Mathematics In Our World PDF

MODULE 1 MATHEMATICS IN OUR WORLD Prepared by: ABEL E. SADJI Math 111 Instructor  MODULE 1: MATHEMATICS IN OUR WORLD MODULE OBJECTIVES At the end of this module, challenge yourself to: 1. argue about the nature of mathematics, what it is, how it is expressed...

MODULE 1 MATHEMATICS IN OUR WORLD Prepared by: ABEL E. SADJI Math 111 Instructor  MODULE 1: MATHEMATICS IN OUR WORLD MODULE OBJECTIVES At the end of this module, challenge yourself to: 1. argue about the nature of mathematics, what it is, how it is expressed, represented and used; 2. discuss the concept of Fibonacci and its application; 3. identify patterns in nature and regularities in the world; 4. appreciate the nature and uses of mathematics in everyday life; 5. establish the relationship between the Fibonacci sequence with the golden ratio; 6. investigate the relationship of the golden ratio and Fibonacci number in natural world; and 7. determine the application of the Golden ratio in arts and architecture.  Lessons in the Module LESSON 1: Patterns and Numbers in Nature and in the World LESSON 2: Fibonacci Sequence LESSON 3: Patterns and Regularities in the World  Lesson 1 PATTERNS AND NUMBERS IN NATURE AND IN THE WORLD  PATTERNS AND NUMBERS IN NATURE AND IN THE WORLD There is much beauty in nature's clues even without any How will you mathematical training we can all define pattern? recognize it. Mathematical stories have its own beauty 1 which start from the clues and 2 deduce the underlying rules and regularities, but it is a 3 different kind of beauty, applying to ideas rather than things.  PATTERNS AND NUMBERS IN NATURE AND IN THE WORLD The development of new mathematical theories begins to reveal the secret of nature's patterns. We have already seen a practical impact as well as an intellectual one of our newfound understandings of nature's secret regularities. But most important of all, it is giving us a deeper vision of the universe in which we live, and of our own place in it. The modern understanding of visible patterns is developed gradually through the years.  PATTERNS AND NUMBERS IN NATURE AND IN THE WORLD Patterns can be observed even in stars which move in circles across the sky each day. The weather seasons cycle each year (e.g., winter, What are the patterns spring summer, fall). All snowflakes contain that you can find in sixfold symmetry which no two are exactly the your surroundings? same. There are evidences presented by mathematician that hexagonal snowflakes have 1 an atomic geometry of ice crystals. 2 3  PATTERNS AND NUMBERS IN NATURE AND IN THE WORLD Patterns can be seen in fish patterns like spotted trunkfish, spotted puffer, blue spotted stingray, spotted moral eel, coral grouper, red lion fish, yellow boxfish, and angel fish. These animals and fish stripes and Spotted Trunkfish Spotted Puffer spots attest to mathematical regularities in biological growth and form. These evolutionary and functional arguments explain why these animals need their patterns, but it is not explained how the patterns are formed. Blue Spotter Stingray Spotted Moray Eel  PATTERNS AND NUMBERS IN NATURE AND IN THE WORLD Zebra, tigers, cats and snakes are covered in patterns of stripes; leopards, and hyenas are covered in patters of spots; and giraffes are What do you think are the covered in patterns of blotches. reasons behind the patterns formed in these animals? 1 2 3  PATTERNS AND NUMBERS IN NATURE AND IN THE WORLD Natural patterns like the intricate waves across the oceans; sand dunes on deserts; formation of typhoon; water drop with ripple; and others. These serve as clues to the rules that govern the flow of water, sand, and air. One of the most strikingly mathematical landscapes on Earth is to be found in the Ocean Waves Dessert Dune great ergs, or sand oceans, of the Arabian and Sahara deserts. When wind blows steadily in a fixed direction, sand dunes form and the simplest pattern is the transverse dunes, which looks like ocean waves. If the sand is slightly moist, and there is a little vegetation to bind it together, then you may find parabolic dunes. Typhoon Water Ripple  Lesson 2 FIBONACCI SEQUENCE  Fibonacci Sequence The Fibonacci sequence is named after Leonardo of Pisa, also known as Fibonacci, who first observed the pattern while investigating how fast rabbits could breed under ideal circumstances. Fibonacci’s 1202 book Liber Abaci introduced the sequence to Western European mathematics, although it had been described earlier in Indian Mathematics. By definition, the first two numbers in the Fibonacci sequence are 1 and 1, and each subsequent number is the sum of the previous two. In mathematical terms, the sequence 𝐹𝑛 of Fibonacci numbers is defined by the recurrence relation 𝐹𝑛 = 𝐹𝑛 − 1 + 𝐹𝑛 + 2 , with seed values 𝐹1 = 1 and 𝐹2 = 1.  