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MATHEMATICS IN THE MODERN WORLD MODULE THREE PROBLEM SOLVING AND REASONING CORE IDEA Module three is basically showing that mathematics is not just about numbers but much of it is problem solving and reasoning. Learning...

MATHEMATICS IN THE MODERN WORLD MODULE THREE PROBLEM SOLVING AND REASONING CORE IDEA Module three is basically showing that mathematics is not just about numbers but much of it is problem solving and reasoning. Learning Outcome: 1. State different types of reasoning to justify statements and arguments made about mathematics and mathematical concept. 2. Write clear and logical proofs. 3.Solve problems involving patterns and recreational problems following Polya’s four steps. 4. Organize one’s methods and approaches for proving and solving problems.  Time Allotment: Six (6) lecture hours Lesson 3.1 Inductive and Deductive Reasoning Specific Objectives At the end of this lesson, the student should be able to: 1. Define inductive and deductive reasoning. 2. Differentiate inductive reasoning from deductive reasoning. 3. Demonstrate the correct way in using the two kinds of reasoning. 4. Apply the concept of patterns in mathematics to solve problems in inductive and deductive reasoning which 96  MATHEMATICS IN THE MODERN WORLD lead into correct conjecture by creating their own reasoning. In mathematics, sometimes we need to use inductive and deductive reasoning to be able to solve some practical problems that we may encounter in our daily lives. During your senior high school, your teacher taught you on how to solve problems in a most scientific way and there are steps to be followed in order to solve problems in a particular math subject, specifically in Algebra. Some of these problems are the number problem, age problem, coin problem, work problem, mixture problem, etc. In this module, we will be studying on how to solve problems in a different way. We will be using what we called an inductive and deductive reasoning way. But before we give an example on how to use this method, let us define first what inductive and deductive reasoning is. A. Inductive Reasoning The type of reasoning that forms a conclusion based on the examination of specific examples is called inductive reasoning. The conclusion formed by using inductive reasoning is often called a conjecture, since it may or may not be correct or in other words, it is a concluding statement that is reached using inductive reasoning. Inductive reasoning uses a set of specific observations to reach an overarching conclusion or it is the process of recognizing or observing patterns and drawing a conclusion. So in short, inductive reasoning is the process of reaching a general conclusion by examining specific examples. Take note that inductive reasoning does not guarantee a true result, it only provides a means of making a conjecture. Based on the given definition above, we could illustrate this by means of a diagram. Also, in inductive reasoning, we use the “then” and “now” approach. The “then” idea is to use the data to find pattern and make a prediction and the “now” 97  MATHEMATICS IN THE MODERN WORLD idea is to make a conjecture base on the inductive reasoning or find a counter- example. Definition for counter example will be discussed on the latter part of our lecture. Let us have some examples on how to deal with this kind of reasoning. Examples: 1. Use inductive reasoning to predict the next number in each of the following list: 3, 6, 9, 12, 15, ? Explanation The given sequence of number is clearly seen that each successive number is three (3) larger than the preceding number, which is if the first number is increased by 3 the result is 6. Now, when this 6 is increased by 3 the next number would be 9. If we are going to continue the process, if 15 is increased by 3 then the next number would be 18. Hence the required number is 18. 2. Write a conjecture that describe the pattern 2, 4, 12, 48, 240. Then use the conjecture to find the next item in the sequence. Step 1. Look for a pattern 2 4 12 48 240 … ? Step 2. Analyze what is happening in the given pattern. The numbers are multiplied by 2, then 3, then 4, then 5. The next number will be the product of 240 times 6 or 1,140. Step 3: Make a conjecture Now, the answer is 1,140 98  MATHEMATICS IN THE MODERN WORLD 3. Write a conjecture that describes the pattern shown below. How many segments could be formed on the fifth figure? Step 1. Look for a pattern 3-segments 9-segments 18-segments Step 2. Analyze what is happening in the given pattern. 3 9 18 30 ? +6 +9 +12 +15 This could be written in a form of: (3)(2) (3)(3) (3)(4) (3)(5) The figure will increase by the next multiple of 3. If we add 15, the next or the fifth figure is made of 45 segments. Step 3. Make a conjecture Hence the fifth figure will have 45 segments. Application of Inductive Reasoning (Using inductive reasoning to solve a problem) Inductive reasoning is very essential to solve some practical problems that you may encounter. With the use of inductive reasoning, we can easily predict a solution or an answer of a certain problem. 99  MATHEMATICS IN THE MODERN WORLD Here, we can see an illustrative examples on how to solve a certain problem using inductive reasoning. Example 1. Use the data below and with the use of inductive reasoning, answer each of the following questions: 1. If a pendulum has a length of 49 units, what is its period? 2. If the length of a pendulum is quadrupled, what happens to its period? Note: The period of a pendulum is the time it takes for the pendulum to swing from left to right and back to its original position. Length of Period of Pendulum in Pendulum in Units heartbeats 1 1 4 2 9 3 16 4 25 5 36 6 Solution: 1. In the table, each pendulum has a period that is the square root of its length. Thus, we conjecture that a pendulum with a length of 49 units will have a period of 7 heartbeats. 