MMW Module 1 PDF

# Module 1: The Nature of Mathematics ## GE 1 - Mathematics in the Modern World Author: Mary Jane B. Calpa ### 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...

# Module 1: The Nature of Mathematics ## GE 1 - Mathematics in the Modern World Author: Mary Jane B. Calpa ### 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 the environment. It is a rich language governed by logic and reasoning, and an application of both inductive and deductive reasoning. This module is composed of the following: 1. **Mathematics in our World**: This section will focus on the core idea that mathematics is a useful way to think about nature and the world. It will explore how pattern-seeking can help us understand the substantial interconnection between the world and mathematics, and appreciate mathematics as a discipline full of beauty and essence. 2. **Mathematics Language and Symbols**: 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 study different methods of reasoning used to justify mathematical statements and arguments. 3. **Problem Solving and Reasoning**: This section will discuss mathematics as a tool for decision-making and problem-solving. It explores the various skills and approaches used in mathematical problem-solving, including trial-and-error, seeking patterns, generalities, and the desire to know the truth. ### Learning Outcomes After completing this module, you will be able to: 1. Identify patterns in nature and irregularities. 2. Articulate the importance of mathematics in one's life. 3. Discuss mathematics, what it is, how it is expressed, represented, and used. 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]](https://vimeo.com/9953368))** and write a sentence that describes your impression after watching the video. 2. **Identify patterns observed in the following pictures:** * *(a) An image showing a repeating pattern of a series of X's. The pattern is arranged to form a grid of smaller squares within a larger one. The last square is empty and should be filled in with another X.* * *(b) A close-up of a floral pattern, where the flowers are arranged in a grid.* * *(c) An image containing a series of half-circles, with a plus symbol at the top of each half-circle. This is followed by a full circle, and then a repeating pattern of another full circle. The repeated pattern has a plus symbol on the bottom of the circle. The last image in the sequence is a full circle with a plus symbol on the top of it.* * *(d) An image showing a staggered arrangement of stars with five points.* * *(e) A simple mandala pattern, where a series of repeating pentagons are arranged into a circle. There is a small gap between each pentagon.* ### Questions To Ponder * What are the different kinds and forms of patterns you have seen in the video and/or pictures? * How do these patterns help us understand the connection between our world and mathematics? ### Patterns and Numbers in Nature and the World When we observe the world around us, from the colorful wings of butterflies to the arrangement of flowers and leaves, there are unseen patterns that exist everywhere. These patterns are not only a source of beauty and wonder but are also mathematically significant. Repeated ways or occurrences, such as the cycle of the moon, the changing seasons, or the transmission pattern of a pandemic, can be considered patterns. Mathematics, as a formal system of thought, helps us to recognize, classify, and exploit these patterns. It assists us in making generalizations, identifying relationships, discovering logical connections, and becoming better problem-solvers. ### Example 1: Logic Pattern **Choose the figure that completes the pattern.** * *Image: a simple grid pattern containing three X symbols. The fourth square is empty.* **Solution:** The pattern is built stage by stage. In each stage, a new line is added that never touches the last line. The fourth square should be filled with an X to continue the pattern. ### Example 2: Number Pattern **Find the next number in the sequence.** 1. `3, 8, 13, 18, 23,?` 2. `1, 4, 9, 16, 25, 36,?` **Solution:** 1. The difference between each term in this sequence is 5. Therefore, the next number is `28` (23 + 5). 2. The sequence can be expressed as `1^2, 2^2, 3^2 ... 6^2`. The next number is `7^2`, which equals `49`. ### Geometric and Word Patterns Geometric patterns are visual representations of geometric shapes, while word patterns focus on the morphological rules used in language, such as pluralizing nouns or conjugating verbs. These patterns are found everywhere, from natural designs like circles and polygons to the identification of a particular country or culture. *(Image: 3 geometric patterns representing traditional patterns from the Phillipines: Cordillera tattoos, woven mat "banig" from Basey, Samar, and a beaded T'boli belt.)* ### Self-Assessment Activity 1 1. **For each set of figures, determine what comes next.** * *(a) A sequence of 4 X symbols, where 3 are arranged in a grid pattern, and the last one is a rotated X. The last square in the grid is empty.* * *(b) A sequence of 4 images. 1st image: a half-circle with a plus on top. 2nd image: a full circle. 3rd image: a full circle with a minus on the bottom. The 4th image is a full circle with a plus on top. The last image in the sequence is a full circle with a minus on the bottom.