Mathematics In Our World PDF

MATHETAMITICS IN OUR WORLD MATHEMATICS - study of the relationships among numbers, quantities, and Includes arithmetic, algebra, trigonometry, geometry, statistics and calculus. - nurtures human characteristics like power of creativity, reasoning, critical thinking, spatial thinking and...

MATHETAMITICS IN OUR WORLD MATHEMATICS - study of the relationships among numbers, quantities, and Includes arithmetic, algebra, trigonometry, geometry, statistics and calculus. - nurtures human characteristics like power of creativity, reasoning, critical thinking, spatial thinking and others. - provides the opportunity to solve both simple and complex problems in many real-world contexts using a variety of strategies. - universal way to make sense of the world and to communicate understanding of concepts and rules using the mathematical symbols, signs, proofs, language and conventions. - helps organize patterns and regularities in the world. - helps predict the behavior of nature and phenomena in the world. - helps control nature and occurrences in the world for the good of mankind. - helps students to utilize, recognize and generalize patterns that exist in numbers, in shapes and in the world around them. "Nature is written in mathematical language." - Galileo Galilei THE FIBONACCI SEQUENCE LEONARDO PISANO BOGOLLO lived between 1170 and 1250 in Italy. Famous for Fibonacci Sequence, also recognized as the Golden Ratio Fibonacci Day is November 23 FIBONACCI NUMBERS - Fibonacci numbers can also be found in sneezewort (harangan in Tagalog): found in Cagayan, Nueva Vizcaya, Pampanga, Rizal, Laguna, and Lanao provinces), where a pattern can be observed in the growth of the stem and leaves. If the number of branches are counted in each section, the numbers are all Fibonacci numbers. This is also true for the number of leaves in each stage. PATTERNS AND REGULARITIES IN THE WORLD AS ORGANIZED BY MATHEMATICS - Scientific and mathematical principles undergird spectacular patterns as in rainbows, water waves, cloud formations, - tree branching patterns, mud- crack patterns, butterfly markings, leopard spots and tiger stripes. - Waves on the surface of puddles, ponds, lakes or oceans are governed by mathematical relationships between their speed, their wavelength, and the depth of water.  SPECTACULAR PATTERNS o Rainbow o Butterflies PATTERNS AND REGULARITIES IN THE WORLD AS ORGANIZED BY MATHEMATICS - The world consists of orders (the regular cycle of days and nights, the recurrence of seasons, alternate sunrise and sunset, etc.) and symmetry (the fractal pattern) from which similarity, predictability and regularity in nature and the world consequently exist. - The concept of symmetry fascinates philosophers, astronomers, mathematicians, artists, architects and physicists. The mathematics behind symmetry seems to permeate in most of the things around us. - The motion of a pendulum, the reflection in a plane mirror, the motion of a falling object and the action-reaction pair of forces are all guided and organized by mathematics. They exhibit regularities and symmetry in motion and behavior according to mathematical laws. PHENOMENA IN THE WORLD AS PREDICTED BY MATHEMATICS - The role of mathematics is to describe symmetry-breaking processes in order to explain that the patterns seen in sand dunes and zebra's stripes are caused by processes which, while physically different, are mathematically very similar. PHENOMENA IN THE WORLD AS PREDICTED BY MATHEMATICS - Mathematics solves puzzles in nature (such as why planets move in the way that they do), describes changing quantities via calculus, modeling change (such as the evolution of the eye), predicts and controls physical system. - Mathematics allows us to summarize, formalize, interpolate, and extrapolate from observations that have been recorded. NATURE AND OCCURENCES IN THE WORLD AS CONTROLLED BY MATHEMATICS FOR HUMAN ENDS - The application of mathematics to medicine is an exciting and novel area of research within the discipline of applied mathematics. - A component in which mathematics contributes significantly to health and medicine concerns life expectancy. - Political scientists use math and statistics to predict the behavior of group of people. - Economists explain what causes rise in prices or unemployment and inflation. Explaining the concepts of prices, quantity of goods sold and costs is best done with the use of mathematics. APPLICATIONS OF MATHEMATICS IN THE WORLD - When one buys a product, follow a recipe, or decorate his room, he uses math principles. - Farming and gardening provide rich mathematical opportunities. Mathematics has enabled farming to be more economically efficient and has increased productivity. - Planning a market list and grocery shopping requires math knowledge, the fundamental operations, to estimation and starting from percentages. - Symmetry arrangement of furniture, wall decorations and frames, wine bottles in the bar, plant plots in the garden. - Working in the kitchen requires mathematical knowledge: measuring Ingredients, calculating cooking time, making ratios and proportions in baking, etc. - Long and short travels involves math in various ways: fuel required based on distance, total expenses for toll fees, tire pressure check, time allowance for the trip, short-cut routes alternatives, road map reading, speed limits, etc. - Making accurate measurements of length, width, and angles; projecting detailed material estimate, getting the best value of available resources are applications of mathematics. - The art of applying mathematics to complex real-world problems is called engineering mathematics which combines mathematical theory, practical