Math in the Modern World (Sequences)

1.1 Mathematics in our World (A Study of Patterns) Overview Welcome to the first module of GE 1 (Mathematics in the Modern World)! This course begins with an introduction to the nature of mathematics as an exploration of unseen patterns in nature and environment, a rich language in itself governed by logic and reasoning, and an application of inductive and deductive reasoning. Section 1 is composed of the following: 1.1 Mathematics in our World; 1.2 Mathematics Language and Symbols; and 1.3 Problem Solving and Reasoning. These topics will allow students to go beyond the typical understanding of mathematics as purely a bunch of memorized formulas and duplicated mathematical computations, but as a powerful tool used to understand better the world around us. Moreover, we will discuss and argue about the nature of mathematics, what it is, and how it is expressed, represented, and used. We will study mathematics as a language in order to read and write mathematical texts and communicate ideas with precision and conciseness. We will also justify statements and arguments made about mathematics and mathematical concepts using different methods of reasoning. Mathematics has always been perceived as a study of numbers, symbols, and rules. It is an art of geometric shapes and patterns, a tool in decision-making and problem solving. It has a language that differs from the ordinary speech. It is done with curiosity, with a penchant for seeking patterns and generalities, with the desire to know the truth, with trial and error, and without the fear of facing more questions and problems to solve. The following diagram shows the very nature of mathematics. study of patterns art language Mathematics is a/an … set of problemsolving tools process of thinking Nocon, R. & Nocon, E. In this module, we will focus on Lesson 1.1 - Mathematics in our World (A Study of Patterns). The lesson is anchored by the following core idea: Mathematics is a useful way to think about nature and the world. Our intention is to observe things, in both in nature and the world, through pattern-seeking, understand the substantial interconnection and relationship of the mathematics and the world, and appreciate mathematics as a discipline full of essence and beauty Learning Outcomes After working on this module, you will be able to: 1. identify patterns in nature and irregularities; 2. articulate the importance in mathematics in one’s life; 3. argue about mathematics, what it is, how it is expressed, represented, and used; and 4. express appreciation for mathematics as a human endeavor. Activities To Do 1.) Watch the video “Nature by Numbers” by Cristóbal Vila (link: https://vimeo.com/9953368) and write one (1) sentence that describes your impression after watching the video. 2.) Identify pattern/s observed in the pictures. (a) (b) (d) (e) (c) Sources: (a) https://www.library.illinois.edu/mtx/2018/10/09/mathematics-in-nature/; (b); https://www.weareteachers.com/teacherdresses-ms-frizzle/ (c) https://www.smithsonianmag.com/science-nature/science-behind-natures-patterns-180959033/;(d) and (e) http://mustafacil-online.blogspot.com/2015/08/manmade-patterns.html Questions To Pon Ponder der The video and the pictures leave us questions to think about. • What are the different kinds and forms of patterns you have seen in the video and/or pictures? • How does these patterns help us understand the connection between our world and mathematics? Patterns and Numbers in Nature and the World When we buy clothes, accessories, furniture, house decorations, and other similar objects, we tend to look for beautiful geometric designs or patterns. We appreciate the patterns seen in the colorful wings of butterflies, the arrangement of flowers and leaves, the reflection of the mountain tops to the clear waters of lakes, the different shapes of clouds in the skies, and other patterns seen