* * *(c) 4 images showing a series of dots, with the pattern changing in each image. The first image shows a 3x3 grid of blank squares. Each image following adds a dot to the grid until the final image which is a grid that is 5 squares wide and 4 squares tall, with all the squares filled in.* 2. **What is the next number in the sequence?** * `3, 6, 12, 24, 48, ?` * `1, 4, 10, 22, 46, ?` * `4, -1, -11, -26, -46, ?` ## Symmetry and Angle of Rotation * *Image of the letter A, where the left side is reflected on the right. This is followed By images of a flower with 3 petals, a starfish with 5 points, a geometric image of a Vitruvian Man, and a 4-leaf clover with 45-degree angles between each leaf.* When you draw an imaginary line across an object, and the resulting parts are mirror images of each other, this is called **symmetry**. This figure above, the letter A, is symmetric about the axis indicated by the broken line. This is called **line symmetry** or **bilateral symmetry**, and it is common to animals and humans. When an object is rotated by a certain angle, and it still looks the same as the original position, this is called **rotational symmetry**. The smallest angle an object can be rotated while maintaining its original formation is called the **angle of rotation**. A figure has a rotational symmetry of order n (n-fold rotational symmetry) if `1/n` of a complete turn leaves the figure unchanged. * *Image: Rotating a 3-leaf clover to show the angle of rotation, followed by an image of a 4-leaf clover rotating between 90 degree angles.* The angle of rotation is calculated as follows: $$Angle \; of \; Rotation = \frac{360}{n}$$ ### Snowflakes and Honeycombs * *(Image: A microscopic view of a snowflake where there is 6-fold symmetry. Below, there is an image showing a hexagonal honeycomb with the caption "Why Hexagon? Bees Know Best!" The text then describes how the hexagon pattern is the most efficient way to store honey. Below there is another image, this time of the tail feathers of a peacock, with the caption "Peacock's Tail." The text describes how the patterns on the peacock's tail help with survival and attracting mates.)* Snowflakes, with their intricate six-fold symmetry, demonstrate the presence of mathematical patterns in nature. This symmetry is due to the effects of different atmospheric conditions on ice crystals as they form and descend from the skies. Honeycombs exhibit a near-perfect hexagonal shape arrangement, which is the most efficient way for bees to store honey. This geometric configuration allows them to maximize storage space using the least amount of wax. Animals, like peacocks, use patterns in their external appearance to attract mates, ensure survival, and even for communication. Beautiful colors and symmetric patterns are used to attract mates or blend in for camouflage. ### The Fibonacci Sequence * *Image: A rectangle divided into 8 squares with the following dimensions: 1, 1, 2, 3, 5, 8, and 13, with an empty square on one side that will have a dimension of 8. The text below describes, how this arrangement works like a jigsaw puzzle to create the Fibonacci sequence. Underneath the image is an image of Leonardo Pisano, with a description of the Fibonacci sequence and how it was discovered.* The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, …) is a fascinating numerical pattern where each subsequent number is the sum of the two preceding numbers. This sequence was discovered by Leonardo Pisano, also known as Fibonacci, in his 1202 book *Liber Abaci*. The sequence is often observed in nature, such as in the arrangement of sunflower seeds. ### The Golden Ratio The golden ratio, represented by the Greek letter phi ($\phi$), is an irrational number, approximately equal to **1.618**. The golden ratio is closely related to the Fibonacci sequence. As you divide consecutive numbers in the Fibonacci sequence, the result approaches the golden ratio. $$\phi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887...$$ The golden ratio is considered to be aesthetically pleasing and is often found in art, architecture, and nature. ### Self-Assessment Activity 2 1. **Find the missing quantity in the formula A = Pert by substitution of the given values:** * **(a):** P = 505,050; r = 5% per year; t = 1 year * **(b):** P = 240,100; r = 11% per year; t = 10 years * **(c):** A = 786,000; P = 247,000; t = 17 years 2. **The exponential growth model A = 45e0.19t describes the population of a city in the Philippines in thousands, t years after 1995.