Nature of Mathematics: Mathematics in Our World PDF

THE NATURE OF MATHEMATICS: MATHEMATICS IN OUR WORLD  Objectives Identify Patterns in Nature and in the world. Articulate the importance of Mathematics in one’s life. Describe Fibonacci Sequence. Discover some patterns and occurrences that exist in nature, in our world...

THE NATURE OF MATHEMATICS: MATHEMATICS IN OUR WORLD  Objectives Identify Patterns in Nature and in the world. Articulate the importance of Mathematics in one’s life. Describe Fibonacci Sequence. Discover some patterns and occurrences that exist in nature, in our world, and in our life a)Patterns and Numbers in Nature and the World THE NATURAL ORDER Patterns that we can see in the universe Stars Seasons  Striped animals Snowflakes Spotted animals Waves Dunes 2 Types of Pattern Fractal Chaos  FRACTAL geometric shapes that repeat their structure on ever-finer scales.  CHAOS apparent randomness whose origins are entirely deterministic. The simplest mathematical objects are numbers, and the simplest of nature's patterns are numerical.  phases of the moon - from new moon to full moon (28 days) The year (roughly 365 days) legs petals in flowers b)Patterns and Numbers in Everyday Life Money  How much change will you receive if you give P100 in paying the Php 14.00 Angeles-Manibaug fare?  Compute for John’s commission after selling 10 electronic products, worth P2300 each and 15% of the sale is given to him as the commission. Measurement  Are you physically fit? How good are you in Math? Statistics What kind of assistance shall we give to Filipino teenagers? One in ten young Filipino women age 15-19 has begun childbearing: 8 percent are already mothers and another 2 percent are pregnant with their first child according to the results of the 2013 National Demographic and Health Survey (NDHS). (http://psa.gov.ph/tags/teenage-pregnancy) Construction How to build a strong bridge? Number Patterns 1. Arithmetic sequence 2. Geometric sequence 3. Harmonic sequence  Arithmetic Sequence - an ordered set of numbers that have a common difference between each consecutive term. - given by the formula An = a1 + (n-1)d, where An as the nth term, a1 as first term, n as the number of terms, and d as the common difference.  Examples 1. Find the 21th term of the arithmetic progression, 6, 4, 2, … 2. What is the first term of the arithmetic sequence if the nth term is 45, common difference of 5, and the total number of terms is 12. 3. Find the total number of terms if the nth term is 64, common difference of 4, and an a1 of -12.  Geometric Sequence - an ordered set of numbers that have a common ratio between each consecutive term. n-1 - given by the formula An = (a1)r , where An as the nth term, a1 as first term, n as the number of terms, and r as the common ratio.  Examples 1. Find the 12th term of the geometric progression, 4, 2, 1, … 2. What is the first term of the geometric sequence if the nth term is 1458, common ratio of 3, and the total number of terms is 7. 3. Find the total number of terms if the nth term is 256, common ratio of 2, and an a1 of 1/4. WWW.REALLYGREATSITE. COM Thank you ACTIVITY 1  1. Which term of the sequence 3,8,13 …is 183? 2. Find the 14th term of the arithmetic sequence 1, 3.5, 6, 8.5,... 3. If the given arithmetic sequence is not changing in value, then its common difference will be___ a. Positive b. Negative c. Zero d. One 4. The 11th term of the geometric sequence 1, 2, 6, 18,... is___ 5.If nth term of the geometric sequence 3/2, 3, 6, 12, …. is 1536, then what is the value of n? FIBON A CC I SEQU E N CE  Learning Objectives At the end of this module, the students will be able to: describe Fibonacci Sequence; discover some patterns and occurrences that exist in nature, in our world, and in our life; and discover how Mathematics becomes a tool to quantify, organize and control our world, predict phenomena, and make life  t h , a m a n is n g o f a m o n t h e b e g i n n i b i t s. A f t e r a A t e w b o r n r a b a p a i r o f n c e d n o given i t s h a v e p r o d u n t h t h e r a b b n t h m o r , e v e r y m o i n g ; h o w e v e o d u c e s o f f sp r f r a b b i t s p r r , t h e p a i r o i n g therea f t e. T h e o f f s p r i r o f r a b b i t s n e r. a n o t h e r p a s a m e m a n e x a c t l y t h e d u c e i n w m a n y repro a b b i t s d i e s , h o o n e o f th e r a t t h e s t a r t If n w il l t h e r e b e r s o f r a b b i t s pa i n g m o n t h ? h s u c c e e d i of e a c The Fib o n a c c i m b e r s w e r e fi r s t nu o v e r e d b y a m an dis c named L e o n a r d o o. H e w a s k n o w n LE O N A R D O Pisan PISANO by his ni c k n a m e , Fib o na c c i. The Fibonacci sequence is a sequence in which each term is the sum of the 2 numbers preceding it. The Fibonacci Numbers are defined by the recursive relation defined by the equations Fn = Fn-1 + Fn-2 for all n ≥ 3 where F1 = 1; F2 = 1 where Fn represents the nth Fibonacci number (n is called an index). The Fibonacci sequence can elaborately written as {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,…….