The Nature of Mathematics PDF

The Nature of Mathematics  Mathematics in our World Numbers Quantities Shapes Mathematics is a universal way to make sense of the world and to communicate understanding of concepts and rules using the mathematical…. 1. Symbols 2. Signs 3. Proofs...

The Nature of Mathematics  Mathematics in our World Numbers Quantities Shapes Mathematics is a universal way to make sense of the world and to communicate understanding of concepts and rules using the mathematical…. 1. Symbols 2. Signs 3. Proofs 4. Language 5. Conventions Objectives At the end of the topic, students will be able to: 1. Articulate the importance of mathematics in one’s life; 2. Identify patterns in nature and regularities in the world; 3. Argue about the nature of mathematics, what it is, how it is expressed, represented and used; and 4. Express appreciation of mathematics as a human endeavor. Patterns and Numbers in Nature and the World  Natural Patterns 1. Spiral 2. Symmetries 3. Mosaics 4. Stripes 5. Dots Greek Philosophers who Studied Patterns “I would teach children music, physics and philosophy; but most importantly music for the patterns in music and all “There is geometry in the the arts are humming of the strings, “The nature of God is a circle of keys to there is music in the which the center is everywhere, learning.” spacing of the spheres.” and the circumference is -Plato -Pythagoras nowhere.” -Empedocles Other People who Studied Patterns Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. SOAP FILMS Other People who Studied Patterns Ernst Haeckel painted hundreds of marine organisms to emphasize their symmetry Voronoi diagrams a partition of a plane into regions close to each of a given set of objects. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators) Voronoi diagrams were considered as early as 1644 by philosopher René Descartes and are named after the Russian mathematician Georgy Voronoi, who defined and studied the general n-dimensional case in 1908. This type of diagram is created by scattering points at random on a Euclidean plane. Voronoi diagrams a partition of a plane into regions close to each of a given set of objects. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators) We use this in our everyday lives, from trying to find the closest supermarket, train station, etc. A plane that is divided up into cells, covering a specific region that is close to a particular point. Other People who Studied Patterns W. Gary Smith adopts eight patterns in his landscape work, namely: scattered, fractured, mosaic, naturalist drift, serpentines, spiral, radial and dendritic. These patterns occurs in plants, animals, rock formations, river flow, stars or in human creations. Other People who Studied Patterns D’Arcy Thompson pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth Other People who Studied Patterns Alan Turing predicted mechanisms of morphogenesis which give rise to patterns of spots and stripes.  L-systems were introduced and developed in 1968 by Aristid Lindenmayer, a Hungarian theoretical biologist and botanist at the University of Utrecht. Lindenmayer used L- systems to describe the behaviors of plant Other People cells and to model the growth processes of plant development. who Studied Patterns  Benoit B. Mandelbrot was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". https://www.youtube.com/watch?v=b005iHf8Z3g A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Romanesco Broccoli Other People who Studied Patterns W. Gary Smith adopts eight patterns in his landscape work, namely: scattered, fractured, mosaic, naturalistic drift, serpentine, spiral, radial, and dendritic. These patterns occur in plants, animals, rocks formation, river flow, stars or in human creations. Doodle/Zentangle Patterns Modulo Arts The Fibonacci Sequence Leonardo Pisano Bigollo lived between 1170 and 1250 in Italy. His nickname, “Fibonacci” roughly means “Son of Bonacci”. His father is Guglielmo Bonacci. He helped spread Hindu Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) through Europe in place of Roman numerals which is part of the sequence, which he developed. The Fibonacci Sequence was recognized as the Golden Ratio. The Fibonacci numbers goes like this: 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 ……….  Fibonacci Sequence B/A = 𝝓 A B (Golden Ratio) 2 3 1.5 3 5 1.66666666667 5 8 1.6 8 13 1.625 … … … 144 233 1.6180555556 233 377 1.6180257511 … … … 75025 121393 1.6180339887 121393 196418 1.6180339887 196418 317811 1.6180339887 … … …. Fibonacci Sequence in a Golden Spiral  Activity 1 Show your appreciation of the movie inspired on numbers, geometry and nature by writing an essay to answer the question: What did you learn in the video? (20 pts) (Short movies inspired on numbers, geometry and nature) By Cristobal Vila https://vimeo.com/9953368  Activity 2 Show your appreciation of the movie inspired on numbers, geometry and nature by writing an essay to answer the question: What did you learn in the video? (20 pts) Spirals and the Golden Ratio By Gary Meisner https://www.goldennumber.net/spirals/  Activity 3 Procedure 1. Ask the students to investigate their surroundings to see if they can find the numbers anywhere else – they may be surprised. Students should take notes on what they find to be used in writing the essay. Students may use a digital camera or camera phone to record and demonstrate their understanding on the Fibonacci findings. 