In this activity, your task is to look for patterns in the environment. You have to take a photo of it and upload it to your ACTIVITY Facebook account. Please write a short caption describing the pattern you have captured and explain how it relates to Mathematics. Upon uploading, use “ EXPLORING the hashtag: ” NATURE #Math111ExploringNature Go and explore the nature! Right now, let us try to wrap up your experiences from the previous activity. Let us try to answer the following questions: 1. What are your observations on the photos you have captured in terms of their pattern? 2. How does Mathematics play an important role in determining recurring patterns ANALYSIS in nature? MODULE 1: MATHEMATICS IN OUR WORLD 2  PATTERNS AND NUMBERS IN NATURE AND THE WORLD There is much beauty in nature's clues even without any mathematical training we can all recognize it. Mathematical stories have its own beauty which start from the clues and deduce the underlying rules and regularities, but it is a different kind of beauty, applying to ideas rather than things. The development of new mathematical theories begins to reveal the secret of nature's patterns. We have already seen a practical impact as well as an intellectual one of our newfound understandings of nature's secret ABSTRACTION regularities. But most important of all, it is giving us a deeper vision of the universe in which we live, and of our own place in it. The modern understanding of visible patterns is developed gradually through the years. Different Snowflakes Patterns can be observed even in stars which move in circles across the sky each day. The weather seasons cycle each year (e.g., winter, spring summer, fall). All snowflakes contain sixfold symmetry which no two are exactly the same. There are evidences presented by mathematician that hexagonal snowflakes have an atomic geometry of ice crystals. Spotted Trunkfish Spotted Puffer Blue Spotter Stingray Spotted Moray Eel Coral Grouper Red Lion Fish Yellow Boxfish Angelfish Patterns can be seen in fish patterns like spotted trunkfish, spotted puffer, blue spotted stingray, spotted moral eel, coral grouper, red lion fish, yellow boxfish, and angel fish. These animals and fish stripes and spots attest to mathematical regularities in biological growth and form. These evolutionary and functional arguments explain why these animals need their patterns, but it is not explained how the patterns are formed. MODULE 1: MATHEMATICS IN OUR WORLD 3  Tiger Zebra Leopard Cat Snake Hyena Giraffe Zebra, tigers, cats and snakes are covered in patterns of stripes; leopards, and hyenas are covered in patters of spots; and giraffes are covered in patterns of blotches. Natural patterns like the intricate waves across the oceans; sand dunes on deserts; formation of typhoon; water drop with ripple; and others. These serve as clues to the rules that govern the flow of water, sand, and air. One of the most strikingly mathematical landscapes on Earth is to be found in the great ergs, or sand oceans, of the Arabian and Sahara deserts. When wind blows steadily in a fixed direction, sand dunes form and the simplest pattern is the transverse dunes, which looks like ocean waves. If the sand is slightly moist, and there is a little vegetation to bind it together, then you may find parabolic dunes. Ocean Waves Dessert Dune Typhoon Water Ripple Other patterns in nature can also be seen in the ball of mackerel, the v-formation of geese in the sky, and the tornado formation of starlings. This prevalence of pattern in locomotion extends to the scuttling of insects, the flight of birds, the pulsations of jellyfish, and the wavelike movements of fish, worms, and snakes. Ball of Mackerel V-formation of Geese Tornado formation of Starling MODULE 1: MATHEMATICS IN OUR WORLD 4  Now that you have explored Mathematics in our World, in a ½ crosswise sheet of paper, write reflection 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 the world. c. Mathematics helps control nature and occurrences in the world for our own APPLICATION ends. 0 point the student is unable to elicit the ideas and concepts from the indicating that s/he has not read the prescribed reading or watched the video. 