in the nature. In the busy streets of the cities, we are impressed by the intricate but well-designed modern homes and high-rise buildings. We are wowed by nature and man-made creations because of these repeated designs of geometric visuals. Repeated ways or occurrences that happens or was done are also considered as patterns. For example, the cycle of the moon, the changing seasons, and even the transmission pattern of the COVID 19 pandemic. Patterns surround us. It is everywhere and are in every people’s task or activity. Mathematics, developed by human mind and culture, is a formal system of thought for recognizing, classifying, and exploiting patterns. (Stewart, I.). Mathematics is indeed a study of patterns. Results in mathematics are brought by the generalizations of patterns. The study of patterns allows us to observe and identify relationships, discover logical connections, and make generalizations. Moreover, the use and study of patterns allows us to be logical thinkers and better problem solvers. Now, let us take a look of some of these patterns. Example 1 Logic Pattern Choose the figure that completes the pattern. 1. ? _________ A B C D A B C D 2. Sources: (1) and (2) hhtp://www.jobtestprep.co.uk; Solution 1. D The sketch is being built stage by stage. A new line is added in each stage and it never touches the last line added in the previous stage. 2. B Each figure consists of 3 shapes; namely: external shape, middle shape, and inner shape. Notice that the external shape appears to be the middle shape of the next figure. The middle shape disappears in the next figure. While the inner shape appears to be as the external shape of the figure two steps forward. For example: The external shape of the 4th figure is a circle, The middle shape is a pentagon, and the inner shape is a hexagon. The circle becomes the 4th figure middle shape of the next figure, the pentagon disappears in the next figure, and the hexagon becomes the external shape of the figure two steps forward from the 4th figure. 2 Number Pattern Find the next number in the sequence. Example 1. 3, 8, 13, 18, 23, ? 2. 1, 4, 9, 16, 25, 36, ? Solution 1. For this sequence, the difference between each term is 5. 3, 8, 13, 18, 23, __ ⋁ ⋁ ⋁ ⋁ ⋁ 5 5 5 5 5 Thus, the next number is 28 (23 + 5). 2. The next number in the sequence can be determined in two ways: (1) The sequence 1, 4, 9, 16, 25, 36 can be written as 12, 22, 32, 42, 52, 62. Thus, the next number is 72, that is, 49 49. (2) For this sequence, the difference 1, 4, 9, 16, 25, 36, __ ⋁ ⋁ ⋁ ⋁ ⋁ ⋁ 3 5 7 9 11 13 3 + 2 = 5; 5 + 2 = 7; 7 + 2 = 9; 9 + 2 = 11; 11 + 2 = 13 Thus, the next number is 49 . Examples 1 and 2 are usually seen on aptitude tests. Before we determine the next shape or number, we have to observe the objects, look into their properties, and their relationship on other objects. In such a way, we are allowed to hypothesize, predict, and construct generalizations based on the observed patterns. Patterns, such as geometric and word patterns, are also very common to us. Word patterns focused on the morphological rules in pluralizing nouns, conjugating verbs for tense, and metrical rules of poetry. Examples: baby: babies buy: bought trolley: trollies bring: brought ally: ? catch: ? answer: allies answer: caught https://newsinfo.inquirer.net/941295/bat ok-tattooing-tattooing-mambabatok https://www.our7107islands.com/basey-samarthe-new-banig-capital-of-the-philippines/ http://alvicsbatik.weebly.com/ mindanao-accessories--page2.html While geometric patterns are designs that depict geometric shapes like lines, circles, and polygons. Geometric patterns are observed in nature. These patterns are also associated to the identification of a particular country and culture. Below are samples of geometric patterns that are associated to Philippine ethnic groups and local regions. (1) (3) (1) Tattoos in the Cordillera (2) Woven mat “banig” in Basey, Samar (3) T’boli belt made of beads (2) Self-Assessment Activity 1 I. For each set of figures, what comes next? 1. A B C D 2. A B C D 3. A B C D II. What is the next number in the series? 