Math 113 Lesson 1 Reviewer.pdf

The Nature of Mathematics we progress. Patterns, on the other hand, are more wider and more prevalent wherever and at any At the end of this Chapter, the students must be time. able to: Identify pa...

The Nature of Mathematics we progress. Patterns, on the other hand, are more wider and more prevalent wherever and at any At the end of this Chapter, the students must be time. able to: Identify patterns in nature and regularities in the world; Articulate the importance of mathematics in one’s life; Argue about the nature of mathematics, what it is, how it is expressed, represented and used; Discuss the roles of mathematics in some Patterns can be sequential, spatial, temporal, and disciplines; and even linguistic. All these phenomena create a Express appreciation for mathematics as repetition of names or events is called regularity. human endeavor ⮚ Dates in the Calendar 2000, 2001, 2002, Introduction 2003 ⮚ Days in a Calendar Mathematics is a valuable tool for exploring ⮚ Months in a Year nature and our surroundings. The study of patterns ⮚ Regular Holidays in nature and the environment is a natural Regularity - same things happen in the same extension of mathematics. Mathematics may be circumstances. observed in almost every situation and is used to explain the most frequent phenomena. Pattern - regularity in the world or man-made design. The history and usefulness of patterns and numbers can be traced back to the origins of Pattern in Nature or Natural Pattern - visible mathematics. It is concerned with human-created regularities found in the natural world. thoughts and ideas that were transformed into products. They were created to connect the meaning of a pattern to the perception of Some examples of Patterns in Nature counting, sequence, and regularities. Patterns are the repeated design or recurring Lesson 1: Patterns in Nature sequence. It is also an ordered set of numbers, shapes or other mathematical objects that There is a link between patterns and counting. arranged according to a rule. One of the most When there is a pattern, counting occurs. There is intriguing things we see in nature is patterns. We reasoning when there is counting. As a result, in tend to think of patterns as sequences or designs nature, pattern corresponds to logic or logical that are orderly and that are repeat object. arrangement. A specific pattern has its own set of characteristics. For example, we recognize the spots on a giraffe as a pattern, but they are not regular nor are any The most people believe that mathematics is the of the spots the same size or shape. study of patterns. Mathematics, like patterns in nature, may be found everywhere. Patterns come Types of Patterns in a variety of shapes and sizes. Symmetry - it refers to a sense of Number patterns, such as 2, 4, 6, and 8, are harmonious and beautiful proportion and recognizable to us since they are among the first balance. patterns we learn when we are young. We gain experience both within and outside of school as 1  Spiral - a curve which emanates from a point, moving farther away as it revolves around the point. Meander - one of a series of regular sinuous curves, bends, loops, turns, or winding in the channel of rivers, streams, or other water course. Waves or Riffles - a disturbance that transfer energy through matter or space, with little or no mass transport. Foams or Bubbles - a substance formed by trapping pockets of gas in a liquid or solid. Tessellation - tilling of plane using one or more geometric shapes, called the tiles, with no overlaps and no gaps. Tessellations are patterns that are formed by repeated cubes or tiles. Fractures or Cracks - separations of an object or material into two or more pieces under the action of stress. Stripes and Spots - series of bands or strips and spots, often of the same width and color along the length. Fractals - a never-ending pattern. Infinitely complex pattern that are self-similar across different scales. 2 3 Uses of Mathematics Technology: navigation, prediction Engineering: construction, robotics Media: music, movie, election Medicine and Health: pharmacy, surgery Finance and Business: banking, gambling Pharmacy and Medicine, Population Dynamics, Plastic Surgery, Counting Calories, Crowd Control, Problem Solving, MRI and Tomography, Neurology, Epidemics Analysis Construction, Automotive Design, Building Bridges, Robotics, Roller Coaster Design, Supply Chains, Finance and Banking, Gambling and Betting, Insurance, Loans Interest Mortgages, Fraud Detection, Big Data, Pricing Strategies, Game Theory Fibonacci – introduced in “Book if Counting”, begins with 0 and 1, adding the last two numbers: Reading CDs and DVDs, Digital Music, Making 0, 1, 2, 3 … (petals). Music, Movie Graphics, Polling and Voting, Music Shuffling 4  Finite Sequence – has first and last terms. Infinite Sequence – has first term but no last term. Arithmetic sequence – term is obtained by adding d to the next term. Ex. 3, 6, 9, 12, 15, … a1 = 3 a2 = 6 a6 = 18 a20 = 60 Ex. 8, 11, 14, 17, … a1 = 8 a2 = 11 a6 = 23 a20 = 65 Formula: an = a1 + (n – 1) d an = nth term (missing) a1 = 1st term n = number of terms d = common difference = R – L = 11 – 8 = 3 d = 14 – 11 = 3 a6 = ? a1 = 8 n=6 d=3 a6 = a1 + (n – 1) d *subtract *multiply *add White Calla Lily - 1 petal Euphorbia - 2 petals a6 = 8 + (6 – 1) 3 Trillum - 3 petals Columbine - 5 petals a6 = 8 + (5) 3 Bloodroot- 8 petals Black-eyed Susan - 13 petals a6 = 8 + 15 Shasta Daisy - 21 petals Field Daisies - 34 petals a6 = 23 Golden Ratio – The Golden ratio (or Golden Part, a20 = ? a1 = 8 n = 20 d=3 or Golden Proportion, or Divine Proportion) is a20 = a1 + (n – 1) d *subtract *multiply *add generally denoted by the Greek letter Phi (φ), in lower case, which represents an irrational number a20 = 8 + (20 – 1) 3 1.6180339887..., It is said that Phi is the initial letters of Phidias’ name adopted to designate the a20 = 8 + (19) 3 golden ratio. a20 = 8 + 57 Sequence – set of all numbers in special orders. a20 = 65 Terms – number of a sequence. Exercises: 7, 2, -3, -8, … Series – sum of terms. 5 Find the following: a6, a10, a20 a6 = ? a1 = 7 n=6 d = 2 – 7 = -5 a6 = 7 + (6 – 1) (-5) *subtract *multiply *add a6 = 7 + (5) (-5) a6 = 7 + (-25) a6 = -18 6