Module 1: Patterns and Numbers in Nature - MMW 101 PDF

BULACAN STATE UNIVERSITY COLLEGE OF SCIENCE MMW 101 MATHEMATICS IN THE MODERN WORLD Module 1 Patterns and Numbers in Nature and the World “Seeing a New Yet the Same World Through Mathematics” Overview Have you ever tried...

BULACAN STATE UNIVERSITY COLLEGE OF SCIENCE MMW 101 MATHEMATICS IN THE MODERN WORLD Module 1 Patterns and Numbers in Nature and the World “Seeing a New Yet the Same World Through Mathematics” Overview Have you ever tried counting the petals of flower blossoms around your community? How you ever wondered why bees made honeycombs in such shape and structure? Do you admire the architectural design, the design motifs, or the textiles' intricate patterns? Was there ever a time when you are amazed and expressed appreciation for the beauty of things around you, especially in nature? If not, you are welcome to take a closer look at your surroundings. Discover the patterns, relationships, and connections explored and studied by mathematicians to show the roles that mathematics plays in human beings' lives and undertakings. Let us create a new look and understanding of the world by going through the following modules: 1. Patterns and Numbers in Nature and the World 2. The Fibonacci Sequence 3. The Golden Ratio 4. The Indispensability of Mathematics. Objectives: At the end of modules 1 to 4, you are expected to: 1. identify patterns in nature and regularities in the world, 2. explain the nature of mathematics: what it is, how it is expressed, represented and used, 3. articulate the importance of mathematics in one's life, and 4. express appreciation for mathematics as a human endeavor.1 1 "Mathematics in the Modern World - CHED." https://ched.gov.ph/wp-content/uploads/2017/10/KWF- Mathematics-in-the-Modern-World.pdf. Accessed 12 Sep. 2020. 1  Patterns and Numbers in Nature and the World Objectives of the Module At the end of the module, you should be able to: 1. describe the types of natural patterns, 2. distinguish the other types of patterns from another, 3. create an artwork or design using patterns, and 4. analyze logic and number patterns. To most of us, the common concept of mathematics is that it is all about numbers and their operations. The usual reaction is that mathematics is a difficult subject, disliked, and even hated. However, that concept is just one aspect of mathematics because it also pertains to a study of patterns, a language, a set of problem-solving tools, As we journey in this module, be aware and a process of thinking, and open your mind to the roles of mathematics an art. in your life. 1. Patterns in Nature We see a great diversity of living things all around us, from the microscopic to the gigantic, from the simple to the complex, from bright colors to dull ones. Do you agree that the most intriguing things we see in nature are patterns? But before answering that question, let me give you a hint: Anything that REPEATS with recurring characteristics or a SERIES of a regular or consistent arrangement according to a specific rule or SEQUENCE is considered a PATTERN. Can you now identify patterns in nature? Here are some more tips to help you. Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modeled mathematically. 2 2"Patterns in nature - Wikipedia." https://en.wikipedia.org/wiki/Patterns_in_nature. Accessed 12 Sep. 2020. 2 Types of Natural Patterns 1. Symmetry. There is symmetry if an imaginary line is drawn across an object, the resulting parts are mirrors of each other, like the following figures. 2. Spiral. It is a curved pattern that focuses on a center point and a series of circular shapes that revolve around it.3 This is common in plants and some animals. 3. Meander is a series of regular sinuous curves, bends, loops, turns, or windings in the channel of a river, stream, or other watercourses. It is produced by a stream or river swinging from side to side as it flows across its floodplain or shifts its channel within a valley.4 4. Cracks are linear openings that form in materials to relieve stress. The pattern of cracks indicates whether the material is elastic or not. 5 5. Stripes is a line or band that differs in color or tone from an adjacent area. This may be seen in various living things, especially animals. Examples: Look at some of the typical examples of natural patterns in the following: Symmetry Amazon Lily Pad Butterfly Dragon Fly https://www.reddit.com/r/oddlysatisfying/comment https://www.pinterest.ph/pin/454582774 https://i.pinimg.com/736x/75/98/f0/7598f s/a34icr/bottom_of_an_amazon_lily_pad/ 62777265/ 050a5efe4442259c6f53f1a3932--teal- blue-bokeh.jpg 3 "Math Patterns in Nature | The Franklin Institute." https://www.fi.edu/math-patterns-nature. Accessed 12 Sep. 2020. 4 "Meander - Wikipedia." https://en.wikipedia.org/wiki/Meander. Accessed 10 Aug. 2020. 