4. 3, 6, 12, 24, 48, ? 5. 1, 4, 10, 22, 46, ? 6. 4, – 1, – 11, – 26, – 46, ? Sources: (1) and (2) hhtp://www.jobtestprep.co.uk; (3) www.psychometric-success.com Answers to SAA 1: 1. C, 2. B, 3. C, 4. 96, 5. 94, 6. –71 The world and the universe are full of beautiful patterns and designs that are mathematical in nature. Let us take a closer look on some of these patterns in nature and the world. Symmetry and Ang Angle le of Rotation Consider the figure below. https://sotafoundations9.wordpress.com/2018 /03/08/transferring-your-design-using-radialsymmetry/#jp-carousel-89 http://www.bio.miami.edu/dana/226/226F 09_22.html https://leonardodavinci.stanford.edu/submi ssions/clabaugh/today/health.html When you draw an imaginary line and the resulting parts are mirror images of each other, we have shown a symm symmetry etry etry. The A figure above is symmetric about the axis indicated by the broken line. This is called as line or bilateral symmetry and is common to animals and humans. Here are other images showing symmetry. Leonardo da Vinci’s Vitruvian Man is an image of proportion and symmetry of the human body. This flower has a three-fold symmetry. This starfish has a five-fold symmetry. Images Source: https://biologydictionary.net/radialsymmetry/ Observe that if we rotate the flower and the starfish by several degrees, we can still have the same appearance as the original position. This is called the rotation otational al symmetry symmetry. The smallest angle an object can be rotated while it is preserving its original formation is called the angle of rotation rotation. 1 A figure has a rotational symmetry of order 𝑛 (𝑛-fold rotational symmetry) if of a complete 𝑛 turn leaves the figure unchanged. To compute for the angle of rotation, we use 360° 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 = 𝑛 For example, This three-leaf clover has a 3-fold 120o symmetry. The angle of rotation is 120O. 120o 90o 120o This four-leaf clover has a 4-fold symmetry. The angle of rotation is 90O. https://listverse.com/2013/04/21/10-beautifulexamples-of-symmetry-in-nature/ Snow Snowflakes flakes and Honeyc Honeycombs ombs Look into a microphotographed snowflake below. https://listverse.com/2013/04/21/10-beautifulexamples-of-symmetry-in-nature/ https://www.benefits-of-honey.com/honeycombpattern.html#:~:text=Studies%20on%20the%20geomet ry%20of,and%20square%20makes%20smaller%20area. Notice that it exhibits a pattern on each arm that repeats six times. This snowflake indicates a six-fold symmetry. However, many snowflakes are not perfectly symmetric due to the effects of the different atmospheric conditions such as temperature and humidity on the ice crystals as it forms when they descend from the skies. The angle of rotation for the snowflake with a 6-fold symmetry is 60O. Humans are also marveled with the almost perfect hexagonal shape arrangements in honeycombs. Peacock’s Tail The patterns exhibited in animal’s external appearance has to do with their growth; their survival; and even with their chances to attract their mates. Symmetric and repeated patterns, enhanced with bright, beautiful colors, on the feathers of a peacock’s tail are used to attract their mates. https://www.benefits-of-honey.com/honeycombpattern.html#:~:text=Studies%20on%20the%20geomet ry%20of,and%20square%20makes%20smaller%20area. The image on the right explains why mathematicians believed that hexagon is the most effective way of storing honey. The hexagonal formation allows bees to store the largest possible amount of honey with the use of the least amount of wax. https://www.benefits-of-honey.com/honeycombpattern.html#:~:text=Studies%20on%20the%20geomet ry%20of,and%20square%20makes%20smaller%20area. Sunflower Nature