GE-Math- mathematics in a modern worldL1
- 2. The Nature of Mathematics Module 1
- 3. O B J E C T I V E S Identify patterns in nature and regularities of the universe; Determine the importance of patterns in life; Explain the importance of utilizing mathematical models; Solve basic problems involving sequences and mathematical models
- 4. regular, repeated, or recurring forms or designs Patterns:
- 5. Activity 1: Video Watching https://youtu.be/9mozmHgg9Sk https://youtu.be/SjSHVDfXHQ4 https://www.youtube.com/w atch?v=7GiKeeWSf4s Fibonacci Sequence Golden Ratio Tesselations To mathematically explain patterns in the world, watch a video which contains discussions on the following math concepts:
- 6. Guide Questions 3 points 01 What math concept is fully discussed on each video? 5 points 02 What is about these concepts? Cite few more real-world cases or examples that would show the concepts. 7 points 03 What are your insights about these mathematical truths or certainties?
- 7. Guide Questions 10 points 04 What did Galileo mean when he said, “Mathematics is the alphabet by which God has written the universe”? Do you agree on this adage? Why?
- 8. Fibonacci Sequence This sequence begins with the numbers 1 and 1 or 0 and 1, and then each subsequent number is found by adding the two previous numbers Named for the famous mathematician, Leonardo Fibonacci, this number sequence is simple, yet profound pattern.
- 9. Fibonacci Sequence Illustration After 1 and 1, the next number is 2, that is, 1+1. The next number is 3, taken from 1+2, and then 5, taken from 2+3 and so on. Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, … 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …
- 10. Fibonacci Sequence in Nature
- 11. Fibonacci Sequence in Nature
- 12. Golden Ratio The "golden ratio" is a unique mathematical relationship. Two numbers are in the golden ratio if the ratio of the sum of the numbers (a + b) divided by the larger number (a) is equal to the ratio of the larger number divided by the smaller number (a/b).
- 13. Golden Ratio The golden ratio is about 1.618, and represented by the Greek letter phi, Φ.
- 19. Golden Ratio The ratios of sequential Fibonacci numbers (2/1, 3/2, 5/3, etc.) approach the golden ratio. In fact, the higher the Fibonacci numbers, the closer their relationship is to 1.618.
- 20. Golden Ratio The golden ratio is sometimes called the "divine proportion," because of its frequency in the natural world. The number of petals on a flower, for instance, will often be a Fibonacci number.
- 21. Golden Ratio The seeds of sunflowers and pine cones twist in opposing spirals of Fibonacci numbers.
- 22. Golden Ratio Even the sides of an unpeeled banana will usually be a Fibonacci number—and the number of ridges on a peeled banana will usually be a larger Fibonacci number.
- 23. Tesselations Tessellation (Tiling) is a shape that repeats to form a pattern. It is created when a shape is repeated over and over again covering a plane without any gaps or overlaps.
- 25. Other Patterns in the World
- 29. Concentric Circles in Nature
- 30. Concentric Circles in Nature
- 31. Determine the next 3 terms in the following sequences. 1. 13, 21, 34, 55, __, __, … 2. 55, 89, 144, 233, __, __, … 3. 2, 3, 5, 8, __, __, … 4. 21, 34, 55, 89, __, __, … 5. 89, 144, 233, 377, __, __, … 6. 34, 55, 89, 144, __, __, … 7. 8, 13, 21, 34, __, __, … 8. 3, 5, 8, 13, __, __, … A c t i v i t y 2 Number Patterns
- 32. Determine what completes the series. Activity 3: Abstract Reasoning A C C
- 33. S t u d y i n g P A T T E R N S Helps in identifying relationships Aids in finding logical connections to form generalizations and make predictions
- 34. 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 5.Let Fib(n) be the nth term of a Fibonacci sequence, with Fib(1) = 1, Fib(2) = 1, Fib(3) = 2. Find: a. Fib(8) b. Fib(19) Activity 4: Generating a Sequence
- 35. Mathematics can be used to model population growth. Using the 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 time, and e is Euler’s constant, ≈ 2.718 World Population
- 36. Activity 5: Population Growth The exponential growth model formula, 𝑨 = 𝟑𝟎𝒆𝟎.𝟎𝟐𝒕 , describes the population of a city in the Philippines in thousands, t years after 1995. a. What was the population of the city in 1995? b. What will be the population by the end of 2021?
- 37. 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. 𝑨 = 30𝑒0.02𝑡 = 30𝑒0.02(0) = 30𝑒0 = 30 1 𝑨 = 𝟑𝟎 Therefore, the city population in 1995 was 30,000.
- 38. Solution: b. We need to find A by the end of 2021. To find t, we subtract 2021 and 1995 to get t = 26. Hence, 𝑨 = 30𝑒0.02𝑡 = 30𝑒0.02(26) = 30𝑒0.52 = 30(2.718)0.52 = 30(1.68194) 𝑨 = 𝟓𝟎. 𝟒𝟓𝟖𝟐 Therefore, the city population would be about 50,458 by the end of 2021.
- 39. Assessment Task #1 Answer the following comprehensively: 1. If Fib(22) = 17,711 and Fib(24) = 46,368, what is Fib(23)? 2. 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) = ______ 3. Determine the pattern in the successive sums from the previous question. What will be the sum of Fib(1) + Fib(2) +…+ Fib(10)? 4. Suppose the population of a certain bacteria in a laboratory sample is 100. If it doubles in population every 6 hours, what is the growth rate? How many bacteria will there be in two days?
- 40. References Mathematics in the Modern World by Rex Bookstore, 2018 https://youtu.be/SjSHVDfXHQ4 https://youtu.be/9mozmHgg9Sk https://www.youtube.com/watch?v=7GiKeeWSf4s https://www.mathsisfun.com/numbers/fibonacci-sequence.html