5 "Crack Propagation - an overview | ScienceDirect Topics." https://www.sciencedirect.com/topics/earth-and- planetary-sciences/crack-propagation. Accessed 12 Sep. 2020. 3 Spirals Aloe polyphylla Tendrils Navy red flower https://bozannical.com/2011/09/20/fibonacci- https://www.pinterest.ph/joannehunt22/te https://tse3.mm.bing.net/th?id=OIP.za_hi3OxvVUc fascination/fiboromanesque/ ndrils-and-ferns/ FltuRr4qRQAAAA&pid=Api&P=0&w=198&h=159 Millipede Chameleon’s tail Ram’s horns https://fineartamerica.com/featured/curled- https://www.wikiwand.com/en/Patterns_in_nature https://www.pinterest.ph/pin/2111131045242170/ millipede-james-l-davidson.html Meander https://geography-revision.co.uk/a-level/physical/meander/ https://www.gapyear.com/articles/travel-ideas/the-22-most-deadly- highways-in-the-world Cracks https://stockfresh.com/image/5456010/the-cracks-texture https://www.squaretrade.com/en-gb/node/663 Stripes emperor angelfish zebra tiger heliconius charithonia https://www.fishkeepingworld.com https://www.dailymail.co.uk/sciencetech/a http://www.catherinejenkins.com/cre https://www.pikist.com/free-photo- /wp- rticle-2908552/So-S-zebra-got-stripes- ative-academic-writing-animals- xmvug content/uploads/2018/04/Emperor Alternating-pattern-absorbs-reflects-heat- different-stripes/ -Angelfish-Three.png create-air-conditioning.html 4 Notes: 1. Some patterns cannot be seen because they are already parts of human experience like, for example, the water cycle of evaporation, condensation, and precipitation. With this knowledge, people were able to understand the world better and make intuitive decisions to improve the ability to forecast weather, climate, water resources, and the ecosystem's health. 2. Regardless of the purpose by which natural patterns served, such as for camouflage, for adaptation to the environment, or they are caused by natural phenomena, all of them are considered to be closely related to mathematics. 2. Other Types of Patterns Aside from the natural patterns, you can also come across four (4) main types of patterns in which you are also familiar with or if not, now is the time to know them. 2.1 Logical Patterns Logic reasoning and pattern observing are the first two math standards, which are the most important measurement of IQ and the core component of many careers.6 Logical patterns are usually the first to be observed since making categories or classification comes before numeration. For children, logical patterns include studying shapes and colors. For older ones, logic tests can be seen on aptitude tests wherein takers are shown a sequence of pictures and asked to select which figure comes next among several choices. To identify logic patterns, you have to look out four (4) things, namely: (1) rotating shapes (2) increase and decrease in numbers of shapes or patterns (3) alternating patterns, colors, and shapes (4) mirror images or reflections In solving problems in logical reasoning, you have to look for patterns or rules and identify which object does follow those patterns or rules. 6 "Common Core State Standards for Mathematics." http://www.corestandards.org/wp- content/uploads/Math_Standards.pdf. Accessed 6 Aug. 2020. 5 Here is a list of examples of Logical Patterns and how they are to be identified. 1.) Identify the odd one out. For this question, you may notice that there are three or four items inside the squares. Ask yourself if that is important. On closer look, you may also find that the largest shape is grey while the bottom shape is black. Since in C, the bottom shape is a combination of white and blue, C is the odd one. 2.) Identify the missing square. In this item, you have to look for the following: 1. Relative Positional Rule: This is how the black square is positioned inside each box. 2. Movement Rule: This pertains to how the square moves in each box, in the clockwise direction. 3. The arrows in the first and third columns are reflections of one another. Considering the above conditions, C is the missing square in the last row.7 7 "Logical Reasoning Test ▷ 10 Practice Questions & 5 Key Tips." 27 Jul. 2020, https://www.wikijob.co.uk/content/aptitude-tests/test-types/logical-reasoning/. Accessed 3 Aug. 2020. 6 3.) Which frame comes next? In each frame, black and white squares are placed from top to bottom while the number of squares is increasing by one from the first to the fourth frame. The answer is A. 4.) Which frame will complete the statement? Here you have to consider the rotation of the elements. Notice that the elements rotate in a counterclockwise direction. The answer, therefore, is C. 5.) Which figure completes the grid? 1. Notice the following from the given: The first and the fourth columns and the first and second rows contain one shape only. The diamond and crescent moon shapes are dark white; the circles and the squares are white. 