College of Teacher Education BSEd Mathematics Midterm PDF

D COLLEGE OF TEACHER EDUCATION BSEd A L...

D COLLEGE OF TEACHER EDUCATION BSEd A L U B Lesson 2: Mathematics in Our World H Mathematics is the science of patterns, and nature exploits just about every pattern that A S there is. A – Ian Stewart A N Introduction N G Look carefully at the world around you and you might start to notice that nature is filled with many different types of patterns. All around us, we see a great diversity of L living things, from the microscopic to the gigantic, from the simple to the complex, from U bright colors to dull ones. One of the most intriguing things we see in nature is patterns. N We tend to think of patterns as sequences or designs that are orderly and that repeat but we S O can also think of patterns as anything that is not random. In this lesson you will learn D about different patterns in nature and their mathematical basis. N Text: Mathematics in the Modern World, Aufmann, et. al., Chapter 1 G Reference: Mathematics in the Modern World , Calingasan et. al., Chapter 1 S Lesson Learning Outcomes A N By the end of this lesson, you should be able to: ❖ recognize different patterns found in objects, plants, and animals and that can be P A modelled mathematically. B ❖ Appreciate the beauty of nature through a mathematical point of view L O ❖ Explain why patterns in nature follow mathematical principles. Patterns in Nature Millions of patterns can be found in the environment. These patterns occur in various forms and in different contexts which can be modelled mathematically. Some examples are symmetries, trees, spirals, waves, tessellations, stripes, meanders, cracks, and many more. Symmetry Symmetry comes from a Greek word which means “to measure together" Mathematically, symmetry means that one shape becomes exactly like another shape when you move it in some way. furniture, flip, or slide. For two objects to be symmetrical, they must be of the same size and shape, with one object having a different orientation from the first. Not all objects have symmetry. If an object is not symmetrical, it is called asymmetric. GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U Symmetry indicates that you can draw an imaginary line across an object and the B resulting parts are mirror images of each other. The figure on the left is symmetric about H A the axis indicated by the dotted line. Note that the left and right portions are exactly the S same. This type of symmetry, known as line or bilateral symmetry, is evident in most A animals, including humans. Look in a mirror and see how the left and right sides of your A face closely match. N There are other types of symmetry depending on the number of sides or faces that are N symmetrical. Most of them can be found in certain organisms. Take a look at these G images. L U N S O D N G S A N P A B L O The fruit fly has only one possible line of symmetry. Note that if you rotate the lily flower above by several degrees you can still achieve the same appearance as the original position. This is known as radial or rotational symmetry. The smallest angle that a figure can be rotated while sill preserving the original formation is called the angle of rotation. The Streptococcus bacteria exhibits spherical geometry as anyway you cut it along its center, it will generate two identical halves. An endless, or great but finite, number of symmetry axes can be drawn through the body. Drill 1 Classify the uppercase letters of the English alphabet as to the number of lines of symmetry they can have. No Line of Symmetry Only 1 Line of Symmetry 2 or More Line of Symmetry GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U Order of Rotation B H A figure has a rotational symmetry of order n (n-fold rotational symmetry) if 1/n of A a complete turn leaves the figure unchanged. To compute for the angle of rotation, we use S the following formula A A 360° N 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 = 𝑛 N Consider this image of a snowflake. It can be G observed that the patterns on a snowflake repeat six times, indicating that there is a six-fold symmetry. L To determine the angle of rotation, we simply U divide 360° by 6 to get 60°. Many combinations and N S complex shapes of snowflakes may occur, which O lead some people to think that "no two are alike". If D