Mathematics in Our World PDF

Mathematics in our World MODULE NO. 1 Learning Outcome: 1. Appreciate and recognize patterns in nature, the importance of mathematics in one's life, nature of mathematics, and how it is expressed, represented and used in human endeavors. 2. Predict the number or pattern by applying the concepts of...

Mathematics in our World MODULE NO. 1 Learning Outcome: 1. Appreciate and recognize patterns in nature, the importance of mathematics in one's life, nature of mathematics, and how it is expressed, represented and used in human endeavors. 2. Predict the number or pattern by applying the concepts of patterns and sequence and solve mathematical problems with the used of golden ratio.  What is Mathematics? Mathematics has been defined as the study of numbers and arithmetic operations. Other define mathematics as a set of tools or a collection of skills that can be applied to questions of “how many” or “how much”. Still others view it is a science which involves logical reasoning, drawing conclusions from assumed premises, systematized knowledge, and strategic reasoning based on accepted rules, laws, or probabilities. Mathematics has also been defined as an art which studies patterns for predictive purposes or as a specialized language which deals with form, size, and quantity. Source: Essential Mathematics for the Modern World by Rizaldi Nocon and Ederlina Nocon PATTERNS In the general sense of the word, patterns are regular, repeated, or recurring forms or designs. Guess what’s next! ? Did you know why? The rule behind for this series of figure is the rule of symmetry? If we are going to delete the left portion part of each figure, you will notice that we form a natural or counting numbers such as; 1; 2; 3; 4; 5; and 6.  Guess what’s next! 1. A, A, C, C, E, E, G, ? 2. 1, -1, 2, -2, 3, -3, ? 3. A, C, E, G, I, ? 4. 15, 10, 14, 10, 13, 10, ? 5. 27, 30, 33, 36, 39, ?  Some pictures with patterns!!! Source: https://pixabay.com/en/photos/pattern/ Think! Think! Think! WHAT FIGURE SHOULD BE REPLACED IN A QUESTION MARK? Explanation: All triangles “move” slightly counter-clockwise and outside. A good solving tip for the first sample abstract reasoning question would be to try and disassemble the complete figure to its elements (triangles) and to focus each time on one of the elements. Think! Think! Think! WHAT FIGURE SHOULD BE REPLACED IN A QUESTION MARK? Explanation: The logic behind this free abstract reasoning example: An X shape is dotted with black and white dots. Both sets of dots are independent and follow a similar pattern. In each frame, a black dot is added counter-clockwise in the angles of the X shape, until all the angles are occupied. Then a dot is reduced, also counter-clockwise. The same pattern occurs with the white dots, only in a clockwise manner. Examining the changes before and after, the “question mark” figure should look the same as in frame 2, only with an additional black dot (making all four black dots present) and an additional white dot in the upper right corner, as determined by the pattern. Think! Think! Think! WHAT COMES NEXT? Explanation: The next frame after each step portrays a mirror image of the previous frame. In addition, every two steps a shape is added to the frame. Answer choice 3 is a mirrored version of frame 5 Now, it’s your turn! Think! Think! Think! WHAT WORD COMES NEXT? SNAKE; SHAKE ; SNORE; SHORE; SNIFT ; ? KNIFE; KNIVES ; LIFE; LIVES; WIFE; ? WHAT WORD (EVEN WITHOUT MEANING )COMES NEXT? SANTOL; SANTLO; SANLTO; SALNTO; ? Think! Think! Think! WHAT NUMBER COMES NEXT? a. 1, 3, 5, 7, 9, ? b. 1, 2, 4, 8, 16, ? c. 1, 2, 4, 7, 11, 16, ? d. 1, 4, 9, 16, 25, 36, ? e. 2, 4, 8, 10, 20, ? SEQUENCE What is a sequence? A sequence is an ordered list of numbers, called terms, that may have repeated values. The arrangement of these terms is set by a definite rule.  