Chapter 1 Mathematics In Our World PDF

GEC3 MATHEMATICS IN THE MODERN WORLD CHAPTER 1: MATHEMATICS IN OUR WORLD MATHEMATICS IN THE MODERN WORLD is about mathematics as a system of knowing or understanding our surroundings. 1 NATURE OF MATHEMATICS Patterns and Numbers in Nature and the World 1.1 PATTERN In general sense of the wor...

GEC3 MATHEMATICS IN THE MODERN WORLD CHAPTER 1: MATHEMATICS IN OUR WORLD MATHEMATICS IN THE MODERN WORLD is about mathematics as a system of knowing or understanding our surroundings. 1 NATURE OF MATHEMATICS Patterns and Numbers in Nature and the World 1.1 PATTERN In general sense of the word, patterns are regular, repeated, or recurring forms or designs.  Studying patterns helps students in identifying relationships and finding logical connection to form generalization and make predictions as well as giving us a deeper vision of the universe in which we live, and of our own place in it. “Where there is life there is a pattern, and where there is a pattern there is math.” TAKE A LOOK! What do you think will be the next face in the sequence? ☺  ☺  ☺  __ WHAT IS THE NEXT FIGURE IN THE GIVEN PATTERN? ? (A) (B) WHAT NUMBER COMES NEXT? 1, 3, 5, 7, 9, _ WHAT NUMBER COMES NEXT? 1, 4, 9, 16, 25, _ WHAT NUMBER COMES NEXT? 1, 8, 27, 64, 125, _ A pattern may have a list of numbers in which a constant is added to get the succeeding terms. For other sequences, the terms are increasing because the number to be added is increasing in a predictable way. There are also number patterns whose terms are decreasing, because the number to be added is decreasing in an expected way. Let us refer to Table for the different set of examples. To generate the next term, Example 1 Example 2 the number to be added is repeating 10, 20, 30, 40, 50, 60, … 9, 18, 27, 36, 45, 54, … growing / increasing 14, 15, 17, 20, 24, 29, … 4, 5, 7, 10, 14, 19, 25, … shrinking / decreasing 118, 98, 73, 43, 8, -32, … 200, 190, 170, 140, … Patterns indicate a sense of structure and organization that it seems only humans are capable of producing these intricate, creative, and amazing formations. It is from this perspective that some people see an “intelligent design” in a way that nature forms. Can be observed even in the stars which move in circles across the sky. The weather seasons cycle, can also be seen in fish patterns like the spots and the stripes that attest to mathematical regularities in biological growth and form. RECALL: SYMMETRY It indicates that you can draw an imaginary line across an object and the resulting parts are mirror images of each other. IS IT SYMMETRIC? SNOWFLAKES AND HONEYCOMB LINE OR BILATERAL SYMMETRY This type of symmetry is evident on most animals, including humans. It is like looking in a mirror that makes right and left figures congruent or closely match. LEONARDO DA VINCI’S VITRUVIAN MAN SHOWING THE PROPORTIONS AND SYMMETRY OF THE HUMAN BODY. OTHER TYPES OF SYMMETRY DEPENDING ON THE NUMBER OF SIDES OR FACES THAT ARE SYMMETRICAL. Three-fold symmetry Five-fold symmetry Eight-fold symmetry Thirteen-fold symmetry N-fold symmetry 3-FOLD SYMMETRY Iris 3-FOLD SYMMETRY Spiderwort 3-FOLD SYMMETRY Trillium 5-FOLD SYMMETRY Buttercup 5-FOLD SYMMETRY Columbine 5-FOLD SYMMETRY Hibiscus 8-FOLD SYMMETRY Clematis 8-FOLD SYMMETRY Delphinium 13-FOLD SYMMETRY Ragwort 13-FOLD SYMMETRY Marigold 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. FORMULA: Angle of Rotation = 360°/n CONSIDER THE IMAGE OF SNOWFLAKES SINCE SNOWFLAKE IS A 6-FOLD SYMMETRY, Where n = 6 Angle of rotation = 360°/6 = 60° Hence, snowflake’s angle of rotation is 60°. PACKING PROBLEMS This involve finding the optimum method of filling up a given space such as a cubic or spherical container. The bees have instinctively found the best solution, evident in the hexagonal construction of their hives. These geometric patterns are not only simple and beautiful, but also optimal functional. Let us illustrate this mathematically. Suppose we have circles of radius 1 cm, each of which will then have an area of 1𝜋𝑐𝑚. We are then going to fill 2 a plane with these circles using square packing and hexagonal packing. SQUARE PACKING  For each packing, each square will have an area of 4𝑐𝑚. Note that 2 for each square, it can fit only one circle (4 quarters).  