has gifted us with beautiful flowers. The brilliant colors, fragrant odors, petal arrangements, and different sizes and number of petals make flowers very appealing. If we closely observe these flowers, we can find interesting patterns. For example, let us take a closer look on the orderly arrangement of sunflower seeds. We can see clockwise and counterclockwise spirals extending outward from the center of the flower. Moreover, the sunflower seed arrangement displays a very interesting numerical sequence called the Fibonacci sequ sequence ence ence. The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, … and so on. Each number on the sequence is determined by adding the two preceding numbers. The number of seeds spirals in a sunflower adds up to a Fibonacci number. Spirals of many plants such as pineapple and pinecones also add up a Fibonacci number. https://www.bigwalls.net/climb/camf/index.html https://www.benefits-of-honey.com/honeycombpattern.html#:~:text=Studies%20on%20the%20geomet ry%20of,and%20square%20makes%20smaller%20area. Nautilus Shell Another example that shows how nature seems to follow a certain set of rules governed by mathematics is spiral patterns seen in a shell of a nautilus. As the mollusk grows inside the shell, the shell also expands and attempts to maintain the same proportional shape as it grows outward. This growth pattern results to refined spirals on the shell which is very evident when it is sliced. The image on the right is called the logarithmic spiral, also known as equiangular spirals . The image shows a mathematical curve which has the property of maintaining a constant angle between the radius and the tangent to the curve at any point on the curve. Equivalently, the property states that as the distance from the spiral center increases (radius), the amplitudes of the angles formed by the radii to the point and the tangent to the point remain constant. World Population According to the estimates of Worldometers (https://www.worldometers.info), the Philippines will be rank 13 on the Most Populous Countries in year 2050, with population of 144, 488, 158. Questions like “How is this estimate being computed?”; “What are the factors used in the computation of the estimate?” may arise. These questions can be answered by the mathematical model of population growth. The formula for exponential growth is, 𝐴 = 𝑃𝑒 𝑟𝑡 where: 𝐴 = size of the population after it grows; 𝑃 = initial number of people; 𝑟 = rate of growth; 𝑡 = time; 𝑒 ≈ 2.718 (This is the Euler’s constant with an approximate value of 2.718) Example 2 Determine what is being asked in each problem. 1. Substitute the given values in the formula 𝐴 = 𝑃𝑒 𝑟𝑡 to find the missing quantity. a. 𝑃 = 680,000; 𝑟 = 12% per year; 𝑡 = 8 years b. 𝐴 = 731,093; 𝑃 = 525,600; 𝑟 = 3% per year 2. In the midyear of 2020, a country’s population is 109,581,078 with a growth rate of approximately 1.35% per year. What will be the country’s population in 2050? 3. The exponential growth model 𝐴 = 25𝑒 0.02𝑡 describes the population of a town in Northern Samar in thousands, 𝑡 years after 1998. What was the population of the town in 1998? Solution 1. a. To find the missing quantity, 𝐴, we substitute the given values to the formula: we let 𝑟 = 12% = 0.12 𝐴 = 𝑃𝑒 𝑟𝑡 = (680,000) 𝑒 (0.12)(8) 𝐴 = (680,000) 𝑒 0.96 𝐴 = (680,000) (2.611696) 𝑒 0.96 ≈ 2.611696 𝑨 = 1,775,953 b. To find the missing quantity, 𝑡, we use the formula of 𝐴 to derive a formula for 𝑡. 𝐴 Dividing both of the formula by P to isolate 𝑒 𝑟𝑡 . 𝐴 = 𝑃𝑒 𝑟𝑡 ⇔ 𝑃 = 𝑒 𝑟𝑡 731,093 525,600 = 𝑒(0.03)𝑡 1.390968 = 𝑒 (0.03)𝑡 𝑙𝑛 1.390968 = 0.03𝑡 0.330000= 0.03𝑡 11 = 𝒕 Substitute the given values and let 𝑟 = 3% = 0.03 731,093 ≈ 1.390968 Definition of logarithm 𝑙𝑛 1.390968 ≈ 0.330000 Divide both sides by 0.03 525,600 Checking of the solution by substitution of the given values and the obtained answer will serve as your exercise. 