100  MATHEMATICS IN THE MODERN WORLD 2. In the table, a pendulum with a length of 4 units has a period that is twice that of pendulum with a length of 1 unit. A pendulum with a length of 16 units has a period that is twice that of pendulum with a length of 4 units. It appears that quadrupling the length of a pendulum doubles its period. Example 2. The diagram below shows a series of squares formed by small square tiles. Complete the table below. Let us make a table. Figure 1st 2nd 3rd 4th 5th 6th 10th 15th Number of 4 8 12 16 Tiles Solution: 1. Based on the given figures from the first up to fourth, we need to observe and analyse what is really happening in the said figures. 2. Next, take a look if there is a pattern. Is there any pattern that you may observe? If so, what it is? For sure you could say that from the first figure, each subsequent square increases by four (4) tiles. How? Let us take a look at this. Number of Tiles : 4 8 12 16 Patterm: +4 +4 +4 So, if each subsequent square increases by four, we could say that the 5th, 6th, 10th, and the 15th figure should have 20, 24, 40 and 60 squares respectively. Hence, the complete table would be; 101  MATHEMATICS IN THE MODERN WORLD Figure 1st 2nd 3rd 4th 5th 6th 10th 15th Number of 4 8 12 16 20 24 40 60 Tiles Example 3. Two stamps are to be torn from the sheet shown below. The four stamps must be intact so that each stamp is joined to another stamp along at least one edge. What would be the possible patterns for these four stamps after the two stamps were torn? Solution: The first possible pattern is if we tear the two right most stamps as shown below. Next is if we tear the two stamps on the lower right portion as shown below. Then, the next possible pattern if we tear the lower rightmost and leftmost stamp as shown below. 102  MATHEMATICS IN THE MODERN WORLD Also, if we tear the upper rightmost and upper leftmost stamp could be another possible pattern as shown below. Next possible pattern is if we tear the two upper right most stamps as shown below. Then, it could be followed two stamps to be torn on the lower leftmost as shown below. Next is the two stamps at the upper rightmost as shown below. 103  MATHEMATICS IN THE MODERN WORLD The eight possible pattern is if we tear one stamp at the upper leftmost and one stamp at the lower rightmost as shown. Lastly, if we tear one stamp at the upper rightmost and another one stamp on the lower leftmost as shown. Hence, below are the different possible pattern based on the given question above. Note: The sequence of these pattern could be interchanged. B. Deductive Reasoning Another type of reasoning is called deductive reasoning. It is a basic form of valid reasoning starts out with a general statement, or hypothesis, and examines the possibilities to reach a specific, logical conclusion. So, we could say also that this kind of reasoning works from the more general to the more specific. 104  MATHEMATICS IN THE MODERN WORLD By definition, deductive reasoning is the process of reaching specific conclusion by applying general ideas or assumptions, procedure or principle or it is a process of reasoning logically from given statement to a conclusion. The concept of deductive reasoning is often expressed visually using a funnel that narrows a general idea into a specific conclusion. GENERAL IDEAS First premise that fits within general truth Second premise that fits within first premise SPECIFIC CONCLUSION Example 1. First Premise: All positive counting numbers whose unit digit is divisible by two are even numbers. Second Premise: A positive counting number 1,236 has a unit digit of 6 which is divisible by two. Conclusion: Therefore, 1,236 is an even number. Example 2. First Premise: If the Department of Education strictly observed health conditions of the students due to Covid 19, then there is no face-to-face teaching and learning activity in a classroom. 105  MATHEMATICS IN THE MODERN WORLD Second Premise: The Philippines is currently experiencing Covid 19 pandemic. Conclusion: Therefore, there will be no face-to-face teaching and learning style in a classroom. Note: Not all arguments are valid! Can you make an example of a deductive reasoning that could be considered as an invalid argument? Self -Learning Activity Directions: Do as indicated. A. Identify the premise and conclusion in each of the following arguments. Tell whether also if the following arguments is an inductive or deductive reasoning. a) The building of College of Informatics and Computing Sciences in BatStateU Alangilan is made out of cement. Both building of the College of Engineering, Architecture and Fine Arts and the College of Industrial Technology in BatStateU Alangilan are made out of cement. Therefore, all building of Batangas State University is made out of cement. b) All birds has wings. Eagle is a bird. Therefore, eagle is a bird. B. Use inductive reasoning to predict the next three numbers on the following series of numbers. a) 3, 7, 11, 15, 19, 23, ____, _____, _____, … b) 1, 2, 6, 15, 31, _____, ______, ______, … c) 1, 4, 9, 16, 25, 36, 49, ______, _____, _____, …. 106  MATHEMATICS IN THE MODERN WORLD C. Write the next possible equation on the following series of an equation. 37 x 3 = 111 37 x 6 = 222 37 x 9 = 333 37 x 12 = 444 _____?______ D. Assume that the figure below is made up of square tiles. a1 a2 a3 a4 What would be the correct formula to determine the number of square tiles in the nth term of a sequence? 107  MATHEMATICS IN THE MODERN WORLD Lesson 3.2 Intuition, Proof and Certainty Specific Objective At the end of the lesson, the student should be able to: 1. Define and differentiate intuition, proof and certainty. 2. Make use of intuition to solve problem. 