Illustrations: Cite the definite rule! a. 2; 5; 8; 11; 14; ? Ans: 17 e. 1; 1; 2; 6; 24; 120; ? b. 5; 14; 27; 44; 65; ? Ans: 720 Ans: 90 f. 2; 6; 9; 27; 30; ? c. 2; 7; 24; 59; 118; 207; ? Ans: 90 Ans: 332 g. 3; 6; 15; 21; 27; ? d. 5; 6; 9; 15; ?; 40 Ans. 36 Ans: 25 What is a Fibonacci Sequence? The Fibonacci numbers or sequence are the numbers in the following integer sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … And on this sequence we could see that there would be what we called the Golden Ratio. What number comes next and could be replaced in the question mark? 1, 1, 2, 3, 5, 8, ?, 21, 34, 55, ?, 144, 233, 377, 610, ?, … 13 89 987  Fibonacci Sequence in our Nature 1 1 2 3 5 8 13 21 34 55 https://www.google.com/search?q=picture+of+flower+with+three+petals&tbm=isch&ved=2ahUKEwjG9oO17sXyAhUIB5QKHRYNDlIQ2-cCegQIABAA&oq=pictur e+of+flower+with+three+petals&gs_lcp=CgNpbWcQAzoFCAAQgAQ6BggAEAcQHlC12ANYo_EDYN_4A2gBcAB4AIABSogB4ASSAQE5mAEAoAEBqgELZ3dzLXdpei1p bWfAAQE&sclient=img&ei=ZugiYcarCoiO0ASWmriQBQ&rlz=1C5CHFA_enPH940PH940  2 1 1 3 5 8 https://www.google.com/search?q=picture+of+plant+with+thirteen+leaves&tbm=isch&ved=2ahUKEwillKWk8sXyAhU7KqYKHQRNCkwQ2-cCegQIABAA &oq=picture+of+plant+with+thirteen+leaves&gs_lcp=CgNpbWcQA1CilgNY4LQDYJy8A2gAcAB4AIABTIgB-weSAQIxNZgBAKABAaoBC2d3cy13aXotaW1nw AEB&sclient=img&ei=dOwiYaWKMbvUmAWEmqngBA&rlz=1C5CHFA_enPH940PH940 FIBONACCI SPIRAL  By definition, the first two numbers in the Fibonacci sequence are 1 and 1, and each subsequent number is the sum of the previous two. In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation: To find the nth Fibonacci number without using the recursion formula, use the formula below with the use of your calculator. This form is known as the Binet form of the nth Fibonacci numbers.  Illustrative Examples: Question 1: Find the next number in the fibonacci series 1, 1, 2, 3, 5, 8, 13,...... ? Solution: The fibonacci formula is given as, Fn = Fn-1 + Fn-2 F8 = F 7 + F 6 F8 = 13 + 8 F8 = 21  Illustrative Examples: Question 2: Find the next number in the fibonacci series 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...... ? Solution: Here, F1 = 0 , F2 = 1 and F3 = 1. The fibonacci formula is given as, Fn = Fn-1 + Fn-2 F12 = F11 + F10 F12 = 55 + 34 F12 = 89 Illustrative Examples: Question 3. What is the 10th term in a Fibonacci sequence using the Binet form? Use your calculator! Solution: The binet form is in a form of:  According to Ivanka Stipancic-Klaic and Josipa Matotek of the University of Osijek, Croatia, the Golden ratio fascinates and intrigues not only mathematicians, but also artists, architects, biologists, philosophers and musicians. This golden ratio was first studied by the ancient Greek because of its frequent appearance in geometry. The development of the idea of the golden ratio is usually attributed to Pythagoras (580-497 BC) and his students. It is often represented by the Greek letter τ (tau) which means “the cut” or “the section” in Greek. But, Mark Barr (early 18th century) represented the golden ratio as ϕ (phi) because it is the first letter in the name of Greek architect and sculptor Phidias who’s work often symbolized the golden ratio. THE GOLDEN RATIO The ratio is came from to a line segment divided according to the golden ratio where the larger part is to the smaller part as the whole part is to the larger part, thus; But the value of Golden ratio can be seen also in the Fibonacci number. Sequence Fibonacci Number Method Process Result 1st 1 Divide 2nd to 1st 1/1 1 2nd 1 Divide 3rd to 2nd 2/1 2 3rd 2 Divide 4th to 3rd 3/2 1.5 4th 3 Divide 5th to 4th 5/3 1.6666... 5th 5 Divide 6th to 5th 8/5 1.6 6th 8 Divide 7th to 6th 13/8 1.625 7th 13 Divide 8th to 7th 21/13 1.6153844... 8th 21 Divide 9th to 8th 34/21 1.619047... 9th 34 Divide 10th to 9th 55/34 1.6176470588235 10th 55 Divide 11th to 10th 89/55 1.61818... 11th 89 Divide 12th to 11th 144/89 1.6179775280898 12th 144 Divide 13th to 12th 233/144 1.618055... 13th 233 Divide 14th to 35th 377/233 1.6180257510729 And so on... 377... And so on......... TRIVIA  The EPIDAURUS THEATRE The Epidaurus Theatre which was designed by Polykleitos the Younger in the 4th century BC used the golden ratio. The auditorium was divided into two parts; one had 34 rows and the other 21 rows (followed the Fibonacci number). The angle between the theatre and the stage divides a circumference of the basis of an amphitheatre in the ratio 222.5o : 137.5o (the golden proportion).  PARTHENON The famous Greek temple, the Parthenon in the Acropolis in Athens includes golden rectangles in many proportions. But, it is necessary to underline the fact that there are no originals plans for the Parthenon, and there is no documentary evidence that this was deliberately designed. The temple is damaged and all the measures are only approximate.  CN TOWER The CN tower in Toronto is one of the highest tower in the world. The height of the tower is 553.33 meters, and the height of the glass floor is 342 meters. The proportion of these two values is 1.617924 which followed in the golden ratio.