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: Solve the following problem: F1 = 1 0, 1 F2 = 0 + 1 = 1 0, 1, 1 F3 = 1 + 1 = 2 0, 1, 1, 2 1 F10= F4 = 1 + 2 = 3 0, 1, 1, 2, 3 F5 = 2 + 3 = 5 0, 1, 1, 2, 3, 5 F6 = 3 + 5 = 8 0, 1, 1, 2, 3, 5, 8 2 F12= F7 = 5 + 8 = 13 0, 1, 1, 2, 3, 5, 8, 13, … 3 F14=  Fibonacci Sequence To find the nth Fibonacci number without using the recursion formula, the following is evaluated using a calculator: 1+ 5 𝑛 1− 5 𝑛 ( ) −( ) 𝐹𝑛 = 2 2 5 This form is known as the Binet formula of the nth Fibonacci number.  Fibonacci Sequence Use Binet’s formula to determine the 25th and 30th Fibonacci numbers. 1+ 5 𝑛 1− 5 𝑛 ( 2 ) −( 2 ) 1+ 5 𝑛 1− 5 𝑛 𝐹𝑛 = ( 2 ) −( 2 ) 5 𝐹𝑛 = 5 1+ 5 25 1− 5 25 1+ 5 30 1− 5 30 ( ) −( ) ( ) −( ) 2 2 𝐹25 = 𝐹30 = 2 2 5 5 167,761 1,860,498 𝐹25 = 𝐹30 = 5 5 𝐹30 = 832,040 𝐹25 = 75, 025 Find the 20th and 22nd Fibonacci number using the Binet formula  Golden Ratio In mathematics and the arts, two quantities are in a golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. 𝑎 𝑎+𝑏 In symbols, a and b, where 𝑎 > 𝑏 > 𝑜, are in a golden ratio if 𝑏 = 𝑎. The golden ratio is often symbolized by the Greek letter 𝜙. It is the number 𝝓=𝟏.𝟔𝟏𝟖𝟎𝟑…. 1+ 5 and the irrational number. 2  Golden Ratio Golden ratio perhaps is the most important part of human beauty and aesthetics as well as a part of the remarkable proportions of growth patterns in living things such as plants and animals, Fibonacci number frequently appears in the numbers of petals in a flower and in the spirals of plants.  Golden Ratio The positions and proportions of the key dimensions of many animals are based on Phi or φ. Examples include the horn of ram, the wing dimensions and location of eye-like spots on moths, body sections of ants and other insects, body features of animals (e.g., tiger, fish, penguin, dolphin, etc.), and the spirals of sea shells. The growth pattern on branches of trees is Fibonacci. Even the human face contains spirals and the human DNA contains phi proportions.  Golden Ratio The golden ratio shows up in art, architecture, music, and nature. For example, the ancient Greeks thought that rectangles whose sides form a golden ratio were pleasing to look at. Many artists and architects have set their works to approximate the golden ratio, also believing this proportion to be aesthetically pleasing. Monalisa Vitruvian Man Parthenon The Last Supper  Lesson 3 Patterns and Regularities in the World  Patterns and Regularities in the World Patterns indicate a sense of structure and organization that it seems only humans are capable of producing these intricate, creative, and amazing formations. It is from this perspective that some people see an “intelligent design” in the way that nature forms. In this lesson, we will focus on: 1.Symmetry 2.Fractals 3.Spirals  Patterns and Regularities in the World SYMMETRY Symmetry is a sense of harmonious 2 types of Symmetry and beautiful proportion of balance or an object is invariant to any of 1 Bilateral various transformations (reflection, rotation or scaling). There are two 2 Radial main types of symmetry, bilateral and radial.  Patterns and Regularities in the World BILATERAL SYMMETRY Butterfly It is a symmetry in which the left and right sides of the organism can be divided into approximately mirror image of each other along the midline. Symmetry exists in living things Dragonfly such as insects, animals, plants, flowers, and others. Animals mainly have bilateral or vertical symmetry, even leaves of plants and some flowers such as orchids. Plant Leaves  Patterns and Regularities in the World RADIAL SYMMETRY It is also known as rotational symmetry. It is a type of symmetry around a fixed point known as the center and it can be classified as either cyclic or dihedral. Plants often have radial or rotational symmetry, as to flowers and some groups of animals. A five-fold Starfish Sea Anemone symmetry is found in the echinoderms, the group which includes starfish (dihedral-D5 symmetry), sea urchins, and sea lilies (dihedral-D5 symmetry). Radial symmetry suits organisms like sea anemones whose adults do not move and jellyfish (dihedral-D4 symmetry). Radial symmetry is also evident in different kinds of flowers. Symmetries in Kiwi Flowers  Patterns and Regularities in the World FRACTALS Fractal is a curve or geometric figure, each part of which has the same statistical character as the whole. Fractal is one of the newest and most exciting branches of mathematics. It is a class of highly irregular shapes that are related to continents, coastlines, and snowflakes. It is useful in modeling structures in which similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as crystal growth, fluid turbulence, and galaxy formation. Fractal can be seen in some plants, trees, leaves, and others.  