Chapter 1 - Patterns in Nature (PDF)

Chapter 1 Lesson 1: Patterns and Numbers in Nature and in the World. (Week 1) Core Idea: Mathematics is a useful way to think about nature and our world. Have you ever looked around and observed that there are endless variety of forms in nature? Have you recognized that these vari...

Chapter 1 Lesson 1: Patterns and Numbers in Nature and in the World. (Week 1) Core Idea: Mathematics is a useful way to think about nature and our world. Have you ever looked around and observed that there are endless variety of forms in nature? Have you recognized that these variety of forms abounds in spectral colors and intricate shapes called patterns? Have you at a certain time asked yourself, what the world’s environment would look like if there were no colors and patterns? Looking at all of these, we come to know that these miraculous creations not only delight our imagination, but they also challenge our understanding. In this fast-paced society, how often have you stopped to appreciate the beauty of the things around you? Have you ever paused and pondered about the underlying principles that govern the universe? Most people do the same routine tasks every day and the fundamental concepts that make these activities possible are often overlooked. Mathematics is a useful way to think about nature and our world. The nature of mathematics underscores the exploration of patterns (in nature and the environment). Mathematics exists everywhere and it is applied in the most useful phenomenon. Mathematics is an integral part of daily life; formal and informal. It is used in technology, business, medicine, natural data sciences, machine learning, and construction. As you go through, this module will help you look at patterns and regularities in this world, and how mathematics comes into play, both in nature and in human endeavor. Learning Outcomes: At the end of the lesson, the students are expected to: 1. Show patterns and numbers that we can see in nature and the world 2. Understand how mathematics help organize patterns and regularities in the world and identify patterns existing in nature 3. Apply mathematics in predicting the behavior of nature and phenomena in the world 4. Appreciate mathematics in helping control nature and occurrences in the world 5. Inculcate the numerous applications of mathematics in the world Engage: Let’s Try This!  Look at the above pictures, can you identify the numbers of petals a sunflower has? Number of times the snowflakes repeats its shape? Number of contours the valve of the shell has? Number of eyes the pineapple fruit has? Total spikes of the succulent? Or the number of cones of the conifers? Kindly fill out the table. Sunflower Snowflakes Shell Pineapple Succulent Conifers As you noticed, those pictures we found in nature have definite count of shapes, sides, cones and all. These are expressed in numbers. Now, what else can you consider? Check the table below and fill out again your answer. Patterns and Numbers in Nature and the World Explore: Discover This! Patterns and counting are correlative. Counting happens when there is pattern. When there is counting, there is logic. Consequently, pattern in nature goes with logic or logical set-up. There are reasons behind a certain pattern. That’s why, oftentimes, some people develop an understanding of patterns, relationships, and functions and use them to represent and explain real-world phenomena. Most people say that mathematics is the science behind patterns. Mathematics exists everywhere as patterns do in nature. Not only do patterns take many forms within the range of school mathematics, they are also a unifying mechanism. One thing must be clear at this point. Mathematics is not all about number. Rather, it is more about reasoning, of making logical inferences and generalizations, and seeing relationships in both visible and invisible patterns in nature and in the world.  Patterns are regular, repeated, or recurring forms or designs. Patterns are more 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 a snail’s shell, and the number of petals of flowers. Humans are hard wired to recognize patterns. The formal system of thought developed by human mind and culture for recognizing, classifying, and exploiting patterns is called Mathematics. (Ian Stewart, p.1) Patterns surround us in nature. They will tell you how things work on our planet, if you know where to look to find them. Out the window, through a microscope, or in the mirror-patterns surround us. Patterns in nature are visible regularities of form found in the nature world. These patterns recur in different context and can sometimes be modeled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Symmetry - is pervasive in living things. https://mathcurious.com/2020/04/ - Animals mainly have bilateral mirror 08/symmetry-in-nature/ symmetry, as to the leaves of plants and some flowers  Explain: Clarify Your Lesson! Nature