3. Name and prove some mathematical statement with the use of different kinds of proving. Introduction Sometimes, we tried to solve problem or problems in mathematics even without using any mathematical computation and we just simply observed, example a pattern to be able on how to deal with the problem and with this, we can come up with our decision with the use of our intuition. On the other hand, we use another method to solve problems in mathematics to come up with a correct conclusion or conjecture with the help of different types of proving where proofs is an example of certainty. Discussion A. Intuition There are a lot of definition of an intuition and one of these is that it is an immediate understanding or knowing something without reasoning. It does not require a big picture or full understanding of the problem, as it uses a lot of small pieces of abstract information that you have in your memory to create a reasoning leading to your decision just from the limited information you have about the problem in hand. Intuition comes from noticing, thinking and questioning. As a student, you can build and improve your intuition by doing the following: 108  MATHEMATICS IN THE MODERN WORLD a. Be observant and see things visually towards with your critical thinking. b. Make your own manipulation on the things that you have noticed and observed. c. Do the right thinking and make a connections with it before doing the solution. Illustration 1. Based on the given picture below, which among of the two yellow lines is longer? Is it the upper one or the lower one? What are you going to do to be able to answer the question? Your own intuition could help you to answer the question correctly and come up with a correct conclusion. For sure, the first thing that you are going to do is to make a keen observation in the figure and you will be asking yourself (starting to process your critical thinking) which of these two yellow lines is longer compare to other line or is it really the yellow line above is more longer than the yellow line below? But what would be the correct explanation? The figure above is called Ponzo illusion (1911). There are two identical yellow lines drawn horizontally in a railway track. If you will be observing these two yellow lines, your mind tells you that upper yellow line looks longer that the below yellow line. But in reality, the two lines has equal length. For sure, you will be using a ruler to be able to determine which of the two is longer than the other one. The exact reasoning could goes like this. The upper yellow line looks longer because of the converging sides of a railway. The farther the line, it seems look line longer that the other yellow line below. Now, have you tried to use a ruler? What have you noticed? 109  MATHEMATICS IN THE MODERN WORLD Self- Learning Activity Now, let us test your intuition. We have here a set of problems. Make your own conclusion based on the given problem without solving it mathematically. 1. Which of the two have the largest value? Explain it accurately towards to correct conclusion. 103 ; 310 Write your explanation here. __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ 2. Which among of the following has a largest product? 34 x 12 = ; 21 x 43 = ; 54 x 31= __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ 3. Look at the figure below. Are two lines a straight line?. What is your intuition? __________________________________________________________________ __________________________________________________________________ __________________________________________________________________ 110  MATHEMATICS IN THE MODERN WORLD B. Proof and Certainty Another equally important lesson that the student should be learned is on how to deal with mathematical proof and certainty. By definition, a proof is an inferential argument for a mathematical statement while proofs are an example of mathematical logical certainty. A mathematical proof is a list of statements in which every statement is one of the following: (1) an axiom (2) derived from previous statements by a rule of inference (3) a previously derived theorem There is a hierarchy of terminology that gives opinions about the importance of derived truths: (1) A proposition is a theorem of lesser generality or of lesser importance. (2) A lemma is a theorem whose importance is mainly as a key step in something deemed to be of greater significance. (3) A corollary is a consequence of a theorem, usually one whose proof is much easier than that of the theorem itself. METHODS OF PROOF In methods of proof, basically we need or we have to prove an existing mathematical theorem to be able to determine if this theorem is true or false. In addition, there is no need to prove any mathematical definition simply because we assumed that this is already true or this is basically true. Usually, a theorem is in the form of if-then statement. So, in a certain theorem, it consists of hypothesis and conclusion. Let us say P and Q are two propositions. In an if-then statement, proposition P would be the hypothesis while the proposition Q would be our conclusion denoted by: P→Q 111  MATHEMATICS IN THE MODERN WORLD Example: If a triangle is a right triangle with sides a, b, and c as hypotenuse, then a2 + b2 = c2. There are two ways on how to present a proof. One is with the use of an outline form and the other one is in a paragraph form. Either of the two presentations could be used by the student. TWO WAYS ON HOW TO PRESENT THE PROOF a. Outline Form Proposition: If P then Q. 1. Suppose/Assume P 2. Statement 3. Statement... Statement Therefore Q.  b. Paragraph Form Proposition: If P then Q. Assume/Suppose P. ____________. ___________. _____________________. ____________... _____________. _______________. _________________. Therefore Q.  Let us have a very simple and basic example on how to prove a certain mathematical statement. Illustration 1: Prove (in outline form) that “If x is a number with 5x + 3 = 33, then x = 6” 112  MATHEMATICS IN THE MODERN WORLD Proof: 1. Assume that x is a number with 5x + 3 = 33. 2. Adding -3 both sides of an equation will not affect the equality of the two members on an equation, thus 5x + 3 – 3 = 33 – 3 3. Simplifying both sides, we got 5x = 30. 