you look closely, however, many snowflakes are not perfectly symmetric due to the effects of N humidity and temperature on the ice crystal as it forms. G Drill 2 S A Find the angle of rotation of the following images. Write the answer on the blank. N P A B L O Spiderwort with three-fold symmetry ____ Starfish with five-fold symmetry ____ Honeycomb Tiling Another marvel of nature's design is the structure and shape of a honeycomb. People have long wondered how bees, despite 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. It is observed that such formation enables the bee colony to maximize their storage of honey using the smallest amount of wax. GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U B H A S A A N N G L U N S O D You can try it out for yourself. N Using several coins G of the same size, try S to cover as much A area of a piece of N paper with coins. If you arrange the P coins in a square A B formation, there are still plenty of spots that are exposed. Following the hexagonal L formation, however, with the second row of coins snugly fitted between the first row of O coins, you will notice that more area will be covered. Drill 3 From the image above, how many more circles does hexagonal packing have compared to square packing? Translating this idea to three-dimensional space, we can conclude that hexagonal formations are more optimal in making use of the available space. These are referred to as packing problems. Packing problems 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 optimally functional. Let us illustrate this mathematically. Suppose you have circles of radius 1 cm. each of which will then have an area of π cm. We are then going to fill a plane with these circles using square packing and hexagonal packing. GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U B H A S A A N N G For square packing, the square will have an area of 16 cm2. Note from the figure that L the square can fit only four circles (1 whole, 4 halves and 4 quarters). The percentage of U the square's area covered by circles will be N S 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒𝑠 4𝜋 𝑐𝑚2 × 100% = × 100% ≈ 78.54% O 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑞𝑢𝑎𝑟𝑒 16 𝑐𝑚2 D N For hexagonal packing, we can think of each hexagon as composed of six equilateral G triangles with side equal to 2 cm. The area of each triangle is given by S 𝑠𝑖𝑑𝑒 2 √3 (2 𝑐𝑚)2 ∙ √3 4 𝑐𝑚2 ∙ √3 A 𝐴𝑟𝑒𝑎 = = = = √3 𝑐𝑚2 N 4 4 4 P This gives the area of the hexagon as 6√3 cm2. Looking at the figure, there are 3 A circles that could fit inside one hexagon (1 whole and 6 one-thirds), which gives the total B area as 3π cm2. The percentage of the hexagons area covered by the circles will be L O 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒𝑠 3𝜋 𝑐𝑚2 × 100% = × 100% ≈ 90.69% 𝑎𝑟𝑒𝑎 𝑜𝑓 ℎ𝑒𝑥𝑎𝑔𝑜𝑛 6√3 𝑐𝑚2 Comparing the two percentages, we can clearly see that using hexagons will cover a larger area than when using squares. Drill 4 Why is hexagonal tiling better than square tiling? Animal Prints and Patterns Patterns are also exhibited in the external appearances of animals. We are familiar with how a tiger looks-distinctive reddish-orange fur and dark stripes. Hyenas, another predator from Africa, are also covered in patterns of spots. These seemingly random designs are believed to be governed by mathematical equations. According to a theory by Alan Turing chemical reactions and diffusion processes in cells determine these growth patterns. GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U Patterns, as Turing saw them, depend on two components: interacting agents and agent B diffusion. He demonstrated that if you combine these two components in just the right H A way, diffusion could actually drive the system to form spots and stripes. This idea was so S far ahead of its time that we are still working on unravelling its complexity 65 years later. A More recent studies addressed the question of why some species grow vertical stripes A while others have horizontal ones. A new model by Harvard University researchers N predicts that there are three variables that could affect the orientation of these stripes – the N substance that amplifies the density of stripe patterns; the substance that changes one of G the parameters involved in stripe formation; and the physical change in the direction of the origin