Patterns and Regularities in the World SPIRAL A logarithmic spiral or growth spiral is self-similar spiral curve which often appears nature. It was first described by Rene Descartes and was later investigated by Jacob Bernoulli. Spirals are more evident in plants. We also see spirals in typhoon, whirlpool, galaxy, tail of chameleon, and shell among others. Worm Fern Spiral Aloe Galaxy  MODULE ASSESSMENT Access the link below to answer the module assessment. You only have 1 hour to answer it. After, go back to the meeting room for our next module. Good luck! https://forms.gle/xsFsULsqngwVUeKJA  MODULE 1 MATHEMATICS IN OUR WORLD Prepared by: ABEL E. SADJI Math 111 Instructor  Prepared by: ABEL E. SADJI Math 111 Instructor At the end of this module, challenge yourself to: compare and contrast expression and sentences; acknowledge that mathematics is a useful language; discuss the language, symbols and conventions of mathematics; explain the nature of mathematics as a language; identify and discuss the four basic concepts in mathematical language; explore the general concept of set; and differentiate function from relation; Lesson in the Module Expressions VS Sentences Lesson 1 Lesson 2 Language of Set Relation and Function Lesson 3 Lesson 1 Expressions VS Sentences  Mathematical Expression A mathematical expression is the mathematical analogue of an English noun; it is a correct arrangement of mathematical symbols used to represent a mathematical object of interest. It does not state a complete thought and does not make sense to ask if an expression is true or false. The most common expression types are numbers, sets, and functions. EXAMPLES: 5 2+3 ½ 4(6 – 2) 1+1+1+1  Mathematical Sentences A mathematical sentence is the analogue of an English sentence; it is a correct arrangement of mathematical symbols that states a complete thought. Sentences have verbs. In the mathematical sentence “3 + 4 = 7”, the verb is “=”. A sentence can be (always) true, (always) false, or sometimes true/sometimes false.  Mathematical Sentences EXAMPLES: 1+2=3 is true 1+2=4 is false x=2 is sometimes true or sometimes false; if it is true when x is 2, and false otherwise x+3=3+x is (always) true, no matter what number is chosen for x. Lesson 2 Language of Set Set theory is the branch of mathematics that studies sets or the mathematical science of the infinite. The study of sets has become a fundamental theory in mathematics in 1870s which was introduced by Georg Cantor (1845-1918), a German mathematician. A set is a well-defined collection of objects; the objects are called the elements or members of the set. The symbol ∈ is used to denote that an object is an element of a set, and the symbol ∉ denotes that an object is not an element of a set. EXAMPLES: Some examples of sets 𝐴 = 𝑥 𝑥 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 10 2 𝐵 = 𝑥 𝑥 𝑖𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑎𝑛𝑑 𝑥 − 1 = 0 𝐶 = 𝑥 𝑥 𝑖𝑠 𝑎 𝑙𝑒𝑡𝑡𝑒𝑟 𝑖𝑛 𝑡ℎ𝑒 𝑤𝑜𝑟𝑑 𝑑𝑖𝑟𝑡 𝐷 = 𝑥 𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟, 1 < 𝑥 < 8 𝐸 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑠𝑒𝑡 𝑜𝑓 𝑣𝑜𝑤𝑒𝑙 𝑙𝑒𝑡𝑡𝑒𝑟𝑠}  TWO WAYS TO REPRESENT SET Roster method is when the elements of the set are enumerated and separated by a comma, it is also called tabulation method. Example: 𝐸 = 𝑎, 𝑒, 𝑖, 𝑜, 𝑢  TWO WAYS TO REPRESENT SET Rule method or set-builder notation is used to describe the elements or members of the set it is also called set builder notation, symbol is written as {𝑥|𝑃 𝑥 }. Example: 𝐸 = 𝑥 𝑥 𝑖𝑠 𝑎 𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑣𝑜𝑤𝑒𝑙 𝑙𝑒𝑡𝑡𝑒𝑟𝑠  Finite and Infinite Sets A finite set is a set whose elements are limited or countable, and the last element can be identified. On the contrary, an infinite set is a set whose elements are unlimited or uncountable, and the last element cannot be specified.  Finite and Infinite Sets EXAMPLES: Finite Set 𝐴 = 𝑥 𝑥 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 10 𝐶 = 𝑑, 𝑖, 𝑟, 𝑡 Infinite Set 𝐹 = … , −2, −1,0,1,2, … 𝐺 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑠𝑒𝑡 𝑜𝑓 𝑤ℎ𝑜𝑙𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠. }  Unit and Empty Sets A unit set is a set with only one element, it is also called singleton. On contrary, the unique set with no elements is called the empty set (or null set), it is denoted by the symbol ∅ or { }. In addition, all sets under investigation in any application of set theory are assumed to be contained in some large fixed set called the universal set, denoted by the symbol U.  