}. FIBONNACI IN NATURE FLOWERS PLANTS Nature isn’t trying to use the Fibonacci numbers: they are appearing as a by-product of a deeper physical process. GOLD E N A N G L E AN D G O L D EN RATIO GOLDEN RATIO G O L D E N AN G L E ORGANS OF HUMAN BODY Humans exhibit Fibonacci characteristics. KNUCKLE ARM  ARCHITECTURE The Golden Ratio is also frequently seen in natural architecture. PYRAMID BINET’S SIMPLIFIED FORMULA  Find the nth term in the Fibonacci sequence using the Binet’s simplified formula. 3rd 11th 15th 22nd 25th MATHEMATICS AS A LANGUAGE  Learning Objectives At the end of this module, the students will be able to: discuss the language, symbols and conventions of Mathematics; explain the nature of Mathematics as a Language; perform operations on Mathematical Expressions correctly; and acknowledge that Mathematics is a useful “MATHEMATICS IS THE LANGUAGE IN WHICH GOD HAS WRITTEN THE UNIVERSE.”  It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.  A LANGUAGE CONTAINS THE FOLLOWING COMPONENTS: There must be a vocabulary of words or symbols. Meaning must be attached to the words or symbols. A language employs grammar, which is a set of rules that outline how  A syntax organizes symbols into linear structures or propositions. A narrative or discourse consists of strings of syntatic propositions. There must be (or have been) a group of people who use and understand the symbols. The language of mathematics makes it easy to express the kinds of thoughts that mathematicians like to express. precise concise powerful  BASIC CONCEPTS OF MATHEMATICA L LANGUAGE  SETS Sets are any well-defined collection of objects; these objects are called members or elements. To denote membership, we use the symbol  Methods of Describing Sets tabular or roster method – lists the elements of the group rule or set builder notation – states the common properties of the elements of the group Represent the following sets in tabular or roster form: 1. Let A be the set of even natural numbers less than 13. 2. The set of all prime numbers less than 20. Represent the following sets in rule or set-builder form 1. S = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday} 2. S = { 2, 4, 6, 8, 10,... } ACTIVITY 2 Find the missing term using the Binet’s Simpified Formula : 1. F31 2. F24/4 3. F29 4. F42/2 5. F12 + (100/4) Represent the following sets in tabular or roster form: 1. Let X be the set of composite numbers less than or equal to 16. 2. Set of cellphone brands. 3. Set of courses in CCA. Represent the following sets in rule or set-builder form 1. D = {Chaos, Nyx, Tartarus, Erebus, Eros, …} 2. S = { …,-18, -9, 9,18, 27,... }  MODULE 4 - THE NATURE OF MATHEMATICS: PROBLEM SOLVING  Learning Objectives At the end of this module, the students will be able to: Use different types of reasoning to justify statements and arguments made about mathematics and mathematical concepts; Write clear and logical proofs; Solve problems involving patterns and recreational problems following Polya’s four steps; +123-456-7890 and [email protected] www.reallygreatsite.com Organize one’s methods and approaches for proving and solving problems.  Polya’s Four-Step in Problem Solving George Polya (1887-1985) was a prominent modern mathematician who studied problem solving. Polya's basic problem-solving method included the four steps:  Polya’s Four-Step Problem-Solving Strategy Understand the problem. Devise a plan. Carry out the plan. Review the solution.  Understand the Problem Have a clear understanding of the problem. Consider the following questions. Can you restate the problem in your own words? Can you determine what is known about these types of problems? Is there extraneous information that is not needed to solve the problem? What is the goal?  Devise a Plan Use variety of techniques when solving a problem Make a list of the known information. Make an organized list that shows all the possibilities. Make a table or a chart. Look for a pattern. Write an equation. If necessary, define what each variable represents.  Carry Out the Plan Work carefully. Keep an accurate and neat record of all your attempts. Realize that some of your initial plans will not work and that you may have to devise another plan or modify your existing plan.  Review the Solution Ensure that the solution is consistent with the facts of the problem. Interpret the solution in the context of the problem.  Example 1 A baseball team won two out of their last four games. In how many different orders could they have two wins and two losses in four games?  Understand the Problem There are many different orders. The team may have won two straight games and lost the last two (WWLL).  Devise a Plan We will make an organized list of all the possible orders.  Carry Out the Plan Each entry in our list must contain two Ws and two Ls.  Review the Solution The list has no duplicates and the list considers all possibilities. Thus, there are six different orders in which a baseball team can win exactly two out of four games.  Example 2 If six people greet each other at a meeting by shaking hands with one another, how many handshakes will take place?  