4. Now, dividing both member of the equation by 5 will not be affected the 5𝑥 30 equality so 5 = 5. 5. Working the equation algebraically, it shows that x = 6. Therefore, if 5x + 3 = 33, then x = 6.  Let us have a very simple and basic example on how to prove a certain mathematical statement in paragraph form. Illustration 2: Prove (in paragraph form) that “If x is a number with 5x + 3 = 33, then x = 6” Proof: If 5x + 3 = 33, then 5x + 3 − 3 = 33 − 3 since subtracting the same number from two equal quantities gives equal results. 5x + 3 − 3 = 5x because adding 3 to 5x and then subtracting 3 just leaves 5x, and also, 33 − 3 = 30. Hence 5x = 30. That is, x is a number which when multiplied by 5 equals 30. The only number with this property is 6. Therefore, if 5x + 3 = 33 then x = 6.  Note: It is up to the student which of the two forms would be their preferred presentation. KINDS OF PROOF 1. DIRECT PROOF DEFINITION. A direct proof is a mathematical argument that uses rules of inference to derive the conclusion from the premises. In a direct proof, let us say we need to prove a given theorem in a form of P → Q. The steps in taking a direct proof would be: 113  MATHEMATICS IN THE MODERN WORLD 1. Assume P is true. 2. Conclusion is true. Example 1: Prove that if x is an even integer, then x2 – 6x + 5 is odd. Proof: (by outline form) 1. Assume that x is an even integer. 2. By definition of an even integer, x = 2a for some a  Z. 3. So, x2 – 6x + 5 = (2a)2 – 6(2a) + 5 = 4a2 – 12a + 4 + 1 = 2(2a2 – 6a + 2) + 1 where 2a2 – 6a + 2  k. 4. Therefore, 2(2a2 – 6a + 2) + 1 = 2k + 1, so x2 – 6x + 5 is odd.  Example 2: With the use of direct proving, prove the following in both form (outline and paragraph). Prove: (in an outline form) If a and b are both odd integers, then the sum of a and b is an even integer. Proof: 1. Assume that a and b are both odd integers. 2. There exists an integer k1 and k2 such that a = 2k1 + 1 and b = 2k2 + 1 (by definition of an odd number). 3. Now, a + b = (2k1 + 1) + (2k2 + 1) = 2k1 + 2k2 + 2. Factoring 2, it follows that a + b = 2(k1 + k2 + 1). 4. So; a + b = 2(k1 + k2 + 1). Let k1 + k2 + 1 = k  Z, hence a + b = 2k. 5. Therefore, if a and b are both odd integer, then a + b is even.  Prove: (in paragraph form) Assume that a and b are both odd integers. By definition of an odd number, there exists an integer k1 and k2 such that a = 2k1 + 1 and b = 2k2 + 1. Now, adding a and b, that is, a + b = (2k1 + 1) + (2k2 + 1) = 2k1 + 2k2 + 2. Factoring 2, it follows that a + b = 2(k1 + k2 + 1). So; a + b = 2(k1 + k2 + 1) and let k1 + k2 + 1 = k 114  MATHEMATICS IN THE MODERN WORLD  Z, hence a + b = 2k. Therefore, if a and b are both odd integers, their sum is always and even integer.  Example 3: With the use of direct proving, prove the following in paragraph form. Prove: If x and y are two odd integers, then the product of x and y is also an odd integer. Proof: Assume that x and y are two different odd integers. There exists k1 and k2  Z such that x = 2k1 + 1 and y = 2k2 + 1 by definition of an odd number. Now, taking the product of x and y, we got xy = (2k1 + 1)(2k2 + 1) = 4k1k2 + 2k1 + 2k2 + 1 = 2(2k1k2 + k1 + k2) + 1. Let 2k1k2 + k1 + k2 = k  Z. Hence (2k1+1)(2k2+1) = 2k + 1. Therefore, xy = 2k + 1 where the product of two odd integers is also an odd integer.  Example 4. Prove the proposition (in outline form) that is “ If x is an positive integer, then x2 is also an odd integer”. Prove: (In outline form) 1. Suppose x is odd. 2. Then by definition of an odd integer, x = 2a + 1 for some a  Z. 3. Thus x2 = (2a + 1)2 = 4a2 + 4a +1 = 2(2a2 + a) + 1. 4. So x2 = 2b+1 where b is the integer b = 2a2 + 2a. 5. Thus x2 = 2b + 1 for an integer b. 6. Therefore x2 is odd, by definition of an odd number.  Example 5. Prove: Let a,b and c be integers. If a| b and b | c, then a | c. Proof (in outline form) 1. Suppose a, b and c are integers and a | b and b | c. 2. We all know that if a | b, there is a certain integer say d which is b = ad. 3. Similarly, when b | c, there is an integer say e which is c = be. 4. Now, since b = ad, substitute the value of b in c = be, it follows that c = (ad)e = a(de). 5. So, c = a(de) = ax for x = de Z. 115  MATHEMATICS IN THE MODERN WORLD 6. Therefore a | c.  Now, it’s your turn to do some direct proving. You can use any of the two forms of presentation for proving. Self- Learning Activity Direction: Prove the following propositions with the use of direct proving. Show your answer on the space provided after each item. (5 marks each) 1. If a is an odd integer, then a2 +3a + 5 is odd. 2. Suppose x, y  Z. If x3 and y3 are odd, then (xy)3 is odd. 3. Suppose x, y  Z. If x is even, then xy is even. 116  MATHEMATICS IN THE MODERN WORLD 4. If n – m is even, then n2 – m2 is also an even. 5. If x is odd positive integer then x2 – 1 is divisible by 4. 6. If x is an odd integer, then 8 is a factor of x2 – 1. 117  MATHEMATICS IN THE MODERN WORLD 2. INDIRECT PROOF (CONTRAPOSITIVE PROOF) DEFINITION: Indirect proof or contrapositive proof is a type of proof in which a statement to be proved is assumed false and if the assumption leads to an impossibility, then the statement assumed false has been proved to be true. Recall that the proposition p →q is a conditional statement. This proposition is logically equivalent to q →p. Now, the expression q →p is the contrapositive form of the statement p →q. In an indirect proof, let us say we need to prove a given theorem in a form of P → Q. The steps or outline in taking an indirect proof would be: Assume/Suppose Q is true.... Therefore P is true.  Example 1. Using indirect/contrapositive proof, prove that “If x is divisible by 6, then x is divisible by 3”. Here in example 1, we let that p : x is divisible by 6 and q : x is divisible by 3. So, this original statement to become a contrapositive could be transformed into “If x is not divisible by 3, then x is not divisible by 6”. Note that, we let p: x is divisible by 6 and q: x is divisible by 3. With the use of indirect proof, we assume that q is true and the conclusion p is also true. 118  MATHEMATICS IN THE MODERN WORLD So, the formal proof would be; Proof: 1. Assume x is not divisible by 3. 