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Chapter 1 - Nature of Mathematics.pptx
- 2. Nature of Mathematics The emergence of digital technology has sparked a monumental rise in the rate at which we consume and produce data. In the fastpace society, how often have you stopped and appreciate the beauty of the things around you? Have you ever paused and pondered about the underlying principles that govern the universe? How about the contemplating about the processes and mechanism that make our lives easier, if not more comfortable? As national creatures, we need to identify and follow pattern whether consciously or subconsciously. Responding patterns feel natural like our brain is hardwired to response them. We will be looking at patterns and regularities in the world and how mathematics comes into play, both in nature and in human endeavor.
- 3. Patterns and Numbers in Nature and the World In the general sense of the word patterns are regular, repeated or recurring forms or designs. Patterns are commonly observed in natural objects such as sixfold symmetry of snowflakes. Lets take a look at this pattern
- 4. Patterns and Numbers in Nature and the World What is the next figure in the pattern below?
- 5. Patterns and Numbers in Nature and the World What is the next figure in the pattern below?
- 6. Snowflakes and Honeycombs Symmetry – the quality of being made up of exactly similar parts facing each other or around an axis.
- 7. Snowflakes and Honeycombs The figures on the previous slide is symmetric about the axis indicated by dotted line. The left and the right is exactly the same. This is known as line or bilateral symmetry. Take a look of these images:
- 8. Order of Rotation To compute the angle of rotation Angle of rotation = 3600 𝑛 Consider the image of snowflakes
- 9. Honeycombs Another marvel of nature’s design is the structure of a honeycomb. People have long wondered how bees, despite of their very small size, are able to produce such arrangement while humans would generally need the use of a ruler and compass to accomplish the same feat.
- 10. Honeycombs
- 11. Square Packing 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 x 100% = 𝛑 𝑐𝑚2 4 𝑐𝑚2 x 100% = 78.54%
- 12. Hexagonal Packing A = 𝑠𝑖𝑑𝑒2 • 3 4 = (2 𝑐𝑚)2• 3 4 = 4 𝑐𝑚2• 3 4 = 3 𝑐𝑚2
- 13. Hexagonal Packing This gives the area of hexagon as 6 3 𝑐𝑚2. There are 3 circles that could fit inside the hexagon (the whole circle in the middle, and 6 one-thirds of a circle), which gives the total area of 3𝛑 𝑐𝑚2. The percentage of the hexagon’s area covered by circles will be 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 ℎ𝑒𝑥𝑎𝑔𝑜𝑛 x 100% = 3𝛑 𝑐𝑚2 6 3 𝑐𝑚2 x 100% = 90.69%
- 14. Tigers’ Stripes and Hyenas’ Spots Patterns are also exhibited in the external appearances of animals. We are familiar with how the tigers looks – distinctive reddish- orange fur and dark stripes. Hyenas, another predator from Africa, are also covered in pattern of spots. These seemingly random design are believed to be governed by mathematical equations.
- 15. The Sunflower Another demonstration of how natures works to optimize the available spaces. This arrangement allows the sunflower seed to occupy the flower head in a way that maximize their access to sunlight and necessary nutrients
- 17. The Snail’s Shell We are also very familiar with spiral patterns. The most common spiral patterns can be seen in whirlpools and in the shell of snails and other similar mollusks. Snails are born with their shells called protoconch, which starts at fragile and colorless. This shells are harden as the snails consume calcium. As the snails grow, their shells also expand proportionately so that they can continue live inside their shells. This process result in a refined spiral structure that is even more visible when the shell is sliced. This figure is called an equiangular spiral, follows the rule that as the distance from the spiral center increases (radius), the amplitudes of the angles formed by the radaii to the point and the tangent to the point remain constant. This another example of how nature seems to follow a certain set of rules governed by mathematics
- 19. Flower Petals Flowers are easily considered as things of beauty. Their vibrant colors and fragrant odor make them very appealing as gifts or decorations. If you look more closely, you will note that different flowers have different numbers of petals.
- 20. Flower Petals Flowers with five petals are said to be most common (buttercup, columbine and hibiscus). Among those flowers with eight petals are clematis and delphinium, while ragwort and marigold are thirteen.
- 21. World Population As of 2017, it is estimated that the world population is about 7.6 billion. World leaders, sociologies and anthropologies are interested in studying population, including its growth. Mathematics can be used as a model population growth. Exponential growth formula 𝐴 = 𝑃𝑒𝑟𝑡, Where: A is the size of population after it grows P is the initial number of people r is the rate of growth t is the time e is Euler’s constant with an approximate value of 2.718 Plugging in values on this formula would result in population size after time r with the growth rate of r.
- 22. Population growth The exponential growth model 𝐴 = 30𝑒0.02𝑡describe the population of the city in the Philippines in thousand, t years after 1995. a. What was the population of the city in 1995? b. What will be the population in 2017? Solution: a. Since our exponential growth model describes the population t years after 1995, we consider 1995 as t=0 and then solve for A, our population size. b. We need to find A for the year 2017 . To find t, we subtract 2017 and 1995 to get t = 22 which we then plug in to our exponential growth.
- 23. Exercise The exponential growth model 𝐴 = 50𝑒0.07𝑡 describe the population of the city in the Philippines in thousand, t years after 1997. a. What is the population after 20 years? b. What is the population in 2037?
