MATHEMATICS IN THE MODERN WORLD (MMW101) Course Description: This course deals with nature of mathematics, appreciation of its practical, intellectual, and aesthetic dimensions, and application of mathematical tools in daily life. The course begins with an introduction to the nature of mathematics as an exploration of patterns (in nature and the environment) and as application of inductive and deductive reasoning. By exploring these topics, students are encouraged to go beyond the typical understanding of mathematics as merely a set of formulas but as source of aesthetics in patterns of nature, for example, and a rich language in itself (and of science) governed by logic and reasoning. Course Description: The course proceeds to survey ways in which mathematics provides a tool for understanding and dealing with various aspects of presentday living, such as managing personal finances, making social choices, appreciating geometric designs, understanding codes used in data transmission and security, and dividing limited resources fairly. These aspects will provide opportunities for actually doing mathematics in a broad range of exercises that bring out the various dimensions of mathematics as a way of knowing, and test the students’ understanding and capacity. Week 1: 1. Patterns and Numbers in Nature and the World 2 . Mathematical Language and Symbols 1. Characteristics of mathematical language 2. Expressions vs. sentences 3. Conventions in the mathematical language Course Outline (Prelim) Week 2-3: 2.4 Four basic concepts: sets, functions, relations, binary operations Week 4: Problem Solving Week 5 2.5 Elementary logic: connectives, quantifiers, negation, variables Week 6 3. Problem Solving and Reasoning 1. Inductive and Deductive Reasoning 2. Intuition, proof, and certainty Course Outline (Midterm) Week 7 Data Management 1. Data: Gathering and organizing data; representing data using graphs and charts; Interpreting organized data 2. Measures of Central Tendency 3. Measures of Dispersion Week 8 4. Measures of Relative Position Week 9 5. Probabilities and Normal Distributions Week 10 6. Linear Regression and Correlation 1. Linear Regression 2. Linear Correlation 3. Least Squares Line Course Outline (Finals) Week 11 -12 6. Linear Programming 1. Linear Inequalities 2. Geometry of Linear Programming 3. Simplex Method Week 13 -14 Optional Topics Lesson 1: PATTERNS AND NUMBERS IN NATURE AND THE WORLD Lesson Objectives Lesson Objectives: At the end of this lesson, you will be able to: Discuss the language, symbols, and conventions of Mathematics Explain the nature of mathematics as a language Convert English expressions to Mathematical sentences Convert Mathematical sentences to English expressions Exercise: 1.Which of the following figures can be used to continue the series given below? Exercise: 2. Which of the following figures can be used to continue the series given below? Exercise: 4. Which of the following figures can be used to continue the series given below? Exercise: 1. What number comes next? 4, 9, 14, 19, 24, 29, 34 .... Exercise: 2. What number comes next? 3, 6, 11, 18, 27,.... PATTERNS Patterns Regular Repeated Recurring forms or designs Identify Relationships Find Logical conncections to form generalizations Symmetry Indicates that you can draw an imaginary line across an object and the resulting parts are mirror image of each other Examples: Butterfly Leonardo Da Vinci's Virtuvian Man Starfish The butterly is symmetric about the axis indicated by the line. Note that the left and right portions are exactly the same. This type of symmetry is called bilateral symmetry Symmetry *Butterfly Leonardo Da Vinci's Virtuvian Man shows the portion and symmetry of the human body Symmetry *Leonardo Da Vinci's Virtuvian Man There are other types of symmetries depending on the number of sides or faces that are symmetrical Note that if you rotate the starfish you can still achieve the same appearance as the original position. This is known as rotational symmetry The smallest measure of angle that a figure can be rotated while still preserving the original position is called the angle of rotation A more common way of describing rotational symmetry is by the order of rotation Symmetry *Starfish Order 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 Angle of Rotation =360/n The pattern on the snowflake repeat six times indicating that there is a six-fold symmetry. Using the formula, the angle of rotation is 60 Tesselations Honeycomb Why bees used hexagon in making in making honeycombs and not any other polygons? Spots and Stripes Tiger's Stripes and Hyena's Spot Patterns exhibited in the external appearances of animals. The tiger looks and hyena's spot, these simingly random designs are believed to be governed by mathematical equation According to the theory of Allan Turing the chemical factors in the cell determine growth patterns, and influence factors like hair color. There are two chemical process: reaction and diffusion Spirals The Sunflower Looking at a sunflower, there is a definite pattern of clockwise or counterclockwise arc of spirals extending outward from the enter of the flower Spirals The Snail Shell As the snails grow the shells also expand proportionately. This process results in a refined spiral structure, it is called equiangular spiral. Let us watch this video https://www.youtube.com/watch?v=J k_1d4mbVyQ Fibonnacci Sequence The Flower Petals Flowers are easily considered as things of beauty Flowers with five petals are said to be the most common Number of petals are all Fibonnaci numbers Fibonnacci Sequence Leonardo of Pisa - Fibonacci Fibonacci is the greatest Europian Mathematician of the middle ages Born in 1170 and died in 1240 He introduced the Arabic number system in Europe Fibonnacci Sequence Origin of the Fibonacci Sequence Fibonacci sequence was discovered after an investigation on the reproduction of rabbits The Fibonacci sequence 1,1,2,3,5,8,13,... of which the first two terms are 1 and 1 and each succeding term is the sum of two immediately preceeding Fibonnacci Sequence How many pair will there be in one year? 1,1,2,3,5,8,13,21,34, 55, 89, 144,... So 144 pairs will be there at the end of 1 year Can you find the 15th term in the sequence? The Golden Ratio Golden Ratio The Golden Ratio is often denoted by the greek letter Phi This is approximately equal to 1.618 The golden ratio can be expressed as the ratio between two numbers The Golden Ratio A wood that is 120inches in length is to be cut into two parts such that the ratio of the parts constitutes the Golden Ratio. What must be the lengths of the wood?