2. Then x  3k for all k  Z 3. It follows that x  (2m)(3) for all m  Z 4. So, x  6m for all m  Z 5. Therefore, x is not divisible by 6.  Example 2: Prove using indirect proof or contraposition. Let x be an integer. Prove that, if x2 is even, then x is even. Note that, we let p: x2 is even and q: x is even. With the use of indirect proof, we assume that q is true and the conclusion p is also true. So, the original statement would become “If x is odd, then x2 is odd”. Now, the formal proof would be; Proof: 1. Assume x is odd. 2. Then x = 2k + 1 for some k  Z 3. It follows that x2 =(2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1 where q = 2k2 + 2k 4. So, x2 = 2q + 1 5. Therefore, x2 is odd.  Again, it is your turn to prove the following propositions with the use of indirect or contrapositive proof. 119  MATHEMATICS IN THE MODERN WORLD Self- Learning Activity Direction: Prove the following propositions with the use of indirect proving. Show your answer on the space provided after each item. 1. If a is an even integer and b is odd integer, then a + b is odd integer. 2. If n is an odd integer, then n3 + 2n2 is also an odd integer. 3. If n – m is even, then n2 – m2 is also an even. 120  MATHEMATICS IN THE MODERN WORLD 4. If x is odd positive integer then x2 – 1 is divisible by 4. 5. If x is an odd integer, then 8 is a factor of x2 – 1. 6. Suppose x, y  Z. If x is even, then xy is even. 121  MATHEMATICS IN THE MODERN WORLD 3. Proof by Counter Example (Disproving Universal Statements) A conjecture may be described as a statement that we hope is a theorem. As we know, many theorems (hence many conjectures) are universally quantified statements. Thus it seems reasonable to begin our discussion by investigating how to disprove a universally quantified statement such as ∀x ∈ S,P(x). To disprove this statement, we must prove its negation. Its negation is ∼ (∀x ∈ S,P(x)) = ∃ x ∈ S,∼ P(x). Things are even simpler if we want to disprove a conditional statement P(x) ⇒ Q(x). This statement asserts that for every x that makes P(x) true, Q(x) will also be true. The statement can only be false if there is an x that makes P(x) true and Q(x) false. This leads to our next outline for disproof. The question is “How to disprove P(x) ⇒ Q(x)”? The answer is simple. Produce an example of an x that makes P(x) true and Q(x) false. In both of the above outlines, the statement is disproved simply by exhibiting an example that shows the statement is not always true. (Think of it as an example that proves the statement is a promise that can be broken.) There is a special name for an example that disproves a statement: It is called a counterexample. Example 1. Prove or disprove: All prime numbers are odd. *Negation : Some prime numbers are even. By counterexample: Let n = 2. By definition of a prime, 2 = (2)·(1). But 2 is even where the only factor of 2 is 2 and 1 so we could say that 2 is a prime number. Since we have found an even prime number so the original statement is not true.  122  MATHEMATICS IN THE MODERN WORLD Example 2. Prove or disprove: For all integers x and y, if x + y is even, then both x and y are even. *Negation : For some integers x and y,if x + y is odd, then x and y is odd. Proof: (x)(y): x + y = 2k1 + 2k2, for k Z x y: x + y = 2k1+ 1 + 2k2 + 1 = 2k1+ 2k2 + 2 = 2(k1 + k2 + 1) By giving a counterexample, if x = 1 and y = 1, then x + y = 2. But x and y are both odd, therefore the theorem is false.  Example 3. Prove that “For every n  Z, the integer f(n) = n2 – n + 1 is prime. Note that the negation of this would be “For some n  Z, the integer f(n) = n2 – n + 1 is composite. We all know that a prime number is a number whose factors are 1 and the number itself, thus if p is prime number then p = (p)(1) where p  Z To be able to resolve the truth or falsity of the above statement, let us construct a table for f(n) for some integers n. If we could find at least one number for f(n) which is not prime (composite), then we could conclude that the statement is false. n 0 1 2 3 4 5 6 7 8 9 10 11 f(n) 11 11 13 17 23 31 41 53 67 83 101 ? In every case, f (n) is prime, so you may begin to suspect that the conjecture is true. Before attempting a proof, let’s try one more n. Unfortunately, f (11) = 112−11+11 = 112 is not prime. The conjecture is false because n = 11 is a counterexample. 123  MATHEMATICS IN THE MODERN WORLD We summarize our disproof as follows: Disproof. The statement “For every n ∈ Z, the integer f (n) = n 2 − n + 11 is prime,” is false. For a counterexample, note that for n = 11, the integer f (11) = 121 = 11·11 is not prime.  Self- Learning Activity Direction: Prove the following propositions with the use of counter-example. Show your answer on the space provided after each item. 1. Prove: For all integer n which is a multiples of 3 are multiples of 6. 2. Prove: For all real numbers a and b, if a2 = b2, then a = b. 124  MATHEMATICS IN THE MODERN WORLD 3. Prove: For all positive integers n, n2 – n + 41 is prime. 4. Prove: For all positive integers n, 22n + 1 is prime. 5. Prove: For all real number n, n2 + 4 < 5. 125  MATHEMATICS IN THE MODERN WORLD 4. Proof by Contradiction Another method of proving is what we called “Proving by Contradiction”. This method works by assuming your implication is not true, then deriving a contradiction. Recall that if p is false then p → q is always true, thus the only way our implication can be false is if p is true and q is false. So, if we let p → q be a theorem, a proof by contradiction is given by this way; 1. Assume p is true. 2. Suppose that q is also true. 3. Try to arrive at a contradiction. 