Mathematics In The Modern World GNED 03 PDF

SLIDESMANIA.COM MATHEMATICS IN THE MODERN WORLD GNED 03 Prepared by: Rachel Mae O. Panganiban SLIDESMANIA.COM Chapter 1. Mathematics in our World Mathema...

SLIDESMANIA.COM MATHEMATICS IN THE MODERN WORLD GNED 03 Prepared by: Rachel Mae O. Panganiban SLIDESMANIA.COM Chapter 1. Mathematics in our World Mathematics – study of the relationships among numbers, quantities, and shapes. It includes arithmetic, algebra, trigonometry, geometry, statistics, and calculus – helps organize patterns and regularities in the world – being a science of patterns, helps students to utilize, recognize, and generalize patterns that exist in numbers, in shapes, and in the world around them SLIDESMANIA.COM A. Patterns and Numbers in Nature and the World Patterns – regular, repeated, or recurring forms or designs 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, spirals, waves, foams, and tesselations. SLIDESMANIA.COM A. Patterns and Numbers in Nature and the World Symmetry – indicates that you can draw an imaginary line across an object and the resulting parts are mirror images of each other. Types of Symmetry 1. Line or Bilateral Symmetry 2. Rotational Symmetry SLIDESMANIA.COM A. Patterns and Numbers in Nature and the World Symmetries SLIDESMANIA.COM A. Patterns and Numbers in Nature and the World Spiral – in mathematics, it is a curve which emanates from a point, moving farther away as it revolves around the point. SLIDESMANIA.COM A. Patterns and Numbers in Nature and the World Wave – in physics, mathematics, and engineering, it is a propagating dynamic disturbance of one or more quantities. SLIDESMANIA.COM A. Patterns and Numbers in Nature and the World Foams – materials formed by trapping pockets of gas in a liquid or solid. SLIDESMANIA.COM A. Patterns and Numbers in Nature and the World Tessellation (or tiling) – the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. SLIDESMANIA.COM A. Patterns and Numbers in Nature and the World Early Greek philosophers studied pattern, with Plato, Pythagoras, and Empedocles attempting to explain order in nature. The modern understanding of visible patterns developed gradually over time. In the 19th century, the Belgian physicist Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface. The German biologist and artist Ernst Haeckel painted hundred of marine organisms to emphasize their symmetry. SLIDESMANIA.COM A. Patterns and Numbers in Nature and the World Scottish biologist D’Arcy Thompson pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth. In the 20th century, the British mathematician Alan Turing predicted mechanisms of morphogenesis which give rise to patterns of spots and stripes. The Hungarian biologist Aristid Lindenmayer and the French American mathematician Benoit Mandelbrot showed how the mathematics of fractals could create plant growth patterns. SLIDESMANIA.COM B. The Fibonacci Sequence One of the most famous examples of mathematical patterns in nature is the Fibonacci sequence. It's a simple series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. So, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. What’s remarkable is that the numbers in the sequence are often seen in nature. SLIDESMANIA.COM B. The Fibonacci Sequence SLIDESMANIA.COM B. The Fibonacci Sequence In mathematics, we can generate patterns by performing one or several mathematical operations repeatedly. Sequence – an ordered list of numbers, called terms, that may have repeated values. The arrangement of these terms is set by a definite rule. SLIDESMANIA.COM B. The Fibonacci Sequence The Fibonacci sequence is named after the Italian mathematician Leonardo of Pisa, who was better known by his nickname Fibonacci. Starting with 0 and 1, the succeeding terms in the sequence can be generated by adding the two numbers that came before the term. It is interesting to note that the ratios of successive Fibonacci numbers approach the number Φ (Phi), also known as the Golden Ratio. This is approximately equal to 1.618. SLIDESMANIA.COM B. The Fibonacci Sequence 1 13 Finding 𝑭𝒏 = 1.0000 = 1.6250 1 8 Binet’s Formula 2 = 2.0000 21 = 1.6154 1 13 𝑛 𝑛 3 34 1+ 5 1− 5 = 1.5000 = 1.6190 − 2 21 𝐹𝑛 = 2 2 5 5 = 1.6667 55 = 1.6176 3 34 8 89 = 1.6000 = 1.6182 5 55 SLIDESMANIA.COM B. The Fibonacci Sequence Shapes and figures that bear the Golden Ratio are generally considered to be aesthetically pleasing. As such, this ratio is visible in many works of art and architecture such as in the Mona Lisa, the Notre Dame Cathedral, and the Parthenon. (For additional information regarding the Golden Ratio, please read the article found in the following link: https://www.elegantthemes.com/blog/design/the-golden- ratio-the-ultimate-guide-to-understanding-and-using-it ) SLIDESMANIA.COM B. The Fibonacci Sequence SLIDESMANIA.COM C. Mathematics for our World Mathematics for Organization Mathematics helps organize patterns and regularities in the world. Mathematics for Prediction Mathematics helps predict the behavior of nature and phenomena in the world. Mathematics for Control Mathematics helps humans exert control over occurrences in the world for the advancement of our civilization. SLIDESMANIA.COM Sequences 1. Arithmetic Sequence – ordered set of numbers that have a common difference between each term. If we add or subtract by the same number each time to make the sequence, it is an arithmetic sequence. 2. Geometric Sequence – ordered set of numbers that progresses by multiplying or dividing each term by a common ratio. If we multiply or divide by the same number each time to make the sequence, it is a geometric sequence. SLIDESMANIA.COM Sequences 3. Quadratic Sequence – a sequence of numbers in which the second difference between any two consecutive terms is constant. 4. Harmonic Sequence – a sequence of numbers such that the difference between the reciprocals of any two consecutive terms is constant. In other words, it is formed by taking the reciprocals of every term in an arithmetic sequence. SLIDESMANIA.COM Sequences 5. Mixed Sequence – a sequence where more than one arithmetic operations are part of the rule according to which the sequence is formed. For example, a rule that has addition, subtraction, and even one more operation as part of the rule that forms the sequence, is a mixed sequence. 6. Square Number Sequence – sequence made up of squares of integers. A square is a number multiplied by itself. 7. Cube Number Sequence – sequence made up of cube of integers. SLIDESMANIA.COM Sequences Examples: Analyze the given sequence for its rule and identify the next three terms. 1. 1, 10, 100, 1000 2. 2, 5, 9, 14, 20 3. 16, 32, 64, 128 4. 0, 1, 1, 2, 3, 5, 8 SLIDESMANIA.COM Sequences Additional Examples: 1. 1, 3, 5, 7, 9 2. 1, 4, 9, 16, 25 3. 3, 6, 9, 12, 15 4. 4, 10, 16, 22, 28 5. Find the missing terms in the following sequence: 8, __, 16, __, 24, 28, 32 6. What is the value of 𝑛 in the following number sequence? 12, 20, 𝑛, 36, 44 SLIDESMANIA.COM Sequences Additional Examples: 7. 0.22, 0.32, 0.42, 0.52, 0.62 8. 5, 3, 1, −1, −3 3 5 7 9 11 9. , , , , 4 4 4 4 4 10. 10, 22, 46, 94, 190 1 1 1 1 11. 1, , , , 2 3 4 5 12. 0, 10, 30, 70, 150  Thank you! SLIDESMANIA.COM