2. Given are the following quantities: 𝑃 = 109,581,078; 𝑟 = 1.35% = 0.0135; 𝑡 = 30 years (Subtract: 2050 – 2020) 𝐴 = 𝑃𝑒 𝑟𝑡 = (109,581,078)𝑒 (0.0135)(30) = 164,295,239. Thus, the population of the country in year 2050 is estimated to be 164, 64,295,239 295,239 . 3. Since the exponential growth model describes the population 𝑡 years after 1998, we consider 1998 as 𝑡 = 0 year and solve for the population size. 𝐴. 𝐴 = 25𝑒 0.02𝑡 = 25𝑒 0.02(0) = 25𝑒 0 = 25(1) = 25 Therefore, the population of the town in 1998 is 25,000. Self-Assessment Activity 2 Answer completely. (Use 6 significant digits in the approximated values) 1. Find the missing quantity in the formula 𝐴 = 𝑃𝑒 𝑟𝑡 by substitution of the given values: a. 𝑃 = 505,050; 𝑟 = 5% per year; 𝑡 = 1 year b. 𝑃 = 240,100; = 11% per year; 𝑡 = 10 years c. Find 𝑟 correct to 4 significant digits. 𝐴 = 786,000; 𝑃 = 247,000; 𝑡 = 17 years 2. The exponential growth model 𝐴 = 45𝑒 0.19𝑡 describes the population of a city in the Philippines in thousands, t years after 1995. a. What is the population of the city in 1995? b. What is the population after 25 years? c. What is the population in 2045?  STOP  Break Time (10 – 15 minutes) Look around you. Try to observe for patterns in your bedroom, house, or backyard. Do they have geometric patterns or numbers patterns? What makes these patterns appealing? The Fibonacci Sequence To start the new discussion, try to answer the puzzle below: Form a rectangle using all of the squares. The measures of the sides are indicated in each square. Answers to SAA 2: 1. (a) 530,945, (b). 721,300, (c). 6.809%, 2. (a) 45,000, (b). 5,201,293, (c). 601,187,707 One of the answers to the puzzle is shown below: Notice that the number pattern formed when the squares are placed side by side. The sequence gives as 1, 1, 2, 3, 5, 8. Does this sequence looks familiar? In the previous discussion, it is said that sunflower seed arrangement displays a very interesting numerical sequences called the Fibonacci sequence . The sequence is 1, 1, 2, 3, 5, 8, 13, 21, … was discovered by an Italian named Leonard Pisano Bigollo who is known in mathematical history by several names: Leonardo of Pisa and Fibonacci. Fibonacci’s 1202 book “Liber Abaci” introduced the sequence to Western European mathematics, although there are some claims that the sequence has been discovered earlier in Indian mathematics. It is said that Fibonacci discovered the number sequence through a practical problem involving the growth of a hypothetical population of rabbits based on idealized assumptions. This problem has an assumption that a pair of rabbits will be born each month and will reproduce a new pair of rabbits two months after they were born. The assumption is illustrated below: https://www.storyofmathematics.com/medieval_fibonacci.html Clearly, each number on the sequence is determined by adding the two preceding numbers. Notice that if we divide two consecutive terms in the Fibonacci sequence, the quotient approaches a particular number. It is the number phi, 𝜙 = 1.6180339887… or the 1+ √5 irrational number . This number is called the golden 2 ratio ratio. In mathematics and in arts, two quantities are in golden ratio if their ratio is the same as their sum to the larger of the two quantities. In symbols, 𝑎 and 𝑏, where 𝑎 > 𝑏 > 0, are in a golden ration 𝑎 𝑎+𝑏 if 𝑏 = . 𝑎 𝑛 𝐹𝑖𝑏(𝑛 + 1) 𝐹𝑖𝑏(𝑛) Fibonacci Number 𝐹𝑖𝑏(𝑛) 1 1 2 1 3 2 4 3 5 5 6 8 7 13 8 21 9 34 10 55 11 89 1 1 2 1 3 2 5 3 8 = 1.00000 =2.00000 =1.50000 ≈1.66667 =1.60000 5 13 8 21 13 34 21 55 =1.62500 ≈1.61538 ≈1.61905 ≈1.61765 34 89 ≈1.61818 55 144 ≈1.61798 89 Here are some of man’s greatest works that would reminds us of the Fibonacci sequence and the golden ratio. Example 3 Answer completely. 