- 24. Exercise Set Determine what comes next in the given patterns 1. A, C, E, G, I, __ 2. 15 10 14 10 13 10 __ 3. 3 6 12 24 48 96 __ 4. 27 30 33 36 39 __ 5. 41 39 37 35 33 __ Substitute the given value in the formula 𝐴 = 𝑃𝑒𝑟𝑡 to find the missing quantity 6. P= 680,000, r = 12% per year, t = 8 years 7. A = 1,240,000, r = 8% per year, t = 30 years 8. A = 786,000, P = 247,000, t = 17 years
- 25. Solving for r • Inputting the scientist's information into the equation for exponential growth, A = Pert, he has: • 550 = 50er5 • The only unknown left in the equation is k, or the rate of exponential growth. • Solve for r • To begin solving for r, first divide both sides of the equation by 50. This gives you: • 550/50 = (50er5)/50, which simplifies to: • 11 = e_r_5
- 26. Solving for r • Next, take the natural logarithm of both sides, which is notated as ln(x). This gives you: • ln(11) = ln(er5) • The natural logarithm is the inverse function of ex, so it effectively "undoes" the ex function on the right side of the equation, leaving you with: • ln(11) = r5 • Next, divide both sides by 5 to isolate the variable, which gives you: • r = ln(11)/5
- 27. Fibonacci Sequence 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. Generating a sequence Analyze the given sequence and identify the next three terms a. 1 10 100 1000 b. 2 5 9 14 20
- 28. Solution a. Looking at he set of numbers, it can be observed that each term is a power of 10: 1 = 100, 10 = 101, 100 = 102, and 1,000 = 103. Following this rule, the next three terms are: 104 = 10,000, 105 = 100,000, and 106 = 1,000,000. b. The difference between the first and second terms (2 and 5) is 3. The difference between the second and third terms (5 and 9) is 4. The difference between the third and fourth terms (9 and 14) is 5. The difference between the fourth and the fifth terms is 6. Following this rule, it can be deduced that to obtain the next three terms, we should add 7, 8, 9, respectively, to the current term. Hence, the next three terms are 20+7 = 27, 27+8 = 35, 35 +9 = 44.
- 29. Check your progress 1 Analyze the given sequence for its rule and identify the next three terms. a. 16 32 64 128 b. 1 1 2 3 5 8
- 30. Fibonacci Sequence The sequence in Check Your Progress 1 Item B is a special sequence called the 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 reproduced. The problem involved having a single pair of rabbits and then finding out how many pairs of rabbits 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 at 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.
- 31. Fibonacci Sequence 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 8 + 13 = 21 0, 1, 1, , 3, 5, 8, 13, 21 Ex. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on….
- 32. Golden Ratio 1 1 = 1.0000 13 8 = 1.6250 2 1 = 2.0000 21 13 = 1.6154 3 2 = 1.5000 34 21 = 1.6190 5 3 = 1.6667 55 34 = 1.6177 8 5 = 1.6000 89 55 = 1.6182
- 33. Exercise Set Let Fib(n) be the nth term of the Fibonacci sequence, with Fib(1) = 1, Fib(2) = 1, Fib(3) = 2, and so on. 1. Find Fib (8). 2. Find Fib (19). 3. If Fib (22) = 17,711 and Fib(24) = 46,368, what is Fib 23)? 4. Evaluate the following sums: a. Fib (1) + Fib (2) = ______ b. Fib (1) + Fib (2) + Fib (3)= ______ c. Fib (1) + Fib (2) + Fib (3) + Fib (4) = _____ 5. Determine the pattern in the successive sums from the previous question. What will be the sum of Fib (1) + Fib(2) +…+ Fib(10)? Answer completely. 6. If you have a wooden board that is 0.75 meters wide, how long should you cut it such that the Golden Ratio is observed? Use 1.618 as the value of the Golden Ratio.
- 34. Mathematics for our world Mathematics help organize patterns and regularities in the world. Mathematics helps predicts the behavior of nature and phenomena in the world, as well as helps human exert control over occurrences in the world for advancement of our civilization.
- 35. Exercise Set 1. Vlad had a summer job packing sweets. Each pack should weight 200 grams. Vlad had to make 15 packs of sweets. He checked the weights, in grams, correct in the nearest gram. Following are his measurements: 212 206 203 206 199 196 197 197 209 206 198 191 196 206 207 What is the frequent data? 2. A certain study found that the relationship between the students’ exams scores (y) and the number of hours they spent studying (x) is given by the equation y = 10x + 45. Using this information, what will be the estimated score of a student who spent 4 hours studying? 3. The distance traveled by an object given its initial velocity and acceleration over a period of time is given by the equation d = v0t + ½at2. Find the distance traveled by an airplane before it takes off if it starts from rest and accelerates down a runway 3.50 cm/s2 for 34.5 s.
- 36. Chapter 1 Summary 1.1 Patterns and Numbers in Nature and the World Patterns are regular, repeated, or recurring forms or designs. Patterns are commonly observed in natural objects, such as the six-fold symmetry of snowflakes, the hexagonal structure and formation of honeycombs, the tiger's stripes and hyena's spots, the number of seeds in a sunflower, the spiral of a snail's shell, and the number of petals of flowers. Humans are hard wired to recognize patterns and by studying them, we discovered the underlying mathematical principles behind nature's designs. Exponential Growth Model Population growth and bacteria decay can be modeled by the exponential growth or decay formula A = Pert.
- 37. Chapter 1 Summary 1.2 The Fibonacci Sequence 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. The terms of a sequence could be generated by applying the rule to previous terms of the sequence. Fibonacci Sequence The Fibonacci sequence is formed by adding the preceding two numbers, beginning with 0 and 1. Ratios of two Fibonacci numbers approximate the Golden Ratio, which is considered as the most aesthetically pleasing proportion.
- 38. Chapter 1 Summary 1.3 Mathematics for our World Mathematics helps organize patterns and regularities in the world. Mathematics helps predict the behavior of nature and phenomena in the world, as well as helps humans exert control over occurrences in the world for the advancement of our civilization.