Understand the Problem There are six people, and each person shakes hands with each of the other people.  Devise a Plan Each person will shake hands with five other people. Since there are six people, we could multiply 6 times 5 to get the total number of handshakes. However, this procedure would count each handshake exactly twice, so we must divide this product by 2 for the actual answer.  Carry Out the Plan 6 times 5 is 30. 30 divided by 2 is 15.  Review the Solution Denote the people by the letters A, B, C, D, E, and F. Make an organized list. Remember that AB and BA represent the same people shaking hands, so do not list both AB and BA.  Example 3 If two ladders are placed end to end, their combined height is 32.5 feet. One ladder is 7.5 feet shorter than the other ladder. What are the heights of the two ladders?  Inductive and Deductive Reasoning Inductive reasoning is the process of reaching a general conclusion by examining specific examples.  Examples I see beetles in my lawn every summer. Last winter, there is no beetle around. This summer, I will probably see beetles in my lawn. My friend’s dog is friendly. Our neighbor’s dog is friendly. Our house dog is friendly. All dogs  Inductive and Deductive Reasoning Deductive reasoning is the process that uses premise as grounds to draw a certain conclusion.  Examples All birds lay eggs. Chickens are bird. Therefore, chickens lay eggs. Anime series are fun to watch. Attack on Titan is an anime. Hence, Attack on Titan is fun to watch.  Examples Determine whether each of the following arguments is an example of inductive reasoning or deductive reasoning. During the past 10 years, a tree has produced plums every other year. Last year the tree did produce plums, so this year the tree will produce plums. All home improvements cost more than the estimate. The contractor estimated that my home improvement will cost P350,000. Thus, my home improvement will cost more than P350,000.  Activity A. Apply Polya’s four-step strategy +58 to solve +250 the following Font Design problems. Visual Design +140 Interface Design +164 Digital Product  1. A frog is at the bottom of a 29-foot well. Each time the frog leaps, it moves up 3 feet. If the frog has not reached the top of the well, then the frog slides back 1 foot before it is ready to make another leap. How many leaps will the frog need to escape the well? +58 Font Design +250 Visual Design 2. The number of ducks and pigs in a field totals 34. The +140 total number of legs Interface Design among them is+164 112. Assuming Digital Product each duck has exactly two legs and each pig has exactly four legs, determine how many ducks and how many pigs are in B. Determine whether each of the following arguments is an example of inductive reasoning or deductive reasoning. 1. Janet Evanovich novels are worth reading. The novel Twelve Sharp is a Janet Evanovich novel. Thus, Twelve +58Sharp is worth reading. Font Design +250 Visual Design +140 2. I know I will win a jackpot Interface Design on this slot+164 machine Digital Product in the next 10 tries, because it has not paid out any money during the last 45 tries.  3. Tony is a grandfather and he is bald. Hence, grandfathers are bald. 4. Rizal’s novels are worth emulating. The novel The Reign of Greed is one of his work. Thus, The Reign of Greed is +58 worth emulating.+250 Font Design Visual Design +140 +164 5. Aloe are plants, and all plants are autotroph. Therefore, Interface Design Digital Product aloe is an autotroph.  Sudoku Abbreviated from suuji wa dokushin ni kagiru, meaning “the numbers must remain single”. How to solve a Sudoku  KenKen Puzzle KenKen is an arithmetic-based logic puzzle that was invented by the Japanese mathematics teacher Tetsuya Miyamoto in 2004. The noun “ken” has “knowledge” and “awareness” as synonyms. Hence, KenKen translates as knowledge squared, or awareness squared. Here is a 4 by 4 puzzle and its solution. Properly constructed puzzles have a unique solution. Here is a 4 by 4 puzzle and its solution. Properly constructed puzzles have a unique solution.  The Einstein’s Riddle Situation: There are 5 houses in five different colors. In each house lives a person with a different nationality. These five owners drink a certain type of beverage, smoke a certain brand of cigar and keep a certain pet. No owners have the same pet, smoke the same brand of cigar or drink the same beverage. The question is: Who owns the fish?  Hints: 1. the Brit lives in the red house 2. the Swede keeps dogs as pets 3. the Dane drinks tea 4. the green house is on the left of the white house 5. the green house's owner drinks coffee 6. the person who smokes Pall Mall rears birds 7. the owner of the yellow house smokes Dunhill 8. the man living in the center house drinks milk 9. the Norwegian lives in the first house 10. the man who smokes blends lives next to the one who keeps cats 11. the man who keeps horses lives next to the man who smokes Dunhill 12. the owner who smokes BlueMaster drinks beer 13. the German smokes Prince 14. the Norwegian lives next to the blue house 15. the man who smokes blend has a neighbor who drinks water