7  2. Divide the grid by columns, by rows, or by four groups of four squares and look for relationships. Therefore, to complete the grid, the answer is A.8 Aside from the examples presented above, there are still numerous examples under the first type of pattern. Can you think of your own example of this type of pattern? 2.2 Geometric Patterns Geometric patterns are a collection of shapes, repeating, or altered to create a cohesive design.9 These patterns are observed regularly. They appear in paintings, drawings, tapestries, wallpapers, tiling, and carpets. https://lh4.googleusercontent.com/proxy/5CWbjaQXb5 4oupH_KVT624dGVlD7fEO2YLf- 6771fdkhdrZ0NurdLazHAMOC- bPT6vVWxyYkpqSHaMbsGYekAbegL4jl906g- pCPBcDHQJ-2rIrLRU4Scc_YnLzKe1Q=w1366-h625 Presented in the succeeding pages are examples of Geometric Patterns. Tessellations Tessellations are repeating patterns of polygons that cover a plane with no gaps or overlaps. Some examples showing tessellations are the honeycombs made by honey bees and scales of fish. http://www.slate.com/content/dam/slate/articles/he https://i.pinimg.com/736x/37/ee/55/37ee5583d alth_and_science/science/2015/07/150721_SCI_H 54187db47e3d7a2aaf592f5.jpg ex-Honeycomb.jpg.CROP.promo-large.jpg 8 "Abstract Reasoning Tests: 90 Free Questions With... - WikiJob." 12 Jun. 2020, https ://www.wikijob.co.uk/content/aptitude-tests/test-types/abstract-reasoning/. Accessed 3 Aug. 2020. 9 "40 Beautiful Geometric Patterns and How to Apply Them to...." https://visme.co/blog/geometric-patterns/. Accessed 5 Aug. 2020. 8  It may be a regular tessellation (composed of regular polygons symmetrically tiling the plane), semi-regular tessellation (made of two or more regular polygons), or demi- regular tessellation (or polymorph).10 https://www.mathsisfun.com/geometry/tessellation.html Fractals Fractals are mathematical constructions characterized by self-similarity. Two objects are self-similar if they can be turned into the same shape by stretching or shrinking (and sometimes rotating). They are some of the most beautiful and most bizarre objects in all of mathematics.11 This means as one examines finer and finer details of the object, the magnified area is similar to the original but is not identical to it. Some famous fractals are the Sierpinski Triangle, Pascal's Triangle, Koch Snowflake, and Fractal Tree. Sierpinski Triangle The Sierpinski triangle begins as an equilateral triangle. The recursive procedure is to replace the triangle with three smaller congruent equilateral triangles such that each smaller triangle shares a vertex with the large triangle. To draw the Sierpinski triangle easier, start with an equilateral triangle. Then mark the midpoint of each side and connect these points. Repeat the procedures to create the triangle. https://fractalformulas.wordpress.com/2017/12/18/sierpinski-triangle/ 10 "Tessellation -- from Wolfram MathWorld." https://mathworld.wolfram.com/Tessellation.html. Accessed 5 Aug. 2020. 11 "Britannica." https://www.britannica.com/. Accessed 5 Aug. 2020. 9  Pascal's Triangle The Pascal's triangle contains the numerical coefficients of binomial expansions. The triangle below shows the coefficients of (𝑥 + 𝑦)0 up to (𝑥 + 𝑦)15. In Pascal's triangle, the Sierpinski triangle can also be drawn by connecting or shading the odd numbers. https://upload.wikimedia.org/wikipedia/commons/thumb/8/87/Sierpinski_Pascal_triangle.svg/220px-Sierpinski_Pascal_triangle.svg.png Can you make a work of art or design using the Sierpinski triangle? 10  Fractal Tree In making a fractal tree, start at some point and move a certain distance in a particular direction. At that point, make a branch (two branches in this example). Turn some angle to the right (and left) and then repeat the previous step using a shorter distance. Then, do the same in making the succeeding branches.12 https://coefs.uncc.edu/ksmit351/files/2018/08/FractalTree2-1.png Koch Snowflake In drawing a Koch Snowflake, one needs to start by drawing an equilateral triangle. Then, divide each side into three equal parts. After that, draw an equilateral triangle on each middle part. https://i.pinimg.com/originals/3b/9d/58/3b9d5827fbc4b0291eca7ec233c42663.jpg Then, divide each outer side into thirds and again, draw an equilateral triangle, but draw on each middle part. 12 (2020, March 1). How to Make a Tree With Fractals | WIRED. Retrieved August 6, 2020, from https://www.wired.com/story/how-to-make-a-tree-with-fractals/ 11 Repeat until you're satisfied with the number of iterations, like the example below. https://orderinchoas.files.wordpress.com/2013/05/biomimicry-koch-snowflake-537x402.jpg 12 2.3 Word Patterns Often, in mathematics education, we forget how many connections we can make to language arts. The metrical patterns of poems and the syntactic patterns of making nouns plural or verbs past tense are both word patterns13. Each supports mathematical and natural language understanding. Patterns can also be found in languages like morphological rules and metrical rules in poetry. These are examples of word patterns. 