4. Therefore q is true So, in practice then, we assume our premise is true but our conclusion is false and use these assumptions to derive a contradiction. This contradiction may be a violation of a law or a previously established result. Having derived the contradiction you can then conclude that your assumption (that p → q is false) was false and so the implication is true. Be careful with this method: make sure that the contradiction arise because of your original assumptions, not because of a mistake in method. Also, if you end up proving ~p then you could have used proof by contraposition. Example 1: Prove by contradiction that “If x + x = x, then x = 0. Proof: 1. Assume that x + x = x. 2. Suppose that x  0. 3. Now, x + x = x, so 2x = x and since x  0, we could multiply both sides of the equation by the reciprocal of x, i.e., 1/x. 4. Multiplying by the reciprocal of x, it follows that 2 = 1 which is a contradiction. 5. Therefore, the original implication is proven to be true.  126  MATHEMATICS IN THE MODERN WORLD Example 2: Prove by contradiction that “If x is even then x + 3 is odd. Proof: 1. Assume x is even, so x = 2k. 2. Suppose x + 3 is even. Since x + 3 is even, there exist k Z such that x + 3 = 2k. 3. It follows that x = 2k -3. We can rewrite this as x = 2k – 4 + 1. Now, x = 2(k – 2) + 1. Let k – 2 = q. So, x = 2q + 1. It is clearly seen that x is an odd number. This is a contradiction to the assumption. 4. Therefore, x + 3 is odd.  Self- Learning Activity Direction: Prove the following propositions with the use of contradiction. Show your answer on the space provided after each item. 1. There are no natural number solutions to the equation x2 - y2 = 1. 2. For all integers n, if n3 + 5 is odd then n is even. 3. If x2 is irrational then x is irrational. 127  MATHEMATICS IN THE MODERN WORLD Lesson 3.3 Polya’s Four Steps in Problem Solving Specific Objective At the end of this lesson, the student should be able to: 1. Tell all the Polya’s four steps in problem solving. 2. Select the appropriate strategy to solve the problem. 3. Solve problems with the use of Polya’s four step. Introduction One of the major problems of a student in mathematics is on how to solve worded problems correctly and accurately. Sometimes, they have difficulty understanding in grasping the main idea of a problem on how to deal with it and to solve it. It is very important that there is always a clear understanding on how to solve problems most especially in a Mathematics as a course. When you were in your senior high school, your teacher in mathematics especially in the course of Algebra taught you on how to solve problem using scientific method. Some of these problems are number problem, age problem, coin problem, work problem, mixture problem, etc. But not all problems in mathematics could be solve on what you have learned in your senior high school. Here, in this “Polya’s Four Steps in Solving Problem”, we will be learning on how to solve mathematical problem in a different way. Discussion Maybe you would ask yourself that who is Polya? Why do we need to use his four steps in solving a mathematical problem? How are we going to use this to be able to solve problems? The answer for these question will be answered in this lesson. 128  MATHEMATICS IN THE MODERN WORLD George Polya is one of the foremost recent mathematicians to make a study of problem solving. He was born in Hungary and moved to the United States in 1940. He is also known as “The Father of Problem Solving”. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory. He is also noted for his work in heuristics and mathematics education. Heuristic, a Greek word means that "find" or "discover" refers to experience-based techniques for problem solving, learning, and discovery that gives a solution which is not guaranteed to be optimal. The George Polya’s Problem-Solving Method are as follows: Step 1. Understand the Problem. This part of Polya’s four-step strategy is often overlooked. You must have a clear understanding of the problem. To help you focus on understanding the problem, consider the following questions. These are some questions that you may be asked to yourself before you solve the problem. a. Are all words in a problem really understand and clear by the reader? b. Do the reader really know what is being asked in a problem on how to find the exact answer? c. Can a reader rephrase the problem by their own without deviating to its meaning? d. If necessary, do the reader can really visualize the real picture of the problem by drawing the diagram? e. Are the information in the problem complete or is there any missing information in a problem that could impossible to solve the problem? Step 2. Devise a Plan Sometimes, it is necessary for us that to be able to solve a problem in mathematics, we need to devise a plan. Just like a Civil Engineer that before he construct a building, he needs to do a floor plan for a building that he wants to build. To be able to succeed to solve a problem, you could use different techniques or way in order to get a positive result. Here are some techniques that could be used. You could one of these or a combination to be able to solve the problem. 