of the stripe. L U N S O D N G S A Drill 5 N How does mathematics influence the formation of animal patterns like stripes and spots? P A B L O Number Patterns Mathematics is very useful in making predictions. Working with number patterns leads directly to the concept of functions in mathematics. It is important that students are able to recognize number patterns to help them develop their problem-solving skill. In your tenth grade you have learned patterns generated through different ways. Arithmetic sequences are generated by adding a common difference to the next term. Geometric sequences are generated by multiplying a common ratio. Other sequences you have learned were harmonic and Fibonacci sequences. You will know more about Fibonacci sequence in the next lesson. The following table summarizes the properties of the most common progression types. Type of Definition Example Progression It is a sequence of numbers such that the Arithmetic difference between the consecutive terms is 2, 5, 8, 11, 14, 17, 20, … Sequence constant. The constant is called a common difference. GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U It is a sequence of numbers where each term B Geometric after the first is found by multiplying the H 3, 6, 12, 24, 48, 96, 192, … A Sequence previous one by a fixed, non-zero number S called the common ratio. A Harmonic It is a sequence formed by taking the 1 1 1 1 1 1 1 A , , , , , , ,… N Sequence reciprocals of an arithmetic sequence. 4 9 14 19 24 29 34 It is a sequence such that each number is the N Fibonacci G sum of the two preceding ones, starting from 0, 1, 1, 2, 3, 5, 8, … Sequence 0 and 1. L U N Drill 6 S Identify the type of progression shown by the following sequences. O D 5 5 _______________1. -3, 0, 3, 6, 9, … _______________2. 20, 10, 5, 2 , 4, … N 1 1 1 1 1 _______________3. 34, 55, 89, 144, 233, … _______________4. 10 , 15 , 20 , 25 , 30, … G 1 1 1 1 1 3 6 S _______________5. , , , , ,… _______________6. 6, 3, 2, 2 , 5, … 2 6 18 54 162 A N Fibonacci Sequence in Nature P Leonardo of Pisa, also known as Fibonacci (c. 1170-1250) is one of the best-known A B mathematicians of medieval Europe. In 1202, after a trip that took him to several Arab and L Eastern countries, Fibonacci wrote the book Liber Abaci. In this book Fibonacci explained O why the Hindu-Arabic numeration system that he had learned about during his travels was a more sophisticated and efficient system than the Roman numeration system. This book also contains a problem created by Fibonacci that concerns the birth rate of rabbits. Here is a statement of Fibonacci's rabbit problem. “At the beginning of a month, you are given a pair of new-born rabbits. After a month the rabbits have produced no offspring; however, every month thereafter, the pair of rabbits produces another pair of rabbits. The offspring reproduce in exactly the same manner. If none of the rabbits dies, how many pairs of rabbits will there be at the start of each succeeding month?” The solution to this problem is a sequence of numbers that we now call the Fibonacci sequence. The following figure shows the numbers of pairs of rabbits on the first day of each of the first five months. The large rabbits represent mature rabbits that produce another pair of rabbits each month. The numbers in the blue region 1, 1, 2, 3, 5, 8 are the first six terms of the Fibonacci sequence. GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U Fibonacci B discovered H A that the S number of A pairs of A rabbits for N any month N after the first G two months can be L determined U by adding the N S numbers of O pairs of D rabbits in each of the two previous months. For instance, the number of pairs of rabbits at the start of N the sixth month is 3 + 5 = 8. G A recursive definition for a sequence is one in which each successive term of the S sequence is defined by using some of the preceding terms. If we use the mathematical A notation Fn to represent the nth Fibonacci number, then the numbers in the Fibonacci N sequence are given by the following recursive definition. P 𝐹1 = 0, 𝐹2 = 1, 𝑎𝑛𝑑 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 𝑓𝑜𝑟 𝑛 ≥ 3 A B Drill 7 L O From the definition list down the first 20 Fibonacci numbers. ________________________________________________________________________ The numbers in the Fibonacci sequence often occur