1. Find the next three terms of the sequence: 1, 1, 2, 3, 5, 8, … 2. Let 𝐹𝑖𝑏(𝑛) be the nth term of the Fibonacci sequence, with 𝐹𝑖𝑏(1) = 1, 𝐹𝑖𝑏(2) = 1, 𝐹𝑖𝑏(3) = 2, and so on. a. Find 𝐹𝑖𝑏(10) b. Find 𝐹𝑖𝑏(17) 3. If 𝐹𝑖𝑏(22) = 17,711 and 𝐹𝑖𝑏(24) = 46,368, what is 𝐹𝑖𝑏(23) Solution 1. To find the next three terms, we add the two preceding numbers: 5 + 8 = 13; 8 + 13 = 21; 13 + 21 = 34 Thus, the next three terms of the sequence are 13, 21, 34. Completing the sequence, we have1, 1, 2, 3, 5, 8, 13, 2 21, 1, 34. 2. a. 𝐹𝑖𝑏(1) = 1, 𝐹𝑖𝑏(2) = 1, 𝐹𝑖𝑏(3) = 2, 𝐹𝑖𝑏(4) = 3, 𝐹𝑖𝑏(5) = 5, 𝐹𝑖𝑏(6) = 8, 𝐹𝑖𝑏(7) = 13 , 𝐹𝑖𝑏(8) = 21, 𝐹𝑖𝑏(9) = 34 𝑭𝒊𝒃(𝟏𝟎) = 𝑭𝒊𝒃(𝟖) + 𝑭𝒊𝒃(𝟗) = 𝟐𝟏 + 𝟑𝟒 = 𝟓𝟓 𝑏. 𝐹𝑖𝑏(10) = 55, 𝐹𝑖𝑏(11) = 89, 𝐹𝑖𝑏(12) = 144, 𝐹𝑖𝑏 (13) = 233, 𝐹𝑖𝑏(14) = 377, 𝐹𝑖𝑏(15) = 610, 𝐹𝑖𝑏(16) = 987 𝑭𝒊𝒃(𝟏𝟕) = 𝑭𝒊𝒃(𝟏𝟓) + 𝑭𝒊𝒃(𝟏𝟔) = 𝟔𝟏𝟎 + 𝟗𝟖𝟕 = 𝟏𝟓𝟗𝟕 3. 𝐹𝑖𝑏(23) = 𝐹𝑖𝑏(24) − 𝐹𝑖𝑏(22) = 46,368 – 17,711 = 28,657 Self-Assessment Activity 3 Answer completely. 1. Let 𝐹𝑖𝑏(𝑛) be the nth term of the Fibonacci sequence, with 𝐹𝑖𝑏(1) = 1, 𝐹𝑖𝑏(2) = 1, 𝐹𝑖𝑏(3) = 2, and so on. a. Find 𝐹𝑖𝑏(20) b. Find 𝐹𝑖𝑏(25) 2. Evaluate the following sum. a. 𝐹𝑖𝑏(1) + 𝐹𝑖𝑏(2) b. 𝐹𝑖𝑏(1) + 𝐹𝑖𝑏(2) + 𝐹𝑖𝑏(3) c. 𝐹𝑖𝑏(1) + 𝐹𝑖𝑏(2) + 𝐹𝑖𝑏(3) + 𝐹𝑖𝑏(4) 3. What will be the sum of 𝐹𝑖𝑏(1) + 𝐹𝑖𝑏(2) + ⋯ + 𝐹𝑖𝑏(10)? 4. If we construct a number sequence using the following: 𝐹𝑖𝑏(1) + 𝐹𝑖𝑏(2), 𝐹𝑖𝑏(1) + 𝐹𝑖𝑏 (2) + 𝐹𝑖𝑏(3) +….+ 𝐹𝑖𝑏(1) + 𝐹𝑖𝑏(2) + ⋯ + 𝐹𝑖𝑏(10), what pattern can be observed? Math for our World The world and the universe are full of complexities and uncertainties. These are very evident in our day-to-day living. From the simplest household chores to challenging and laborious tasks, from observation and prediction of weather conditions and natural phenomena to its survival from the aftermath, from legislation to implementation of governing laws, from theater shows to high definition videos and films; from ancient navigational methods to global national satellite systems, these and the like are some of the intricacies of the universe. With this, people of the ancient times and the modern world have learned to live and to cope. 4. The differences of two preceding numbers form a Fibonacci sequence. Answers to SAA 3: 1. (a) 6,765, (b) 75,025, 2. (a) 2, (b) 4, (c) 7, 3. 143, Technologies were invented and areas of knowledge were cultivated for better understanding of the underlying aspects that would lead to the world’s development. Furthermore, contexts of the real world and the universe were revealed for varied reasons through science and mathematics. Man’s exploration of the world began with its curiosity of the objects within and outside Earth’s surface. The Babylon were successful in predicting eclipses. Aristarchus of Samos, determined that the sun is the center of the universe through mathematical computation. Hipparchus calculated the size of the moon and its distance from Earth, and determined specific locations of geographical points through the use of longitude and latitude measurements. The earliest civilizations of Babylon, China, and Egypt devised calendars and completed public works, such as irrigational canals for agricultural purposes. Tax collections were made possibly easy when Blaise Pascal invented the world’s first digital calculator when the first mechanical computer was invented by Charles Babbage. (Todd, 2003) Computations done by scientists and mathematicians paved the way to greater discoveries and creations of expedient methods, devices, and tools. Galileo Galilei who invented the first thermometer was able to measure temperatures of water and air. Similarly, Gabriel Daniel Fahrenheit introduced the boiling and freezing points of liquid when he invented the first mercury thermometer. James Gregory’s invention of reflecting telescope, and Ole Roemer’s calculations of measuring the speed of light made great contributions to physics and other related fields. (Todd, 2003) Isaac Newton’s explanation of the Universal Law of Gravitation and Laws of Motion, and Albert’s Einstein Theory of Relativity, the motion particles when suspended within a liquid, and the mathematical formula e = mc2 are some of the notable contributions in the field of science and mathematics. In