129  MATHEMATICS IN THE MODERN WORLD a. As much as possible, list down or identify all important information in the problem b. Sometimes, to be able to solve problem easily, you need to draw figures or diagram and tables or charts. c. Organized all information that are very essential to solve a problem. d. You could work backwards so that you could get the main idea of the problem. e. Look for a pattern and try to solve a similar but simpler problem. f. Create a working equation that determines the given (constant) and variable. g. You could use the experiment method and sometimes guessing is okay. Step 3. Carry out the plan After we devised a plan, the next question is “How are we going to carry out the plan?” Now, to be able to carry out the plan, the following suggestions could help us in order to solve a problem. a. Carefully and accurately working on the problem. b. There must be a clear and essential information or data in the problem. c. If the first plan did not materialize, make another plan. Do not afraid to make mistakes if the first plan that you do would not materialize. There is a saying that “There is a second chance.” Step 4. Look back or Review the Solution Just like on what you do in solving worded problems in Algebra, you should always check if your answer is correct or not. You need to review the solution that you have made. How will you check your solution? The following could be your guide. a. Make it sure that your solution is very accurate and it jibed all important details of the problem. b. Interpret the solution in the context of the problem. c. Try to ask yourself that the solutions you’ve made could also be used in other problems. 130  MATHEMATICS IN THE MODERN WORLD As it was mentioned in this lesson, there are different strategies that you could employ or use to solve a problem. These strategies will help you to solve the problem easily. These are the following strategies that you could be used: 1. Draw a picture, diagram, table or charts. Label these with correct information or data that you could see in the problems. Sometimes, there are hidden information that is very much important also to solve the problem. So, be cautious. 2. Identify the known and unknown quantities. Choose appropriate variable in identifying unknown quantities. For example, the unknown quantity is height. You could use “h” as your variable. 3. You have to be systematic. 4. Just like on what we have in devising a plan, look for a pattern and try to solve a similar but simpler problem. 5. Sometimes, guessing is okay. There is no problem in guessing and it is not a bad idea to be able to begin in solving a problem. In guessing, you could examine how closed is your guess based on the given problem. Illustrative examples will be solved with the use of Polya’s four step method. 1. The sum of three consecutive positive integers is 165. What are these three numbers? Step 1. Understand the problem When we say consecutive numbers, these are like succeeding numbers. Say, 4, 5, 6 are three consecutive numbers for single-digit numbers. For the two- digit number, example of these three consecutive is 32, 33, and 34. Noticing that the second number added by 1 from the first number and the third number is increased by 2 from the first number. Step 2. Devise a plan From the previous discussion of this lesson, devising a plan is very essential to solve a problem. We could use an appropriate plan for this kind of problem and that is formulating a working equation. Since we do not know what are these three consecutive positive integers, we will be using a variable, say x to represent a particular number. This variable x could be the first number. Now, since it is consecutive, the second number will be increased by 1. So, the possible 131  MATHEMATICS IN THE MODERN WORLD presentation would be x + 1. The third number was increased by 2 from the first number so the possible presentation would be x + 2. Since, based on the problem that the sum of these three consecutive positive integers is 165, the working equation is: (x) + (x + 1) + (x + 2) = 165 where x be the first positive integer, x + 1 be the second number and x + 2 be the third number. Step 3. Carry out the plan We already know the working formula. To be able to determine the three positive consecutive integers, we will be using the concept of Algebra here in order to solve the problem. Manipulating algebraically the given equation; x + x + 1 + x + 2 = 165 Combining similar terms; 3x + 3 = 165 Transposing 3 to the right side of the equation; 3x = 165 – 3 Simplifying; 3x = 162 Dividing both side by 3 to determine the value of x; x = 54 and this would be the first number. Now, the second number is x + 1 and we already know the value of x = 54. So, the next number is 55. Then the third number would be x + 2 and again we know that x = 54 so the third number is 56. Hence, the three positive consecutive integers whose sum is 165 are 54, 55 and 56. 132  MATHEMATICS IN THE MODERN WORLD Step 4. Look back and review the solution We need to review our solution to check if the answer is correct. How are we going to do that? Just simply add the identified three consecutive positive integers and the result should be 165. So, adding these three numbers, 54 + 55 + 56 will give us a sum of 165. 2. There are ten students in a room. If they give a handshake for his classmate once and only once, how many handshakes can be made? Step 1. Understand the problem Let us say that you are one of those ten students in a room. To know how many handshakes could be made with of those ten students, we need name those ten students as A, B, C. …, I, and J. If you would be the A student you can give a handshake to different nine students. Now, if you give your hand to B, i.e., A to B, it is can be said the giving handshake by B to A is just only one count. Obviously, you cannot make a handshake to yourself as well as B to himself and so on. Step 2. Devising a plan To determine the total number of handshakes, it is very easy if we are going to create a table. Say, A B C D E F G H I J A B C D E F G H I J The x symbol represent that you cannot make a handshake to yourself and  symbol meaning that a handshake was made. 133  MATHEMATICS IN THE MODERN WORLD Step 3. Carry out the plan A B C D E F G H I J A x          B x x         C x x x        D x x x x       E x x x x x      F x x x x x x     G x x x x x x x    H x x x x x x x x   I x x x x x x x x x  J x x x x x x x x x x Total 1 2 3 4 5 6 7 8 9 So, adding this 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45. Hence, there are a total of 45 handshakes if these ten students give a handshake for his classmate once and only once. Step 4. Look back and review the solution Trying to double check the diagram, it is clearly seen that the total number of handshakes that could be made which is 45 is correct. 