in nature. For instance, the seeds on a sunflower are arranged in clockwise or counterclockwise spirals that curve from the center of the sunflower's head to its outer edge. In many sunflowers, the number spirals are consecutive Fibonacci numbers. For instance, in the sunflower shown at the left, the numbers of spirals are 21, 34 and 55 which are consecutive Fibonacci numbers. It has been conjectured that the seeds on a sunflower grow in spirals that involve Fibonacci numbers because this arrangement forms a uniform packing. At any stage in the sunflower's development, its seeds are packed so that they are not too crowded in the center and not too sparse at the edges. This allows the sunflower seeds to occupy the flower head in a way that maximizes the access to light and necessary nutrients. GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U B H A S A A N N G Pineapples have spirals L formed by their hexagonal nubs. U N The nubs on many pineapples S form 8 spirals that rotate O diagonally upward to the left, 13 D spirals that rotate diagonally upward to the right and 21 N G spirals that are almost vertical. The numbers 8, 13 and 21 are S consecutive Fibonacci numbers. What other objects from nature exhibit this property? A N Flowers are easily considered as things of beauty. Their vibrant P colors and fragrant odors make them A very appealing as gifts or B L decorations. If you look more O closely, you will note that different flowers have different number of petals. Some flowers have three petals such as the iris and the trillium Flowers with five petals are said to be the most common. These include buttercup, columbine, and hibiscus. Among those flowers with eight petals are clematis and delphinium, while ragwort and marigold have thirteen. Some sunflowers have 21 petals. These numbers are all Fibonacci numbers. Drill 8 Why do many plants follow the Fibonacci numbers? What benefits do they derive from it? GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U Spirals B H We are also very familiar with spiral A patterns. Spiral patterns can be seen in S whirlpools, plants, and in the shells of snails A and other similar molluscs. Most spirals in A N nature are logarithmic or equiangular spiral which follows the rule that as the N distance from the spiral center increases G (radius), the amplitudes of the angles formed by the radii to the point and the L U tangent to the point remain constant. N The iconic image of Fibonacci numbers in S nature is the Fibonacci spiral or golden spiral O D which is a type of logarithmic spiral. It’s made up of square blocks arranged in order by N Fibonacci numbers. The vertices are then joined G together as show in the image on the left to S generate the spiral. A N Snails are born with their shells, P called protoconch, which start out as A B fragile and colorless. Eventually, L these original shells harden as the O snails consume calcium. As the snails grow, their shells also expand proportionately so that they can continue to live inside their shells This process results in a refined spiral structure that is even more visible when the shell is sliced. This is another example of how nature seems to follow a certain set of rules governed by mathematics. Spirals can be seen from the microscopic to the astronomic. Some cells inside organism arrange in spiral formation. Objects in outer space can also exhibit spirals. There also many other objects in between that show spirals. The following images show some of the places they can be found. GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U B H A S A A N N G L U Cancer cell (HeLa) undergoing cell Rose flower N division S O D N G S A N P A B L O Human ear Hurricane Irene The Whirlpool Galaxy What other objects in nature have spirals? GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U Drill 9 B H Connect the needed point to construct a Fibonacci spiral. A S A A N N G L U N S O D N G Golden Ratio S A The ratios of successive Fibonacci numbers approach the N number φ (Phi), also known as the golden ratio. This is approximately equal to 1.618. P A The golden ratio can also be expressed as the ratio B between two numbers, if the latter is also the ratio between the L sum and the larger of the two numbers. Geometrically, it can O also be visualized as a rectangle perfectly formed by a square and another rectangle, which can be repeated infinitely inside each section. This rectangle is called the golden rectangle. The figure above is a golden rectangle. If (a + b) is the length of the whole side, a is the longer part and b is the shorter part. The following formula gives the value of the golden ratio. 