its purest sense, mathematics is an abstraction. It is extremely useful in describing and predicting events of the world. It has the ability to model effectively numerous aspects of the world by creating abstract structures that have properties or attributes to its real-world counterparts. Models, if they behave in a manner that truly parallel with originals, are used to make conclusions and/or predictions about the real world. (Post, 1981) Mathematics finds wide applications in arts, nature, music, medicine, chemistry, biology, astronomy, and in other disciplines. The artist of antiquity and of the modern times described their works using the Golden Ratio and Fibonacci Numbers. They are very evident in paintings, architecture, sculpture, dance, and even in music. Throughout the ages, music and mathematics go hand-in-hand when Pythagoras was able to establish a fundamental relationship between vibrating strings and harmony. Today, music is stored and played digitally. The sound production, recording, and engineering use advanced technology equipped with mathematics and science. Nicolas Copernicus, by using uniform circular motion, modelled that the planets revolved around the sun. The Heliocentrism paved the way to greater discoveries and explorations of the solar system. Mathematics calculations are used to launch satellites, rockets and other space robes, and used to describe the natural order and occurrences of the universe. In medicine, mathematics is use to: 1) predict complex medical situations; 2) model biological processes that underlie a disease, and 3) develop formulas from chemistry and physics, and medical technologies. (Lerner and Lerner, 2006). In this section, we have just seen how mathematics help our lives better and make the world a better place to live in. As we learn mathematical concepts, we also apply them in our day-to-day interactions and in solving societal problems. Moreover, mathematics is useful in making conclusions and/or predictions of the events of the world. It is used to organize patterns and regularities as well as irregularities. It is, therefore, mathematics is considered an integral part of our world. Summary The lessons in Section 1 (1.1 Mathematics in our World) allow us to “get -to-know” mathematics. Far from the idea that it is full of difficulties and complexities, mathematics is a study of patterns, an art, a language, a process of thinking, and a set of problem-solving tools. We were able to see through the beauty of the world through observed patterns that are mathematical in nature. Patterns that are in the natural objects and man-made creations. The ratios of the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, …) converge to the Golden Ratio that made these creations more aesthetically pleasing. With a careful understanding, we were able to see the beauty and the significance of mathematics in our day-to-day living. We are reminded how evident mathematics in the world. The module has bought us to realization the usefulness of mathematics that makes our lives better and our world a better place to live. Responses To Consider Congratulations! You have finished Module 1. After working with this module, what are the new ideas about mathematics did you learn? What is it about mathematics that might have changed your thoughts about it? What is most useful about mathematics for humankind? You are encouraged to provide a lecture notebook where you can write all your responses and solutions to the activities and SAAs. Answers to SAAs are provided at the bottom part of the page. If you have difficulty in obtaining the correct answer, you can go over again with the examples. To be successful in mathematics, you have to do mathematics. Do it without the fear of facing more problems and questions to solve. For further understanding, use the references, suggested readings, and other materials indicated in the module.