1 point the student is able to elicit the ideas and concepts from the readings but shows erroneous understanding of these. 2 points the student is able to elicit the ideas and concepts from the readings and shows correct understanding of these. 3 points the student not only elicits the correct ideas from the readings but also shows evidence of internalizing these. RUBRICS 4 points the student elicits the correct ideas from the readings, shows evidence of internalizing these and consistently contributes additional thoughts to the Core Idea. Well done! You have just finished this lesson. Keep working and enjoy! See you in our next lesson! MODULE 1: MATHEMATICS IN OUR WORLD 5  MODULE 1 LESSON 2 Fibonacci Sequence At the end of this lesson, you should be able to: a. discuss the concept of Fibonacci and its application; b. establish the relationship between the Fibonacci sequence with the golden ratio; c. investigate the relationship of the golden ratio and Fibonacci number in natural world; and d. determine the application of the Golden ratio in arts and architecture LEARNING OUTCOMES Mathematics is universal. Mathematical concepts and influences can be seen in both nature and human creations, such as art. Seeing patterns in nature has led many to believe that such patterns are evidence of divine creation. This lesson will show you the mathematics of art-related concepts like the Fibonacci sequence and the Golden Ratio. INTRODUCTION Get a ruler and measure the following by centimeters (cm): 1. Distance from the ground to your navel 2. Distance from your navel to the top of your head ACTIVITY 3. Distance from the ground to your knees “ 4. Length of your hand MEASURE 5. Distance from your wrist to your elbow ” IT! 6. Distances A, B, and C as indicated in the figure to the right Now, calculate the following ratios and write the results in the table below: 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑔𝑟𝑜𝑢𝑛𝑑 𝑡𝑜 𝑦𝑜𝑢𝑟 𝑛𝑎𝑣𝑒𝑙 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝐶 1. = 4. 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝐵 = 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑦𝑜𝑢𝑟 𝑛𝑎𝑣𝑒𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑡𝑜𝑝 𝑜𝑓 𝑦𝑜𝑢𝑟 ℎ𝑒𝑎𝑑 𝐷𝐼𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑔𝑟𝑜𝑢𝑛𝑑 𝑡𝑜 𝑦𝑜𝑢𝑟 𝑛𝑎𝑣𝑒𝑙 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝐵 2. = 5. 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝐴 = 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑔𝑟𝑜𝑢𝑛𝑑 𝑡𝑜 𝑦𝑜𝑢𝑟 𝑘𝑛𝑒𝑒𝑠 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑦𝑜𝑢𝑟 𝑤𝑟𝑖𝑠𝑡 𝑡𝑜 𝑦𝑜𝑢𝑟 𝑒𝑙𝑏𝑜𝑤 3. = 𝐿𝑒𝑛𝑔𝑡 𝑜𝑓 𝑦𝑜𝑢𝑟 ℎ𝑎𝑛𝑑 MODULE 1: MATHEMATICS IN OUR WORLD 6  Right now, let us try to wrap up your experiences from the previous activity. Let us try to answer the following questions: 1. What pre-requisite concept did you use in getting your answers? 2. What are your observations on the ratios? ANALYSIS FIBONACCI SEQUENCE The Fibonacci sequence is named after Leonardo of Pisa, also known as Fibonacci, who first observed the pattern while investigating how fast rabbits could breed under ideal circumstances. Fibonacci’s 1202 book Liber Abaci introduced the sequence to Western European mathematics, ABSTRACTION although it had been described earlier in Indian Mathematics. Leonardo of Pisa By definition, the first two numbers in the Fibonacci sequence are 1 and 1, and each subsequent number is the sum of the previous two. In mathematical terms, the sequence 𝐹𝑛 of Fibonacci numbers is defined by the recurrence relation 𝐹𝑛 = 𝐹𝑛 − 1 + 𝐹𝑛 + 2 , with seed values 𝐹1 = 1 and 𝐹2 = 1. Starting with 0 and 1, the succeeding terms in the sequence can be generated by adding the two numbers that came before the term: 0+1=1 0, 1, 1 1+1=2 0, 1, 1, 2 1+2=3 0, 1, 1, 2, 3 2+3=5 0, 1, 1, 2, 3, 5 3+5=8 0, 1, 1, 2, 3, 5, 8 5 + 8 = 13 0, 1, 1, 2, 3, 5, 8, 13, … To find the nth Fibonacci number without using the recursion formula, the following is evaluated using a calculator: 1 + √5 𝑛 1 − √5 ( ) − ( 2 )𝑛 𝐹𝑛 = 2 √5 This form is known as the Binet formula of the nth Fibonacci number. MODULE 1: MATHEMATICS IN OUR WORLD 7  Use Binet’s formula