The percentage of the square’s area are covered by the circles will be 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒𝑠 1𝜋𝑐𝑚2 × 100% = 2 × 100% ≈ 78.54% 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 4𝑐𝑚 HEXAGONAL PACKING For hexagonal packing, we can think of each hexagon as composed of six equilateral triangles with side equal to 2 cm. The area for each triangle is given by 𝑠𝑖𝑑𝑒 2 3 𝐴= 4 3 3𝑠2 Note: Area of hexagon = 2  Then, 𝑠𝑖𝑑𝑒 2 3 (2𝑐𝑚)2 3 4𝑐𝑚2 3 𝐴= = = = 3𝑐𝑚2 4 4 4 This gives the area of hexagon as 6 3𝑐𝑚2 Looking at the figure, there are 3 circles that could fit inside one hexagon (1 whole circle in the middle and 6 one-thirds of a circle), which gives the total area as 3π𝑐𝑚 2  The percentage of the hexagon’s area covered by circles will be 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒𝑠 3𝜋𝑐𝑚2 × 100% = × 100% ≈ 90.69% 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 ℎ𝑒𝑥𝑎𝑔𝑜𝑛 6 3𝑐𝑚2 Therefore, by comparing the two percentages, we can clearly see that using hexagons will cover larger area than using squares. 1.2 SEQUENCE It is an ordered list of numbers called terms, that may have repeated values. The arrangement of these terms is set by a definite rule. GENERATING A SEQUENCE 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) 1, 1, 2, 3, 5, 8, FIBONACCI SEQUENCE It is named after the Italian mathematician Leonardo of Pisa, who was better known by his nickname Fibonacci.  He is said to have discovered this sequence as he looked at how a hypothesized group of rabbits bred and produced. The problem involved having a single pair of rabbits and then finding out how many pair of rabbit will be born in a year.  With the assumption that a new pair of rabbits is born each month and this new pair, in turn, gives birth to additional pairs of rabbits beginning of two months after they were born. He noted that the set of numbers generated from this problem could be extended by getting the sum of the two previous terms. Starting with 0 and 1, the succeeding terms in the sequence can be generated by adding the two numbers that came before the term: 0+1=1 0, 1, 1 1+1=2 0, 1, 1, 2 1+2=3 0, 1, 1, 2, 3 2+3=5 0, 1, 1, 2, 3, 5 3+5=8 0, 1, 1, 2, 3, 5, 8 5 + 8 = 13 0, 1, 1, 2, 3, 5, 8, 13... 0, 1, 1, 2, 3, 5, 8, 13,... FIBONACCI IN PASCAL TRIANGLE If you have noticed, adding the two consecutive terms will result to another term of the Fibonacci sequence, Hence, it could be expressed as: Fn = Fn-1 + Fn-2 Where: Fn = Fibonacci number Fn-1 = previous term Fn-2 = the term before Fn-1 Apply this concept to be able to find the 10th to 15th term of the Fibonacci Sequence. Given the 8th and 9th term of the sequence, 21 and 34 respectively. 10th term: ____________ 11th term: ____________ 12th term: ____________ 13th term: ____________ 14th term: ____________ 15th term: ____________ It is interesting to note that the ratios of successive Fibonacci numbers approach the number ɸ (phi), also known as the Golden Ratio. This is approximately equal to 1.618. GOLDEN RATIO 1 = 1.0000 1 2 ▪ = 2.0000 1 3 ▪ = 1.5000 2 5 ▪ ≈ 1.6667 3 8 ▪ = 1.6000 5 GOLDEN RATIO 13 = 1.6250 8 21 ▪ ≈ 1.6154 13 34 ▪ ≈ 1.6190 21 55 ▪ ≈ 1.6177 34 89 ▪ ≈ 1.6182 55  8 𝑐𝑚2 13 𝑐𝑚2 1 2 𝑐𝑚2 1 5 𝑐𝑚2 3 𝑐𝑚2  8 𝑐𝑚2 13 𝑐𝑚2 1 2 𝑐𝑚2 1 5 𝑐𝑚2 3 𝑐𝑚2 Have you considered if you’re being asked what is the 80th term of the Fibonacci Sequence? It may take you an hour or more to find it. However, one may calculate the Fibonacci number using the Golden Ratio. It is usually denoted using the Greek letter “phi” 𝜑 or ∅. The Golden Ratio is a special number, which is approximately equal to 1.618034 and may be used to find the nth term of a Fibonacci Sequence using the Binet’s formula. 𝑛 𝑛 1+ 5 1− 5 − 𝐹𝑛 = 2 2 5 Use this formula to find the following terms of the Fibonacci Sequence. a. 15th term: _______________ b. 20th term: _______________ c. 25th term: _______________ Aside from the famous painting Mona Lisa, Leonardo da Vinci was also known for his Vitruvian Man. It was one of the most important works of the Italian Renaissance. According to Vitruvius (1492), the 15th century drawing was also known as “the proportions of the human body”. A LOGARITHMIC SPIRAL