1.) An analogy compares two different things, but they do it by breaking them into parts to see how they are related. The colons stand for words; single colon reads as is to, double colon reads as. Analogy How to read mother: home:: teacher: school mother is to home as a teacher is to school black: white:: out: in black is to white as out is to in obese: fat:: slender: thin obese is to fat as slender is to thin 2.) Rhyme Scheme is the rhymes' pattern at the line of a poem or song (often in nursery rhymes). Can you recall some nursery rhymes? It is typically referred to by using letters to indicate which lines rhyme; lines designated with the same letter all rhyme with each other. A Haiku may be considered as a pattern concerning words. It is a Japanese poem with 17 syllables divided into three lines of 5, 7, and 5 syllables. 14 Clouds murmur darkly, 5 it is a blinding habit— 7 gazing at the moon. 5 2.4 Number Patterns A number pattern is a list of numbers that follow a particular sequence or order. Consider the example below. The given sequence of numbers is 11, 17, 23, 29, 35, 41, 47, and 53. The following figure helps to understand the relationship between the numbers. 13 "Examples 2 Number Patterns It is a list of numbers that follow...." https://www.coursehero.com/file/proqt9j/Examples-2-Number-Patterns-It-is-a-list-of-numbers-that-follow-a- certain/. Accessed 14 Sep. 2020. 14 "Haiku: Definition and Examples | LiteraryTerms.net." https://literaryterms.net/haiku/. Accessed 6 Aug. 2020. 13  11 17 23 29 35 41 47 53 +6 +6 +6 +6 +6 +6 +6 In the given pattern, the sequence is increased by 6. It means the addition of the number 6 to the previous number gives the next number. Also, the difference between the two consecutive numbers is 6. In answering number patterns, it is important to examine the interval or the difference between the consecutive numbers. Often, these intervals reveal the correct responses. In constructing a pattern, we must know the rules and the nature of the sequence. Other Examples of Number Patterns 1.) What comes next in the sequence 1, 2, 5, 10, 17, 26,…? Solution: Given sequence: 1, 2, 5, 10, 17, 26,… +1 +3 +5 +7 +9 +11 1, 2, 5, 10, 17, 26, 37 The number in the sequence is increasing by adding consecutive odd numbers. 2.) What comes next in the sequence 50, 49, 47, 44, 40, 35,…? Solution: Given sequence: 50, 49, 47, 44, 40, 35,… -1 -2 -3 -4 -5 -6 50, 49, 47, 44, 40, 35, 29 Here, you may observe that the numbers in the sequence are decreasing by consecutive integers. 3.) What is the missing number in the sequence 1, 4, 9, x, 25, 36,…? Solution: Given sequence: 1, 4, 9, x, 25, 36,… This item follows the same pattern as the first example, but here we are looking for a number in the middle of the sequence. It takes us to the idea that the answer should be 16. The numbers are also belonging to the set of perfect squares listed in increasing order. 4.) Determine the value of R and S in the following pattern. 90, 86, 82, 78, 74, 70, 66, 62, R, 54, 50, S 14  Solution: Given sequence: 90, 86, 82, 78, 74, 70, 66, 62, R, 54, 50, S You can observe that each number in the sequence is decreasing by 4. Since the number before R is 62, then R = 62 - 4 = 58. For S, we have S = 50 - 4 = 46. Item number 4 is an example of an arithmetic sequence, where the difference between two consecutive terms is called the common difference. A geometric sequence is a sequence where a term is multiplied by a constant, called the common ratio, to get the next term. The following image shows geometric sequences with three (a whole number) and one-half (a fraction) as common ratios. There are other types of number patterns aside from arithmetic and geometric sequences. These other types of number patterns are as follows: 15 Triangular Numbers: The terms of a triangular sequence are related to the number of dots needed to create a triangle. Begin forming a triangle with three dots; one on top and two on the bottom. The next row would have three dots, making a total of six dots. The next row in the triangle would have four dots, making a total of 10 dots. The following row would have five dots, for a total of 15 dots. Therefore, a triangular sequence begins: "1, 3, 6, 10, 15…" Square Numbers: In a square number sequence, the terms are the squares of their position. A square sequence would begin with "1, 4, 9, 16, 25…" Cube Numbers: In a cube number sequence, the terms are the cubes of their position. Therefore, a cube sequence starts with "1, 8, 27, 64, 125…" 15 Fibonacci Numbers. (An in-depth lesson for this number pattern is in the next module.) 15 "Types of Number Patterns in Math - Sciencing." https://sciencing.com/types-number-patterns-math- 8093943.html. Accessed 4 Aug. 2020. 16