Unit and Empty Sets EXAMPLES Unit Set 𝐼 = 𝑥 𝑥 𝑖𝑠 𝑎 𝑤ℎ𝑜𝑙𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 1 𝑏𝑢𝑡 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 3 𝐽= 𝑤 Empty Set 𝐿 = 𝑥 𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 2 𝑏𝑢𝑡 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 1 𝑀 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑎𝑛𝑑𝑎 𝑏𝑒𝑎𝑟 𝑖𝑛 𝑀𝑎𝑛𝑖𝑙𝑎 𝑍𝑜𝑜}  CARDINAL NUMBER The cardinal number of a set is the number of elements or members in the set, the cardinality of set A is denoted by n(A). For example, given set 𝐸 = (𝑎, 𝑒, 𝑖, 𝑜, 𝑢), the cardinal number of 𝐸 is 5 or 𝑛 𝐸 = 5 while the set 𝐴 = {𝑥|𝑥 𝑖𝑠 𝑎 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 10} , which can also be written as {1, 2, 3, 4, 5, 6, 7, 8, 9} has a cardinal number of 𝐴 is 9 or 𝑛(𝐴) = 9.  Operations on Sets UNION OF SETS For two given sets A and B, A∪B (read as A union B) is the set of distinct elements that belong to set A and B or both. The number of elements in A ∪ B is given by n(A∪B) = n(A) + n(B) − n(A∩B), where n(X) is the number of elements in set X. Example: If A = {1, 2, 3, 4} and B = {4, 5, 6, 7}, then the union of A and B is given by A ∪ B = {1, 2, 3, 4, 5, 6, 7}.  Operations on Sets INTERSECTION OF SETS For two given sets A and B, A∩B (read as A intersection B) is the set of common elements that belong to set A and B. The number of elements in A∩B is given by n(A∩B) = n(A)+n(B)−n(A∪B), where n(X) is the number of elements in set X. Example: If A = {1, 2, 3, 4} and B = {3, 4, 5, 7}, then the intersection of A and B is given by A ∩ B = {3, 4}.  Operations on Sets SET DIFFERENCE The set operation difference between sets implies subtracting the elements from a set which is similar to the concept of the difference between numbers. The difference between sets A and B denoted as A − B lists all the elements that are in set A but not in set B. Example: If A = {1, 2, 3, 4} and B = {3, 4, 5, 7}, then the difference between sets A and B is given by A - B = {1, 2}.  Operations on Sets COMPLEMENT OF A SET The complement of a set A denoted as A′ or Ac (read as A complement) is defined as the set of all the elements in the given universal set(U) that are not present in set A. Example: If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 2, 3, 4}, then the complement of set A is given by A' = {5, 6, 7, 8, 9}. U = {x| x is a positive integer lesser than or equal to 30} A = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30} B = {4, 8, 12, 16, 20, 24, 28} Solve: 1. A∪B 2. A∩B 3. A-B 4. A’ U = {x| x is a positive integer lesser than or equal to 30} A = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30} B = {4, 8, 12, 16, 20, 24, 28} Solve: 1. A∪B 2. A∩B 3. A-B 4. A’ Lesson 3 Relation and Function  Relation A relation is a set of ordered pairs. If x and y are elements of these sets and if a relation exists between x and y, then we say that x corresponds to y or that y depends on x and is represented as the ordered pair of (x, y). A relation from set A to set B is defined to be any subset of A×B. If R is a relation from A to B and (a, b)∈R, then we say that "a is related to " and it is denoted as a 𝑹 b.  Relation Example: Let 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑} be the set of car brands, and 𝐵 = {𝑠, 𝑡, 𝑢, 𝑣} be the set of countries of the car manufacturer. Then 𝐴 × 𝐵 gives all possible pairings of the elements of A and B, let the relation R from A to B be given by  Relation Example: Let Y = {0, 1, 2} and Z = {0, 1}, define a relation R from Y to Z as follows: Given any (x, y) ∈ Y x Z, 𝑥+𝑦 (x, y) ∈ R means that is an integer. 2 A function is a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component. The set X is called the domain of the function. For each element of x in X, the corresponding element y in Y is called the value of the function at x, or the image of x. The set of all images of the elements of the domain is called the range of the function. A function can also be expressed as a correspondence or mapping from one set to another.  Domain Range The mapping in figure at the a s right is a function that assigns to x to y. The domain of the b t function is {a,b,c,d}, while the c u range is {s,t,u,v}. d v Let us have another example by using set of ordered pairs of the relations. Now let us determine whether each of the following relations is a function. 𝐴 = { 1,3 , 2,4 , 4,6 } 𝐵 = { −2,7 , −1,3 , 0,1 , 1,5 , 2,5 } 𝐶 = { 3,0 , 3,2 , 7,4 , 9,1 } 𝐴 = { 1,3 , 2,4 , 4,6 } Domain Range 1 3 2 4 4 6 𝐵 = { −2,7 , −1,3 , 0,1 , 1,5 , 2,5 } Domain Range -2 1 -1 3 0 5 1 7 2 𝐶 = { 3,0 , 3,2 , 7,4 , 9,1 } Domain Range 0 3 1 7 2 9 4  Prepared by: ABEL E. SADJI Math 111 Instructor  MODULE 3 Problem Solving and Reasoning Prepared by: ABEL E. SADJI Math 111 Instructor At the end of this module, challenge yourself to: compare and contrast inductive and deductive reasoning; use different types of reasoning to justify statements and mathematics and mathematical concepts; apply the Polya's four-step in problem solving to justify statements and arguments made about mathematics and mathematical concepts; and organize one's methods and procedures for proving and solving problems.  