to determine the 25th and 30th Fibonacci numbers. Solution: a) 25th Fibonacci number b) 30th Fibonacci number 𝑛 𝑛 𝑛 𝑛 1+√5 1−√5 1+√5 1−√5 ( ) −( ) ( ) −( ) 2 2 𝐹𝑛 = 𝐹𝑛 = 2 2 √5 √5 EXAMPLES ( 1+√5 2 ) 25 1−√5 −( 2 ) 25 ( 1+√5 ) 30 −( 1−√5 ) 30 𝐹25 = 𝐹30 = 2 2 √5 √5 (1.61803)25 −(−0.61803)25 (1.61803)30 −(−0.61803)30 𝐹25 = 𝐹30 = √5 √5 167,761 1,860,498 𝐹25 = 𝐹30 = √5 √5 𝐹25 = 75, 025 𝐹30 = 832,040 The Fibonacci sequence occurs many times in nature. Take a look at sunflowers. In particular, pay attention to the arrangement of the seeds in its head. Do you notice that they form spirals? In certain species, there are 21 spirals in the clockwise direction and 34 spirals in the counterclockwise direction. Heart of the Sunflower You can observe the phenomenon in the number of petals in daisies, cauliflower florets, spirals in pine cones, and the bands winding around in pineapples. Core of a Daisy Blossom GOLDEN RATIO In mathematics and the arts, two quantities are in a golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. 𝑎 𝑎+𝑏 In symbols, a and b, where 𝑎 > 𝑏 > 𝑜, are in a golden ratio if =. The golden ratio is often symbolized by the 𝑏 𝑎 1+√5 Greek letter 𝜙. It is the number 𝝓=𝟏.𝟔𝟏𝟖𝟎𝟑…. and the irrational number. 2 MODULE 1: MATHEMATICS IN OUR WORLD 8  Golden ratio perhaps is the most important part of human beauty and aesthetics as well as a part of the remarkable proportions of growth patterns in living things such as plants and animals, Fibonacci number frequently appears in the numbers of petals in a flower and in the spirals of plants. Plants have distinct characteristics of Golden Ratio where they establish a Fibonacci sequence in the number of leaves. Even the eyes of a pineapple follow the golden ratio and golden spiral. The spiral happens naturally when each new cell is formed after a turn, as plants grow new cells in spirals format and this pattern is seen on the seed’s arrangement of the beautiful sunflower, rose petals, comfrey flowers, fern fiddleheads, flower buds, spiral aloe, and pine cones. Leaves, branches and petals grow in spiral form in order for the new leaves not to block the older leaves from the sun ray or the maximum amount of rain or dew gets directed down to the roots. If a plant has spirals, the rotation tends to be a fraction made with two successive Fibonacci numbers, for example, 1-2, 3-5 or even 5-8 also common that getting closer to the golden ratio. Rose Calla Lilies Comfrey Flowers Fern Fiddleheads The positions and proportions of the key dimensions of many animals are based on Phi or φ. Examples include the horn of ram, the wing dimensions and location of eye-like spots on moths, body sections of ants and other insects, body features of animals (e.g., tiger, fish, penguin, dolphin, etc.), and the spirals of sea shells. The growth pattern on branches of trees is Fibonacci. Even the human face contains spirals and the human DNA contains phi proportions. Spirals on Ram’s Horn Golden Ratio on Butterfly and Ants Body Golden Ratio on Tiger’s Face, Fish, Penguin and Dolphin The golden ratio shows up in art, architecture, music, and nature. For example, the ancient Greeks thought that rectangles whose sides form a golden ratio were pleasing to look at. Many artists and architects have set their works to approximate the golden ratio, also believing this proportion to be aesthetically pleasing. Monalisa Parthenon The Last Supper Vitruvian Man MODULE 1: MATHEMATICS IN OUR WORLD 9  Direction: In a short bond paper, find the nth Fibonacci number using the Binet formula. 1. n = 14 2. n = 22 3. n = 20 4. n = 25 5. n = 50 APPLICATION Well done! You have just finished this lesson. Keep working and enjoy! See you in our next lesson! MODULE 1: MATHEMATICS IN OUR WORLD 10  MODULE 1 Patterns and Regularities LESSON 3 in the World At the end of this lesson, you should be able to: a. identify patterns in nature and uses of mathematics in everyday life; and b. solve problems involving patterns and regularities in the world. LEARNING OUTCOMES Mathematics is all around us. As we discover more about our environment, we can mathematically describe nature. The beauty of a flower, the majestic tree, even the rock formation exhibits nature’s sense of symmetry. There are also examples of microscopic level of nature such as snowflakes. There are different types of patterns such as symmetry, fractals and spirals. INTRODUCTION Instruction: Determine the next figure in the pattern below. A. ACTIVITY “ WHAT COMES B. ” NEXT? A B Right now, let us try to wrap up your experiences from the previous activity. Let us try to answer the following questions: 1. What are your observations on the patterns? 