𝑎+𝑏 𝑎 = = 𝜑 ≈ 1.61803 𝑎 𝑏 Shapes and figures that bear this proportion 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. In fact, the human DNA molecule also contains Fibonacci numbers, being 34 ångstroms long by 21 ångstroms wide for each full cycle of the double helix spiral. (1 ångstrom = 100 meter or 0.1 nanometer). GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U B H A S A A N N G L U N S O D N G S A N P A B L O Drill 10 Confirm if the ratios of the successive Fibonacci numbers approach the value of the golden ratio. 0 1 2 3 5 8 13 21 34 55 89 = = = = = = = = = = = 1 1 1 2 3 5 8 13 21 34 55 ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U To learn more, watch the following video on https://youtu.be/VE_RU0fNjt0 (Part 1) and B https://youtu.be/n2WHNMfRmHE (Part 2). H A S Answer the following questions about the lesson you have read. A A 1. Which objects in nature that exhibits mathematical pattern did you like the most? N Why? N G L U N 2. Why do you think is it important for organisms to have symmetry? S O D N G 3. How do you think bees determine the best shape to use as tiling for honeycombs? S A N P A 4. How do you think Alan Turing was able to come up with the mathematical idea of B animal prints and patterns? L O 5. Many consider mathematics as beautiful. Do you agree? Why or why not? GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U Lesson Exercises B H A. Draw all possible line of symmetries on the following figures. Identify also the A symmetry exhibited by each figure (asymmetry, bilateral sym., radial sym., S spherical sym.). Indicate also the angle of rotation if applicable. A A N N G L U N S O D N G S A N P A B L O B. Write TRUE if the statement is correct otherwise write FALSE. __________1, The Fibonacci sequence was first used by Leonardo of Pisa. __________2. If n is even, then Fn is an odd number. __________3. 2Fn – Fn-2 = Fn+1 for n ≥ 3 __________4. 2Fn > Fn+1 for n ≥ 3 __________5. 2Fn + 4 = Fn+3 for n ≥ 3 C. Check if the following items approximate the golden ratio. Up to 3 decimal places. 89 1 √5 55 1. 2. + 3. 55 2 2 34 𝜋 11984 4. 5. 2 sin 54° 6. 2 7408 Lesson Applications GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd  D COLLEGE OF TEACHER EDUCATION BSEd A L U D. Suppose you are preparing for an art exhibition inspired from the beauty B nature. Create an artwork of any medium that exhibits the beauty of nature H A through mathematics. Create a title and caption of the work produced and S explain the concept in mathematics as it relates to your work. A A References N Aufmann, R., Lockwood, J., Nation, R., Clegg, D., Epp, S. S., & Abad, E. P. (2018). Mathematics in the Modern World. Manila: Rex Book Store, Inc. N G Calingasan, R. M., Martin, M. C., & Yambao, E. M. (2018). Mathematics in the Modern World. Quezon City: C & E Publishing, Inc.,. L Commission on Higher Education. (2017, October 6). Mathematics in the Modern World U Preliminaries. Retrieved from CHED: https://ched.gov.ph/wp- N content/uploads/2017/10/KWF-Mathematics-in-the-Modern-World.pdf S O Drew, E. (2019, September 10). Fibonacci. Retrieved from The Cultivate Blog: D https://www.emilydrewmash.com/blog/2019/9/10/the-cultivate-blog-fibonacci-in- art N Haken, H. (2004). What are Patterns? Synergetic Computers and Cognition. Springer G Series in Synergetics, 50, 9-17. S Jenn. (2016, August 15). Fibonacci and the Good Kind of Math. Retrieved from Roving A Crafters: https://rovingcrafters.com/2016/08/15/fibonacci-and-the-good-kind-of- N math/ Keiren. (2012, December 14). The Fibonacci Sequence In Nature. Retrieved from P Insteading: https://insteading.com/blog/fibonacci-sequence-in-nature/#comments- A B section L Sirug, W. S. (2018). Mathematics in the Modern World. Manila: Mindshapers Co., Inc O Woolley, T. (2017, October 4). How Animals Got Their Spots and Stripes—According to Math. Retrieved from Scientific American: https://www.scientificamerican.com/article/how-animals-got-their-spots-and- stripes-mdash-according-to-math/ GE 103 MATHEMATICS DEPARTMENT MODULE - ------- CTEd