Lessons in this Module Polya’s Four- Inductive and Recreational Step Deductive Problem Using Problem Reasoning Mathematics Solving  Lesson 1 Inductive and Deductive Reasoning Inductive INDUCTIVE REASONING is drawing a general conclusion from a repeated observation or limited sets of observations of specific examples. Basically, there is a given data, then we draw conclusion based from the frame these data or simply from specific case to general case. The conclusion drawn by using inductive reasoning is called conjecture. The conjecture may be true or false depending on the truthfulness of the argument. A statement is a true statement provided that it is true in all cases and it only takes one example to prove the conjecture is false, such example is called a counterexample.  Inductive Example 1: 1 is an odd number. 11 is an odd number. 21 is an odd number. Therefore, all number ending with 1 are odd numbers. Example 2: Use Inductive Reasoning to Predict a Number a) 3, 6, 9, 12, 15, __ Solution: Each successive number is 3 larger than the preceding number. Thus, we predict that the next number in the list is 3 larger 15, which is 18. Deductive DEDUCTIVE REASONING is drawing general to specific examples or simply from general case to specific case. Deductive starts with a general statement (or hypothesis) and examines to reach a specific conclusion.  Deductive Example 1: All birds have feathers. Ducks are birds. Therefore, ducks have feathers. Example 2: All positive counting numbers whose unit digit is divisible by two are even numbers. A positive counting number 1,236 has a unit digit of 6 which is divisible by two. Therefore, 1,236 is an even number. Inductive vs Deductive Deductive GENERAL SPECIFIC Inductive  Exercise DIRECTION: In this activity, your task is to identify if a statement is an example of Deductive or Inductive Reasoning. Write your answers of the space before the number. 1. All cookies are made with sugar. Oreos are cookie so Rosa knows. Oreos are made with sugar. 2. It snows when temperature is below 32°F. The temperature is 35°F. Therefore, it is not snowing. 3. 1, 1, 2, 3, 5, 8, 13…… 4. All science teachers are bald. Mark is a science teacher. Therefore, Mark is bald. 5. Essay test is difficult. Problem1 solving test is difficult. Therefore, all tests are difficult.  Lesson 2 Polya’s Four- Step Problem Solving Polya’s Four-Step Problem One of the foremost recent mathematicians to make a study of problem solving was George Polya (1887- 1985). He was a mathematic educator who strongly believed that the skill of problem solving can be taught. He developed a framework known as Polya’s Four-steps in problem solving that addressed the difficulty of students in solving a problem. He firmly Keep it up. believed that the most efficient way of learning You are doing great. mathematical concepts is through problem solving and students and teachers become a better problem solver. Polya’s Four-Step Problem The Polya’s four step in problem solving are: 1. Understand the problem. 2. Devise a plan. 3. Carry out a plan. 4. Look back. Polya’s Four-Step Problem UNDERSTAND THE PROBLEM This requires reading. The problem 1. What sort of a problem is it? slowly and recognizing the 2. What is being asked? information given in the problem. 3. What do the terms mean? This is achieved when all the words 4. Is there enough information or is more used in stating the problem is fully information needed? understood, it can be restated in 5. What is known or unknown? one own’s word. Polya’s Four-Step Problem DEVISE A PLAN To devise a plan is to come up with 1. Draw a picture/diagram/model a way to solve the problem. There 2. Make an organized list or table are many different types of plans 3. Look for patterns for solving problems. You must 4. Work backwards start somewhere so try something. 5. Act it out 6. Guess and Check 7. Use Logical Reasoning Polya’s Four-Step Problem CARRY OUT THE PLAN This step is where the 1. Be patient. identified plan is applied to solve 2. Work carefully. 3. Modify the plan or try a new plan. the problem. If plan does not work, 4. Keep trying until something works. it can be modified or changed. 5. Implement the strategy and strategies in Step 2. 