engineering and scientific computing to address the fast changing technology. Without strong math skills, people tend to invest, save or spend money based on their emotions. - 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 the others MATHEMATICAL LANGUAGE AND SYMBOLS LANGUAGE - system of words, signs and symbols which people use to express ideas, thoughts and feelings - systematic means of communicating ideas or feelings by the use of conventionalized signs, sounds, gestures or marks having understood meanings MATHEMATICAL LANGUAGE - system used to communicate mathematical ideas - more precise than any other language - has its own grammar, syntax, vocabulary, word order, synonyms, negations, conventions, idioms, abbreviations, sentence structure and paragraph structure - has certain language features such as representation - includes a large component of logic CHARACTERISTICS OF MATHEMATICAL LANGUAGE - The use of language in mathematics differs from the language of ordinary speech in three ways (Jamison, 2000). - Mathematical language is non-temporal. - Mathematical language is devoid of emotional content. - Mathematical language is precise. "Mathematics is the language in which God has written the universe." Galileo Galilei EXPRESSIONS VS SENTENCES A paragraph contains simple sentences that convey ideas. The ideas are understood when the sentences are written in a particular language that we know which is usually English. In mathematics, ideas are likewise given, however, some difficulties arise when it is presented in a language that not everyone could comprehend. This is what we call language of mathematics. In English, nouns are used to name things or everything of interest like people or places. A sentence expresses a complete thought. In mathematics, the equivalent of a noun is an expression and the complete thought is also called a sentence. A mathematical sentence states a fact or a complete idea. PROBLEM SOLVING AND REASONING Logic - The science of correct reasoning. Reasoning - The drawing of inferences or conclusions from known or assumed facts. When solving a problem, one must understand the question, gather all pertinent facts, analyze the problem i.e. compare with previous problems (note similarities and differences), perhaps use pictures or formulas to solve the problem. Two fundamental types of Reasoning for Mathematicians: 1. INDUCTIVE REASONING refers to the process of making generalized decisions after observing, and/or witnessing, repeated specific instances of something. Inductive Reasoning, Involves going from a series of specific cases to a general statement. The conclusion in an inductive argument is never guaranteed. 2. DEDUCTIVE REASONING refers to the process of taking the information gathered from general observations and making specific decisions based on that information. Deductive Reasoning A type of logic in which one goes from a general statement to a specific instance. It is better to solve one problem five different ways, than to solve five problems one way. ~George Polya PROBLEM SOLVING What is a Problem? - A problem is a statement or a situation where there is an obstacle between what we have and what we want. One of the most popular proponents of problem solving George Polya of Hungary (1965) stated that: "A question is considered a problem if the procedure or method of solution is not immediately known but requires one to apply creativity and previous knowledge in new and unfamiliar situations." What is Problem Solving? - Problem solving is the means by which an individual uses previously acquired knowledge, skills and understanding to satisfy the demands of an unfamiliar situation. - Problem solving means engaging in a task for which the solution method is not known in advance. In order to find a solution, students must draw on their knowledge and through this process, they will often develop new mathematical understandings. George Polya (1887-1985) was a mathematics educator who strongly believed that the skill of problem solving can be taught. He developed a framework known as Polya's Four-Steps in Problem Solving. This process addressed the difficulty of students in problem solving. He firmly believed that the most efficient way of learning mathematical concepts is through problem solving and students and teachers become a better problem solver. POLYA’s FOUR STEPS IN PROBLEM SOLVING 1. Understand the problem. (Analysis) This part of Polya's four-step strategy is often overlooked. You must have a clear understanding of the problem. To help you focus on understanding the problem, consider the following questions: Can you restate the problem in your own words? Can you determine what is known about these types of problems? Is there missing information that, if known, would allow you to solve the problem? Is there extraneous information that is not needed to solve the problem? What is the goal? 2. Devise a plan. (Planning) - Look for a pattern. - Write an equation. If necessary, define what each variable represents. - Perform an experiment. - Guess at a solution and then check your result. 3. Carry out the plan. (Implementation) - Once you have devised a plan, you must carry it out. - Work carefully. - Keep an accurate and neat record of all your attempts. - Realize that some of your initial plans will not work and that you may have to devise another plan or modify your existing plan. 4. Look back. (Review the solution.) - Once you have found a solution, check the solution. - Ensure that the solution is consistent with the facts of the problem. - Interpret the solution in the context of the problem. - Ask yourself whether there are generalizations of the solution that could apply to other problems.