2. Explain how the Fibonacci works in your selected picture. (20 points) Patterns and Regularities in the World as Organized by Mathematics The following constitute a unifying theme of mathematics: Patterns Relations Functions Pattern – constitutes a set of numbers or objects in which all the members are related with each other by a specific rule. It is also known as sequence. There can be finite or infinite number of members in a pattern. Relation – constitutes between two sets is a collection of ordered pairs containing one object from each set. If the object x is from the first set and the object y is from the second set, then the objects are said to be related if the ordered pair (x, y) is in the relation. Function – is a type of relation. But a relation is allowed to have the object x in the first set to be related to more than one object in the second set. Patterns in nature can be seen in the following: Rainbows Water Waves Cloud Formations Tree Branching Mud-crack Butterfly Markings Leopard Spots Tiger Stripes Relationships in nature can be observed from…. Waves on the surface of puddles, ponds, lakes, or oceans are governed by mathematical relationships between their speed, their wavelength, and the depth of the water. Functions can be observed from…. The time turkey cooks (Range) Value) is determined by the weight of the turkey (Domain Value). Similarities in nature can be observed…. Snail Shell and the Swirling Stars of a galaxy Branches of the three and those of a river network Scenes in which regularities exists Definition. Regularities is an attribute of a shape or relation; exact reflection of form on opposite sides of a dividing line or plane Motion of Pendulum – Reflection in the Free Falling Object – Action-Reaction – In Its period of time it mirror plane – the Any object that is every interaction, takes to swing back to image that is exactly moving and being there is a pair of its original position is the same size as the acted upon only be forces acting on the related to its length, object and is far the force od gravity is two interacting but the relationship is behind the mirror as said to be in a state of objects. not linear. the object is distant free fall. from the mirror. Phenomena in the World as predicted by Mathematics Patterns in Nature – The role of mathematics is to describe symmetry- breaking processes in order to explain in a unified way the fact that the patterns seen in san dunes and zebra’s stripes are caused by processes which, while physically different, are mathematically very similar. Puzzles in Nature – Mathematics solves puzzles in nature (such as why planets move in the way that they do), describes changing quantities via calculus, modelling change (such as the evolution of the eye), and predicts and controls physical systems. Nature and Occurrences in the World as Controlled by Mathematics for Human Ends The chief value of mathematics is how it applies to work. Because mathematics plays such a central role in modern culture, some basic understanding of the nature of mathematics is requisite for scientific literacy. To achieve this, students need to….. perceive mathematics as part of the scientific endeavor comprehend the nature of mathematical thinking become familiar with key mathematical ideas and skills. Mathematics in Medicine - Nurses routinely use addition, fractions, ratios and algebraic equations each workday to deliver the right amount of medication to their patients or monitor changes in their health. - In life expectancy, mathematics used to summarizes the remaining years of life that that a person is expected to live. Mathematics in Political Science - Political Scientists use mathematics (statistics) to predict the behavior of group of people. Mathematics in Economics - Analysis and study in economics help explain the interdependent relation between different variables. Economists try to explain what causes rise in prices or unemployment or inflation. Applications of Mathematics in the World Mathematics in Farming and Gardening - Mathematics has enabled farming to be more economically efficient and has increased productivity. - Famers use mathematics as a system of organization to effectively utilized their time and manage they money. - Farmers use numbers everyday for a variety of tasks, from measuring and weighing, to land marking. - Basic geometry, proportions, multiplications and measurement skills are used every day by farmers. Mathematics in Planning a Market and Grocery Shopping - Calculating prick per unit - Figuring percentage discounts - Comparing unit and bulk price of items - Estimating total price - Etc. Mathematics Anywhere in the House - Symmetric arrangement of furniture - Wall decorations and frames - Wine bottles in the bar - Plant pots in the inner garden and even restroom fixtures - Measuring ingredients - Calculating cooking time - Making ratios and proportions in baking Mathematics in Travels - Fuel required based on distance - Total expenses for toll fees - Tire pressure check - Time allowance to the trip - Short-cut routes alternatives - Road map reading - Speed limits and others Mathematics in Construction - Making accurate measurements of lengths, widths, and angles - Projecting detailed material estimate - Getting the best value of valuable resources - Etc. Mathematics in Engineering - It combines mathematical theory, practical engineering and scientific computing to address the fast changing technology. - It is a creative and exciting discipline, spanning traditional boundaries and dealing with today’s technological challenges. - It can be found in an extraordinarily wide range of careers, from designing next generation high-end cars to inventing robotics and automotive devices Mathematics in Investment - Individuals with poor math fundamentals typically make greater financial mistakes like underestimating how quickly interest accumulates. Mathematics in Time - Without a good planning, the day can slip idly, and tasks and duties accrue. - In a swift changing world, creating and following schedule prove beneficial, but it takes more mathematical skills than simply using a clock and calendar to manage time well and be on top of others.