2. How did you come up with the patterns? ANALYSIS MODULE 1: MATHEMATICS IN OUR WORLD 11  PATTERNS AND NUMBERS IN NATURE AND THE WORLD Patterns indicate a sense of structure and organization that it seems only humans are capable of producing these intricate, creative, and amazing formations. It is from this ABSTRACTION perspective that some people see an “intelligent design” in the way that nature forms. SYMMETRY Symmetry is a sense of harmonious and beautiful proportion of balance or an object is invariant to any of various transformations (reflection, rotation or scaling). There are two main types of symmetry, bilateral and radial. BILATERAL SYMMETRY It is a symmetry in which the left and right sides of the organism can be divided into approximately mirror image of each other along the midline. Symmetry exists in living things such as insects, animals, plants, flowers, and others. Animals mainly have bilateral or vertical symmetry, even leaves of plants and some flowers such as orchids. Butterfly Dragonfly Plant Leaves RADIAL SYMMETRY It is also known as rotational symmetry. It is a type of symmetry around a fixed point known as the center and it can be classified as either cyclic or dihedral. Plants often have radial or rotational symmetry, as to flowers and some groups of animals. A five-fold symmetry is found in the echinoderms, the group which includes starfish (dihedral-D5 symmetry), sea urchins, and sea lilies (dihedral-D5 symmetry). Radial symmetry suits organisms like sea anemones whose adults do not move and jellyfish (dihedral-D4 symmetry). Radial symmetry is also evident in different kinds of flowers. Starfish Sea Anemone Symmetries in Kiwi Flowers MODULE 1: MATHEMATICS IN OUR WORLD 12  FRACTALS Fractal is a curve or geometric figure, each part of which has the same statistical character as the whole. Fractal is one of the newest and most exciting branches of mathematics. It is a class of highly irregular shapes that are related to continents, coastlines, and snowflakes. It is useful in modeling structures 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. Fractal can be seen in some plants, trees, leaves, and others. SPIRALS A logarithmic spiral or growth spiral is self-similar spiral curve which often appears nature. It was first described by Rene Descartes and was later investigated by Jacob Bernoulli. Spirals are more evident in plants. We also see spirals in typhoon, whirlpool, galaxy, tail of chameleon, and shell among others. Spiral Worm Fern Spiral Spiral Aloe Galaxy Typhoon Spiky Shell Direction: In this activity, your task is to look for patterns in the environment showing symmetry, fractals and spirals. You have to take a photo of it and upload it to your Facebook account. Please write a short caption describing the pattern you have captured. Upon uploading, use the hashtag: #MATH111PatternsAndRegularities APPLICATION Good job! You have just finished this module. Keep working and enjoy! See you in our module assessment! MODULE 1: MATHEMATICS IN OUR WORLD 13  CLOSING ASSESSMENT, SUMMARY MODULE 1 AND REFERENCES Direction: In a short bond paper, answer the following questions. Some questions need complete solution. 1. What completes the following pattern? CSD, ETF, GUH, _______, KWL 2. What number should come next in this sequence? 22, 21, 25, 24, 28, 27, …. 3. What number comes next in 1, 8, 27, 64, 125, ________? 4. Starting with the first Fibonacci number 𝐹1 = 1 and the second Fibonacci number 𝐹2 = 1, what is the 15th Fibonacci number? ASSESSMENT 5. What is 𝐹20 ? 