6. Try another strategy if the first one isn't working. Common sense and natural 7. Keep a complete and accurate record of your work. 8. Be determined and don't get discouraged if the plan does not thinking abilities can help in this work immediately. step. Polya’s Four-Step Problem LOOK BACK This step is where answers can be 1. Look for an easier solution. verified and checked and where 2. Does the answer make sense? mistakes can be identified. Answers 3. Check the results in the original problem. 4. Interpret the solution with the facts of the problem. should be checked if it is plausible. 5. Recheck any computations involved in the solution. Looking back is an opportunity to make 6. Can the solution be extended to a more general case? 7. Ensure that all the conditions related to the problem are met. connections. This is a time to review 8. Determine whether there is another method of finding the what you have done, what worked and solution. Ensure the consistency of the solution in the context what didn’t. of the problem. Polya’s Four-Step Problem Example Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet? Polya’s Four-Step Problem STEP 1: UNDERSTAND THE PROBLEM We are given in the problem that there are 25 chickens and cows. All together there are 76 feet. Chickens have 2 feet and cows have 4 feet. We are trying to determine how many cows and how many chickens Mr. Jones has on his farm. Polya’s Four-Step Problem STEP 2: DEVISE A PLAN We will going to use Guess and Check along with making a tab many times the strategy below is used with guess and test. Make a table and look for a pattern. Polya’s Four-Step Problem STEP 3: CARRY OUT THE PLAN NO. OF CHICKEN TOTAL NO. CHICKENS COWS NO. COW FEET FEET OF FEET 20 5 40 20 60 21 4 42 16 58 Notice we are going in the wrong direction! The total number of feet is decreasing! NO. OF CHICKEN TOTAL NO. CHICKENS COWS NO. COW FEET FEET OF FEET 19 6 38 24 62 15 10 30 40 70 Better! The total number of feet are increasing! Polya’s Four-Step Problem STEP 3: CARRY OUT THE PLAN NO. OF CHICKEN TOTAL NO. CHICKENS COWS NO. COW FEET FEET OF FEET 12 13 24 52 76 STEP 4: LOOKING BACK Check: 12 + 13 = 25 heads 24 + 52 = 76 feet. We have found the solution to this problem. We could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.  Lesson 3 Recreational Problem Using Mathematics Recreational Problem: One of ancient “square” mathematical recreations of all is the magic square. A Chinese myth, on the time of emperor Yu, came across a sacred turtle with strange marking on its shell known as Lo Shu. The Markings are numbers and they form a square pattern of order 3. Recreational Problem: Magic square of order n is an arrangement of 4 9 2 numbers in squares such that the sum on the n numbers in each row, column, and diagonal is the same number. The magic square in the figure 3 5 7 below (left) has order 3, and the sum of the numbers in each row, column, and diagonal is 15. 8 1 6 Recreational Problem: EXERCISE Complete each magic square. MAGIC SUM = 18 Use any whole numbers 1–15. Each number can be used no more than two times in each magic square. 6 8 7 Recreational Problem: Another fascinating topic is the Palindromes of squares are as palindrome. A palindrome is a number (or follows: word, or phrase) sequence of characters 1. 12 = 1 (or symbols) which reads the same 2. 112 = 121 backward as forward, for example 131, 12,321, 1,234,321, etc. Palindrome maybe in 3. 1112 = 12,321 form of palindromic number, palindromic 4. 1,1112 = 1,234,321 triangle, and palindromic primes. 5. 11,1112 = 123,454,321  MODULE ASSESSMENT Access the link below to answer the module assessment. You only have 1 hour to answer it. After, go back to the meeting room for our next module. Good luck! https://forms.gle/d595zfFw1ydqHjrL8 MATHEMATICAL LOGIC Prepared by: Abel E. Sadji Logic Statement and Quantifiers LOGICAL STATEMENTS AND QUANTIFIERS A proposition (or statement) is a declarative sentence which is either true or false, but not both. The truth value of the propositions is the truth and falsity of the proposition. DIRECTION: Determine which of the following are proposition and not a proposition. 1. Manila is the capital of the Philippines. Proposition 2. What day is it? Not a Proposition 3. Help me, please. Not a Proposition 4. He is handsome. Proposition LOGICAL STATEMENTS AND QUANTIFIERS A propositional variable is a variable which is used to represent a proposition. A formal propositional variable written using propositional logic notation, 𝑝, 𝑞, and 𝑟 are used to represent propositions. LOGICAL STATEMENTS AND QUANTIFIERS Logical connectives are used to combine simple propositions which are referred as compound propositions. A compound proposition is a proposition composed of two or more simple propositions connected by logical connectives "and," "or," "if then," "not," and "if and only if”. A proposition which is not compound is said to be simple (also called atomic).  