** * What is the population of the city in 1995? * What is the population after 25 years? * What is the population in 2045? ### Math for Our World We are surrounded by a world of complexities and uncertainties, from our daily lives to the vastness of the cosmos. Mathematics plays a vital role in helping us understand and navigate this intricate world. From everyday tasks and weather patterns to the development of advanced technologies, mathematical principles guide us in diverse domains. Mathematics has a powerful and wide-ranging influence on our daily lives. It is utilized in various fields, including: * **Science**: Understanding the natural world, from predicting eclipses to the development of new drugs. * **Technology**: Creating devices and systems that improve our lives, from smartphones to spacecraft. * **Engineering**: Designing buildings, roads, and bridges that are safe and efficient. * **Finance**: Managing money and investments. * **Art**: Creating beautiful and harmonious works of art using mathematical principles. By harnessing the power of mathematics, we can explore the world around us, solve problems, make informed decisions, and advance human progress. ### Self-Assessment Activity 3 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 sums:** * **(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? ## The Great Works of Man and the Fibonacci Sequence and the Golden Ratio * *(a) An image of the Great Pyramid of Giza, where the height of the pyramid, the length, and the width are all labelled with their respective dimensions. There is text below explaining how these dimensions are related to the Golden Ratio.* * *(b) An image of the Parthenon in Athens. The image includes several lines connecting key points on the building, along with measurements labelled in Roman numerals. There is text below explaining how the ratio of the length to the height was designed based on the Golden Ratio.* * *(c) An image of the Roman Colosseum. This image includes various lines connecting the architecture of the building to showcase how the Golden Ratio was used in the design.* These images demonstrate how the ancient Greeks and Egyptians incorporated the Fibonacci sequence and the golden ratio into their architectural masterpieces. The dimensions of these structures often reflect a harmonious relationship between height, width, and length, providing a sense of beauty and proportion. ### References * Aufmann, R., Lockwood, J., et.al, *Mathematics in the Modern World*, Rex Bookstore, Inc., 2018. * Lerner, K.L., Lerner, B.W., *Real-life Math, Vol. 2*, Thomson Gale, 2006. * Nocon, R., Nocon, E., *Essential Mathematics for the Modern World*, C & E Publishing, Inc. 2018. * Post, T.R., The Role of Manipulative Materials in the Learning Mathematical Concepts. Retrieved from: [[http://www.cehd.umm.edu/ci/rationalnumberproject/81_4.html]](http://www.cehd.umm.edu/ci/rationalnumberproject/81_4.html) ### Image Sources * hhtp://www.jobtestprep.co.uk * www.psychometric-success.com * [[https://www.library.illinois.edu/mtx/2018/10/09/mathematics-in-nature/]](https://www.library.illinois.edu/mtx/2018/10/09/mathematics-in-nature/) * [[https://www.weareteachers.com/teacher-dresses-ms-frizzle/]](https://www.weareteachers.com/teacher-dresses-ms-frizzle/) * [[https://www.smithsonianmag.com/science-nature/science-behind-natures-patterns-180959033/]](https://www.smithsonianmag.com/science-nature/science-behind-natures-patterns-180959033/) * [[http://mustafacil-online.blogspot.com/2015/08/manmade-patterns.html]](http://mustafacil-online.blogspot.com/2015/08/manmade-patterns.html) * [[https://newsinfo.inquirer.net/941295/batok-tattooing-tattooing-mambabatok]](https://newsinfo.inquirer.net/941295/batok-tattooing-tattooing-mambabatok) * [[https://www.our7107islands.com/basey-samar-the-new-banig-capital-of-the-philippines/]](https://www.our7107islands.com/basey-samar-the-new-banig-capital-of-the-philippines/) * [[http://alvicsbatik.weebly.com/mindanao-accessories---page2.html]](http://alvicsbatik.weebly.com/mindanao-accessories---page2.html) * [[https://www.benefits-of-honey.com/honeycomb-pattern.html#:~:text=Studies%20on%20the%20geometry%20of, and%20square%20makes%20smaller%20area.]](https://www.benefits-of-honey.com/honeycomb-pattern.html#:~:text=Studies%20on%20the%20geometry%20of, and%20square%20makes%20smaller%20area.) * [[https://www.bigwalls.net/climb/camf/index.html]](https://www.bigwalls.net/climb/camf/index.html) * [[https://listverse.com/2013/04/21/10-beautiful-examples-of-symmetry-in-nature/]](https://listverse.com/2013/04/21/10-beautiful-examples-of-symmetry-in-nature/) * [[https://www.storyofmathematics.com/medieval_fibonacci.html]](https://www.storyofmathematics.com/medieval_fibonacci.html) ### Other Materials * [[https://vimeo.com/9953368]](https://vimeo.com/9953368) * [[https://youtu.be/pb0MSMGSley]](https://youtu.be/pb0MSMGSley) (BBC's Documentary "The Language of the Universe") ### Suggested Readings * Stewart, Ian, *Nature's Numbers* * Adam, John A.,* Mathematics in Nature: Modeling Patterns in the natural World* * Adam, John A., *A Mathematical Nature Walk* * Akiyama & Ruis, *A Day's Adventure in Math Wonderland* * Enzensberger, *The Number Devil* ### Note to Students You can discuss the lessons with your GE 1 instructor/professor through different modes of communication (email, Messenger, Moodle, Google Meet, Zoom, Google classroom, etc.). Your GE 1 instructor/professor will contact you using the email address and/or mobile number you have provided the University upon your registration. If you have not received a message from your assigned faculty at least two (2) weeks from the resumption of classes (October 5, 2020), please send your concerns to the Department of Mathematics Chair using the following address: [email protected], or through your respective municipal links. Please include your FULL NAME, STUDENT NUMBER, COURSE - YEAR, and GE 1 CLASS ID NUMBER. **Deadline of submission of Worksheet and Reflection Paper to the Municipal Link:** Department of Mathematics, College of Science, University of Eastern Philippines