embraces mathematics completely. There are many different things around us that have a deep sense of awareness and appreciation of patterns. Nature provides numerous examples of beautiful shapes and patterns, from the nightly motion of the stars and the rainbow that we see in the sky. Some animals show pattern in their body like the tiger’s stripes and hyena’s spots. Snails make https://phys.org/news/2016-09-scientists- universe.html their shells, spiders design their webs, and bees build hexagonal combs. The structured formation of parts of human beings, animals and insects, and the beautiful pattern of plants and flowers are examples of patterns that possess utility and beauty. The patterns that we see are also the keys to understanding the processes of biological growth. It is indeed true that the place we live is a world of https://www.aswangproject.com/rainbows-in- philippine-mythology-folklore/ patterns. In the general sense of the word, patterns are regular, repeated, or recurring forms or designs. We see patterns every day – from the layout of floor tiles, designs of skyscrapers, to the way we tie our shoelaces. Studying patterns help students in identifying relationships and finding logical connections to form generalizations and make predictions. Coat patterns of different species of animals  Patterns indicate sense of structure and organization that it seems only humans are capable of producing these intricate, creative, and amazing formations. Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modelled mathematically. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. Mathematics makes our life orderly and prevents chaos. Certain qualities that are nurtured by mathematics are power of reasoning, creativity, abstract or spatial thinking, critical thinking, problem-solving ability and even effective communication skills. Mathematics is the cradle of all creations, without which the world cannot https://www.quora.com/What-are-mathematics-patterns-in-nature-and- move an inch. Be it a cook or a farmer, a carpenter or a mechanic, a shopkeeper or a doctor, an engineer or a scientist, a musician or a magician, everyone needs mathematics how-does-it-help-organize-lifes-many-challenges in their day-to-day life. Mathematics reveals hidden patterns that help us understand the world around us. Now much more than arithmetic and geometry, mathematics today is a diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems. As a practical matter, mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects, mathematics relies on logic rather than on observation as its standard of truth, yet employs observation, simulation, and even experimentation as means of discovering truth. Mathematics as a human endeavor. For example, consider the math of measurement of time such as years, seasons, months, weeks, days, and so on. Or, consider the measurement of distance, and the different systems of distance measurement that developed throughout the world. Or, think about math in art, dance, and music. There is a rich history of human development of mathematics and mathematical uses in our modern society. https://www.fao.org/in-action/planting-the-seeds-of-recovery-in-the-philippines-after-typhoon-haiyan/en/ Example 1: Let’s take a look at this pattern below. What do you think will be the next face in the sequence? Solution: It should be easy enough to note that the pattern is made up of two smiling faces – one without teeth and one with teeth. Beginning with a toothless face, the two faces then alternate. Logically, the face that should follow is Example 2: What is the next figure in the pattern below? ? A B Solution: Looking at the given figures, the lines seem to rotate at 90-degree intervals in a counterclockwise direction, always parallel to one side the square. Hence, either A or B could be the answer. Checking the other patterns, the length of the lines inside the square follow a decreasing trend. So again, either A or B could be the answer. Finally, looking at the number of the lines inside the box, each succeeding figure has the number of lines increase by 1. This means that the next figure should have five lines inside. This leads to option A as the correct choice. Example 3. How many lines of symmetry does a rhombus have? A. 0 B. 2 C. 4 D. 1 Solution: A rhombus has 2 lines of symmetry which cut it into two identical parts. Both the lines of symmetry in a rhombus are from its diagonals. So, it can be said the rhombus lines of symmetry are both diagonals. Hence, letter B is the correct answer.  