OPERATIONS ON PROPOSITIONS There are three main logical connectives such as conjunction, disjunction, and negation. The following are briefly discussed in this section. Note that T refers to true proposition and F refers to false proposition.  OPERATIONS ON PROPOSITIONS CONJUNCTION p q p∧q The conjunction of the proposition p and q is the compound proposition "p and q." T T T Symbolically, p∧q, where ∧ is the symbol for "and." If p is true and q is true, then T F F p ∧ q is true; otherwise, p ∧ q is false. Meaning, the conjunction of two F T F propositions is true only if each F F F proposition is true.  OPERATIONS ON PROPOSITIONS Common Words Associated with Conjunction p q p∧q T T T p and q T F F p but q p also q F T F p in addition q F F F p moreover q EXAMPLE 2+6=9 and men are mammal. p q p∧q T T T p: 2+6=9 T F F q: men are mammal. F T F F F F Since "2 + 6 = 9", is a false proposition and the proposition "man is a mammal" is true, the conjunction of the compound proposition is false. EXAMPLE Manny Pacquiao is a boxing champion p q p∧q and Gloria Macapagal Arroyo is the first T T T female Philippine President. T F F p: Manny Pacquiao is a boxing champion F T F q: Gloria Macapagal Arroyo is the first female Philippine President. F F F In the proposition "Manny Pacquiao is a boxing champion" is true while the proposition "Gloria Macapagal Arroyo is the first female Philippine President" is false therefore the conjunction of the compound proposition is false. EXAMPLE Abraham Lincoln is a former US President p q p∧q and the Philippine Senate is composed of T T T 24 senators. T F F p: Abraham Lincoln is a former US President F T F q: Philippine Senate is composed of 24 senators. F F F Since both the propositions "Abraham Lincoln is a former US Philippine President" and "Philippine Senate is composed of 24 senators" are both true, thus the conjunction of the compound proposition is true.  OPERATIONS ON PROPOSITIONS DISJUNCTION p q p∨q The disjunction of the proposition p, q is the compound proposition "p or q." T T T Symbolically, p∨q, where ∨ is the symbol for "or". If p is true or q is true or if both p T F T and q are true, then p ∨ q is true; otherwise, p ∨ q is false. Meaning, the F T T disjunction of two propositions is false F F F only if each proposition is false.  OPERATIONS ON PROPOSITIONS Common Words Associated with Disjunction p q p∨q T T T p or q T F T F T T F F F EXAMPLE 2+6=9 or Manny Pacquiao is a boxing p q p∨q champion T T T T F T p: 2+6=9 F T T q: Manny Pacquiao is a boxing champion. F F F Note that the proposition "2 + 6 = 9" is false while the proposition "Manny Pacquiao is a boxing champion" is true; hence the disjunction of the compound proposition is true. EXAMPLE Philippine Senate is composed of 24 p q p∨q senators or Gloria Macapagal Arroyo is T T T the first female Philippine President. T F T p: Philippine Senate is composed of 24 senators F T T q: Gloria Macapagal Arroyo is the first female F F F Philippine President. Since proposition "Philippine Senate is composed of 24 senators" is true and the proposition "Gloria Macapagal Arroyo is the first female Philippine President" is false, therefore the disjunction of the compound proposition is true EXAMPLE Abraham Lincoln is a former US President p q p∨q or man is a mammal. T T T T F T p: Abraham Lincoln is a former US President q: man is a mammal F T T F F F Given that both propositions "Abraham Lincoln is a former US President" and "man is a mammal" are both true, thus the disjunction of the compound proposition is true.  OPERATIONS ON PROPOSITIONS NEGATION The negation of the proposition p is p ∼p denoted by-p, where is the symbol for "not." If p is true, ∼p is false. Meaning, the T F truth value of the negation of a proposition is always the reverse of the F T truth value of the original proposition.  OPERATIONS ON PROPOSITIONS Common Words Associated with Negation p ∼p not p T F It is false that p... F T It is not the case that p... EXAMPLE The following are propositions for p, find the corresponding ∼p. p ∼p a. 3+5=8. b. Sofia is a girl. T F c. Achaiah is not here. F T ANSWER: a. 3+5≠8. b. Sofia is not a girl. c. Achaiah is here.  OPERATIONS ON PROPOSITIONS CONDITIONAL The conditional (or implication) of the proposition p and q is the compound p q p→q proposition "if p then q. Symbolically, p → q, where is the symbol for "if then." p is called T T T hypothesis (or antecedent or premise) and q is called conclusion (or consequent or T F F consequence). The conditional proposition p F T T → q is false only when p is true and q is false; otherwise, p → q is true. Meaning p → F F T q states that a true proposition cannot imply a false proposition.  