6. Given 𝐹30 = 832, 040 and 𝐹28 = 317, 811, what is 𝐹29 ? ESSAY: Answer in 100 words the following open-ended questions below. Write your answer on a short bond paper. For you to be guided, please refer to the rubric found on the next page. 1. How does mathematics help in solving societal problems? 2. How does mathematics help you in everyday living? MODULE 1: MATHEMATICS IN OUR WORLD 14  SCORING RUBRIC FOR ESSAY Category 5 4 3 2 There is one clear, well There is one clear, well focused focused topic. Main ideas There is one topic. topic. Main ideas are clear and are The topic and main Focus & Details are clear but are not well Main ideas are well supported by detailed and ideas are not clear. supported by detailed somewhat clear. accurate information. information. The introduction is inviting, states the The introduction states The introduction There is no clear main topic, and provides an overview the main topic and states the main introduction, Organization of the paper. Information is relevant provides an overview of topic. A conclusion structure, or and presented in a logical order. The the paper. A conclusion is is included. conclusion. conclusion is strong. included. The author’s purpose of writing is The author’s purpose of somewhat clear, The author’s purpose of writing is writing is somewhat clear, and there is very clear, and there is strong and there is some evidence of evidence of attention to audience. evidence of attention to The author’s purpose Voice attention to The author’s extensive knowledge audience. The author’s of writing is unclear. audience. The and/or experience with the topic knowledge and/or author’s knowledge is/are evident. experience with the topic and/or experience is/are evident. with the topic is/are limited. The author uses vivid The writer uses a The author uses The author uses vivid words and words and phrases. The limited vocabulary. words that phrases. The choice and placement choice and placement of Jargon or clichés Word Choice communicate of words seems accurate, natural, words is inaccurate at may be present and clearly, but the and not forced. times and/or seems detract from the writing lacks variety. overdone. meaning. Sentences sound Most sentences are awkward, are well constructed, Most sentences are well distractingly Sentence but they have a constructed and have repetitive, similar structure Structure, All sentences are well constructed varied structure and and/or length. The or are difficult to and have varied structure and length. The author makes author makes understand. The Grammar, length. The author makes no errors a few errors in grammar, several errors in author makes in grammar, mechanics, and/or mechanics, and/or Mechanics, & grammar, numerous errors in spelling. spelling, but they do not mechanics, and/or grammar, Spelling interfere with spelling that mechanics, understanding. interfere with understanding. and/or spelling that interfere with understanding. MODULE 1: MATHEMATICS IN OUR WORLD 15  The following summarizes essential concepts in this module: ▪ Patterns are regular, repeated, or recurring forms or designs, Patterns are commonly observed in natural objects, such as the six-fold symmetry of snowflakes, the hexagonal structure and formation of honeycombs, the tiger’s stripes and hyena’s spots, the number of seeds in a sunflower, the spiral of snail’s shell, and the number of petals of flowers. ▪ The Fibonacci sequence is formed by adding the preceding two numbers, beginning SUMMARY with 0 and 1. Ratios of two Fibonacci numbers approximate the Golden Ratio, which is considered as the most aesthetically pleasing proportion. ▪ Mathematics helps organize patterns and regularities in the world. Mathematics help predict the behavior of nature and phenomena in the world, as well as helps human exert control over occurrences in the world for the advancement of our civilization. REFERENCES Aufmann, R. N., Lockwood, J. S., Nation, R. D. & Clegg, D. K. (2013). Mathematical Excursions. 3rd Ed. CA: Brooks/Cole, Cengage Learning. Mathematics in the Modern World. (2018). Rex Book Store, Inc. Smith, Karl J. (2010) Nature of Mathematics. Tenth Edition. CA: Brooks/Cole, Cengage Learning. MODULE 1: MATHEMATICS IN OUR WORLD 16