Lesson 2: Fibonacci Sequence, Golden Ratio and Golden Rectangle (Week 2) As we have seen in the previous section, the human mind is hardwired to recognize patterns. In mathematics, we can generate patterns by performing one or several mathematical operations repeatedly. Patterns make up the entire universe, and everything in it (both static and dynamic forms) should be the subject of inquiry of every mathematician. Over time, mathematics has triumphantly organized these patterns allowing the human intellect to understand the order and system by which the world operates and then made inferences out of these patterns to predict the behavior of nature as well as other phenomena in the world. This lesson journeys into how mathematicians have defined the course of scientific inquiry through a comprehensive and intensive treatment of the patterns that occur in nature and in the world. This lesson involves the mathematical ideas of Fibonacci sequence, the golden ratio phi and the golden rectangle. Learning Objective: At the end of the lesson, the students are expected to: 1. Discuss the Fibonacci sequence and its application. Engage: Let’s Try This! The pictures below depict the different species of flowers. Count the number of petals for each flower. What sequence of numbers formed from the number of petals? , , , , , , The sequence of numbers formed from the number of petals of the different species of flower is a Fibonacci sequence. The Fibonacci sequence exhibits a certain numerical pattern which has turned out to be one of the most interesting ever written down. Its method of development has led to far-reaching applications such as to model or describe an amazing variety of phenomena, in mathematics and science, and even more fascinating is its surprising appearance in Nature and in Art, in classical theories of beauty and proportion.  Explain: Clarify Your Lesson! The mathematical ideas of the Fibonacci sequence led to the discovery of the golden ratio, spirals and self- similar curves, and have long been appreciated for their charm and beauty, but no one can really explain why they are echoed so clearly in the world of art and nature. Fibonacci sequence The Fibonacci sequence was invented by the Italian Leonardo https://fibanachi.wordpress. Pisano Bigollo (1180-1250), who is known in mathematical history by several names: Leonardo of Pisa and Fibonacci (son of Bonacci). The fibonacci sequence was the outcome of a mathematical problem about rabbit breeding. com/ Fibonacci sequence derived from a problem in the Liber Abaci, which was about how fast rabbits could breed in ideal circumstances. a) A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive? b) Beginning with a male and female rabbit, how many pairs of rabbits could be born in a year? The problem assumes the following conditions: c) Begin with one male rabbit and female rabbit that have just been born. d) Rabbits reach sexual maturity after one month. e) The gestation period of a rabbit is one month. f) After reaching sexual maturity, female rabbits give birth every month. g) A female rabbit gives birth to one male rabbit and one female rabbit h) Rabbits do not die. This is illustrated in the diagram. After one month, the first pair is not yet at sexual maturity and can't mate. At two months, the rabbits have mated but not yet given birth, resulting in only one pair of rabbits. After three months, the first pair will give birth to another pair, resulting in two pairs. At the fourth month mark, the original pair gives birth again, and the second pair mates but does not yet give birth, leaving the total at three pairs. This continues until a year has passed, in which there will be 233 pairs of rabbits. https://commons.wikimedia.org/wiki/File:FibonacciRabbit.svg  The resulting number sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... (Leonardo himself omitted the first term), is the first recursive sequence (in which the relation between two or more successive terms can be expressed by a formula) known in Europe. A recursive definition for a sequence is one in which each successive term of the sequence is defined by using some of the preceding terms. If we sue the mathematical notation 𝐹𝑛 to represent 𝑛𝑡ℎ Fibonacci number, then the numbers in the Fibonacci sequence are given by the following recursive definition 𝐹1 = 1, 𝐹2 = 1, and 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 for 𝑛 ≥ 3. Example 1: Use the definition of Fibonacci numbers to find the seventh and eight Fibonacci numbers. Solution: The first six Fibonacci numbers are 1, 1, 2, 3, 5, and 8. The seventh Fibonacci number is the sum of the two previous Fibonacci numbers. Thus, 𝐹7 = 𝐹6 + 𝐹5 = 8 + 5 = 13 The eight Fibonacci number is 𝐹8 = 𝐹7 + 𝐹6 = 13 + 8 = 21 Fibonacci observed numbers in nature. Pinecones grow in a numerical sequence. https://www.pinterest.ph/pin/1016898790839155932/ https://faculty.math.illinois.edu/~delcour2/LessonPlanSunflowerWorksheets.pdf Count the number of spirals going from the center of the cone (where it attached to the tree) to the outside edge. Count the spirals in both directions. The resulting numbers are usually two consecutive Fibonacci numbers (e.g., 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,...). Fibonacci numbers appear in nature as illustrated below, often enough to prove that they reflect some naturally occurring patterns. Flowers and branches: Some plants express the Fibonacci sequence in their growth points, the places where tree branches form or split. Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. https://botanicamathematica.wordpress.com/2 https://thefibonaccisequence.weebl https://www.deviantart.com/arietes/art/Fibo 014/04/01/fibonacci-tree/ y.com/flowers.html nacci-study-341249834  Explore: Discover This! Golden ratio phi “φ” The relationship of this sequence to the Golden Ratio lies not in the actual numbers of the sequence, but in the ratio of the consecutive numbers. Since the ratio is basically a fraction, we will find the ratios of these numbers by dividing the larger number by the smaller number that fall consecutively in the series. The golden ratio is a mathematical ratio. It is commonly found in nature, and when used in design, it fosters organic and natural looking compositions that are aesthetically pleasing to the eye. It is a number often encountered when taking the ratios of distances in simple geometric figures, such as the pentagon, pentagram, decagon and dodecahedron. It is a ratio or proportion defined by an irrational number Phi (Ф) = 1.618033988749895... It is expressed algebraically as, a+b a https://johnmjennings.com/the-golden-ratio/ = =φ a b It has its unique positive solution with a value One more interesting thing about phi is its reciprocal. If you take the ratio of any number in the Fibonacci sequence to the next number (this is the reverse of what we did before), the ratio will approach the approximation 0.618. This is the reciprocal of Phi: 1/1.618=0.618. It is highly unusual for the decimal integers of a number and its reciprocal to be exactly the same. This only adds to the mystique of the Golden Ratio and leads us to ask: What makes it so special? The ratio between the forearm and the hand also yields a value close to the golden ratio. Measure the length of your forearm and your hand (in centimeters). Divide the length measure of your forearm to the length measure of your hand. What can you say to its ratio? Length of your forearm (in centimeters): _________ Length of your hand (in centimeters): ___________ Ratio of forearm and hand: _______________ https://www.chegg.com/homework-help/collect-lengths-forearm-y-hand-x-15- people-following-picture-chapter-3-problem-73e-solution-9780538733502-exc  According to Markowsky (1992), the ratio of a person’s height to the height of his/her navel is roughly a golden ratio. Measure your height (in centimeters) and the height of your navel (in centimeters). Get the ratio of your height and the height of your navel. What can you say to its ratio? ___________________ https://5minutecrafts.site/improve-life/the-proportions-of-a- human-body-2518/ Golden section can be found in the Great pyramid in Egypt. Perimeter of the pyramid, divided by twice its vertical height is the value of Phi. https://www.semanticscholar.org/paper/Golden-Ratio-and-Its-Effect-on- Handwritings-Using-Solanki/d49b3d6cca34a17a50345e6d0f6b38e4fbdce9e4 A pleasing smile and an attractive eyes https://www.pinterest.ph/pin/7670261859585674/ https://radium-aesthetics.com/is-the-golden-ratio-the-secret-to-beauty/ Other examples of golden ratio in architectures and in nature https://www.livescience.com/42183-snowflake- https://www.responsify.com/golden-ratio-logo- https://www.phimatrix.com/nature-animals-golden-ratio/ formation-explained-video.html design/ https://www.lifecoachcode.com/2018/09/26/divine-ratio-is-found- https://brej.weebly.com/fibonacci-the-golden-ratio.html everywhere-nature/  The Divine Proportion https://www.phimatrix.com/face-beauty-golden-ratio/ The photo below illustrates the following golden ratio proportions in the human face: 1. Center of pupil : Bottom of teeth : Bottom of chin 2. Outer & inner edge of eye: Center of nose 3. Outer edges of lips : Upper ridges of lips 4. Width of center tooth : Width of second tooth: 5. Width of eye : Width of iris The Proportions in the Body The white line is the body’s height. 1. The blue line, a golden section of the white line, defines the distance from the head to the finger tips. 2. The yellow line, a golden section of the blue line, defines the distance from the head to the navel and the elbows. 3. The green line, a golden section of the yellow line, defines the distance from the head to the pectorals and inside top of the arms, the width of the shoulders, the length of the forearm and the shin bone. 4. The magenta line, a golden section of the green line, defines the distance from the head to the base of the skull and the width of the abdomen. The sectioned portions of the magenta line determine the position of the nose and the hairline. 5. Although not shown, the golden section of the https://www.goldennumber. net/human-body/ magenta line (also the short section of the green line) defines the width of the head and half the width of the chest and the hips.