OPERATIONS ON PROPOSITIONS Common Words Associated with Conditional p q p→q If p, then q. p implies q. p only if q. T T T p therefore q. T F F p is stronger than q. p is sufficient condition for q. F T T F F T  OPERATIONS ON PROPOSITIONS Common Words Associated with Conditional p q p→q q if p. q follows p. T T T q whenever p. T F F q is weaker than p. q is a necessary condition for p. F T T F F T EXAMPLE p q p→q If vinegar is sweet, then sugar is sour. T T T T F F p: If vinegar is sweet F T T q: sugar is sour F F T Since the propositions "vinegar is sweet" and the "sugar is sour" are both false, therefore the conditional of the compound proposition is true. EXAMPLE p q p→q 2+5=7 is a sufficient condition for T T T 5+6=1. T F F p: 2+5=7 F T T q: 5+6=1. F F T Note that "2+5=7 " is true and "5+6=1" is false, thus the conditional of the compound proposition is false.  OPERATIONS ON PROPOSITIONS BICONDITIONAL The biconditional of the proposition p p q p↔q and q is the compound proposition "p if and only if q". Symbolically, p↔q, where T T T ↔ is the symbol for "if and only if." If p T F F and q are true or both false, then p↔q is true; if p and q have opposite truth F T F values, then p↔q is false. F F T  OPERATIONS ON PROPOSITIONS Common Words Associated with Biconditional p q p↔q p if and only if q. (p iff q) p is equivalent to q. T T T p is necessary and sufficient for q T F F F T F F F T EXAMPLE p q p↔q 2+8=10 if and only if 6-3=3. T T T T F F p: 2+8=10 F T F q: 6-3=3 F F T Since the statements "2+8=10" and the "6- 3=3" are both true, therefore the conditional of the compound proposition is true. EXAMPLE p q p↔q Manila is the capital of the Philippines is equivalent to fish live in the moon. T T T T F F p: Manila is the capital of the Philippines F T F q: fish live in the moon F F T Note that "Manila is the capital of the Philippines" is true proposition while "fish live in the moon" is false, thus the conditional of the compound proposition is false. EXAMPLE p q p↔q 8-2=5 is a necessary and sufficient for 4+2 = 7. T T T T F F p: 8-2=5 q: 4+2 = 7 F T F F F T Given that "8 -2 = 5" and "4 + 2 = 7" are both false, thus the conditional of the compound proposition is true. MODULE’S QUIZ NO. 1 Access the link below to answer the quiz no. 1. https://forms.gle/W4rxrQF7WEnpS1xh7 MEASUREMENT MATH 111 - UNIT 4 The Metric System The Metric System To convert 4200 cm to meters, write the units in order from largest to smallest. A metric measurement that involves two units is customarily written in terms of one unit. Convert the smaller unit to the larger unit, and then add. To convert 8 km 32 m to kilometers, first convert 32 m to kilometers.  EXAMPLE 1. Convert 0.38 m to millimeters. 0.38 m = 380 mm 2. Convert 3.07 m to centimeters. 3.07 m = 307 cm  Mass and weight are closely related. Weight is a measure of how strongly Earth is pulling on an object. Therefore, an object’s weight is less in space than on Earth’s surface. However, the amount of material in the object, its mass, remains the same. On the surface of Earth, mass and weight can be used interchangeably. The gram is the unit of mass in the metric system to which prefixes are added. One gram is about the weight of a paper clip.  Conversion between units of mass in the metric system involves moving the decimal point to the right or to the left. Listing the units in order from largest to smallest will indicate how many places to move the decimal point and in which direction.  EXAMPLE To convert 324 g to kilograms, write the units in order from largest to smallest.  EXAMPLE Convert 4.23 g to milligrams. 4.23 g = 4230 mg Convert 42.3 mg to grams. 42.3 mg = 0.0423 g The basic unit of capacity in the metric system is the liter. One liter is defined as the capacity of a box that is 10 cm long on each side. The units of capacity in the metric system have the same prefixes as the units of length. Conversion between units of capacity in the metric system involves moving the decimal point to the right or to the left. Listing the units in order from largest to smallest will indicate how many places to move the decimal point and in which direction. To convert 824 ml to liters, first write the units in order from largest to smallest.  EXAMPLE 1. Convert 4 L 32 ml to liters. 32 ml = 0.032 L 4L 32ml = 4L + 0.032L = 4.032 L 2. Convert 2 kl 167 L to liters.  BASIC CONCEPTS OF EUCLIDEAN GEOMETRY At the end of the lesson, the students are expected to be familiar with basic facts about geometry. 𝐴𝐵 ℓ 𝐴𝐵  𝐴𝐵 𝐴𝐵 ℓ  ∥ ∥ 𝐴𝐵 ∥ 𝐶𝐷 ∠ ∠ ∠ ∠ ∠ Find the Measure of the Complement of an Angle x + 38° = 90° 𝒙 = 𝟓𝟐° ∠ ∠  ∠ ∠ ∠ ∠ ∠ ∠ Given that the m∠DEF is 118°, find the measure of ∠a.