3. Five different points, say A, B, C, D, and E are on a plane where no three points are collinear. How many lines can be produced in these five points? Step 1. Understand the problem Based on the given problem, five points are on a plane where no three points are collinear. If you want to determine the number of lines from these five points, remember that the minimum number of points to produce a line is two. So, two points determine a line. Step 2. Device a plan The best way in order to determine the number of lines that can produce in a five different points where no three points are collinear is to plot those five 134  MATHEMATICS IN THE MODERN WORLD points in a plane and label it as A, B, C, D, and E. Then list down all the possible lines. Take note that line AB is as the same as line BA. Step 3. Carry out the plan List: AB BC CD DE AC BD CE AD BE AE Step 4. Look back and review the solution Counting all possible lines connected in five different points where no three points are collinear is still we could check that the total number of line could be produced of these five points are ten lines. 135  MATHEMATICS IN THE MODERN WORLD Learning Activity Directions: Solve the following problems with the use of Polya’s four-step problem solving procedures similar on the presentation on this topic. 1. The sum of three consecutive odd integers is 27. Find the three integers. 2. If the perimeter of a tennis court is 228 feet and the length is 6 feet longer than twice the width, then what are the length and the width? 3. There are 364 first-grade students in Park Elementary School. If there are 26 more girls than boys, how many girls are there? 4. If two ladders are placed end to end, their combined height is 31.5 feet. One ladder is 6.5 feet shorter than the other ladder. What are the heights of the two ladders? 5. A shirt and a tie together cost $50. The shirt costs $30 more than the tie. What is the cost of the shirt? Lesson 3.4 Mathematical Problems Involving Patterns Specific Objective At the end of this lesson, the student should be able to: 1. Demonstrate appreciation in solving problems involving patterns. 2. Show the appropriate strategies in solving problems which involve patterns. 3. Apply the Polya’s 4-step rule method in solving problems with patterns. 4. Make a correct conclusion based on their final result. 136  MATHEMATICS IN THE MODERN WORLD Introduction There are some problems that patterns may involve. One of the examples of problems that patterns are involve is an “abstract reasoning” where this kind of pattern is one of the type of exam that most of the Universities used in their entrance examination. Discussion Solving problems which involve pattern do not follow the steps on how to solve the problem on its traditional way. To be able to solve for this kind of problem, the following may be used as a guide: (i) showing an understanding of the problem, (ii) organising information systematically, (iii) describing and explaining the methods used and the results obtained, (iv) formulating a generalisation or rule, in words or algebraically. The following sample of questions gives an indication of the variety likely to occur in the examination. 1. A group of businessmen were at a networking meeting. Each businessman exchanged his business card with every other businessman who was present. a) If there were 16 businessmen, how many business cards were exchanged? b) If there was a total of 380 business cards exchanged, how many businessmen were at the meeting? Solution: a) 15 + 14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 120 exchanges 120 × 2 = 240 business cards. If there were 16 businessmen, 240 business cards were exchanged. b) 380 ÷ 2 = 190 190 = (19 × 20) ÷ 2 = 19 + 18 + 17 + … + 3 + 2 + 1 If there was a total of 380 business cards exchanged, there were 20 businessmen at the meeting. 137  MATHEMATICS IN THE MODERN WORLD 2. Josie takes up jogging. On the first week she jogs for 10 minutes per day, on the second week she jogs for 12 minutes per day. Each week, she wants to increase her jogging time by 2 minutes per day. If she jogs six days each week, what will be her total jogging time on the sixth week? Solution: Understand We know in the first week Josie jogs 10 minutes per day for six days. We know in the second week Josie jogs 12 minutes per day for six days. Each week, she increases her jogging time by 2 minutes per day and she jogs 6 days per week. We want to find her total jogging time in week six. Strategy A good strategy is to list the data we have been given in a table and use the information we have been given to find new information. We are told that Josie jogs 10 minutes per day for six days in the first week and 12 minutes per day for six days in the second week. We can enter this information in a table: Week Minutes per Day Minutes per Week 1 10 60 2 12 72 You are told that each week Josie increases her jogging time by 2 minutes per day and jogs 6 times per week. We can use this information to continue filling in the table until we get to week six. 138  MATHEMATICS IN THE MODERN WORLD Week Minutes per Day Minutes per Week 1 10 60 2 12 72 3 14 84 4 16 96 5 18 108 6 20 120 Apply strategy/solve To get the answer we read the entry for week six. Answer: In week six Josie jogs a total of 120 minutes. 3. You arrange tennis balls in triangular shapes as shown. How many balls will there be in a triangle that has 8 rows? Solution Understand We know that we arrange tennis balls in triangles as shown. We want to know how many balls there are in a triangle that has 8 rows. Strategy A good strategy is to make a table and list how many balls are in triangles of different rows. 139  MATHEMATICS IN THE MODERN WORLD One row: It is simple to see that a triangle with one row has only one ball. Two rows: For a triangle with two rows, we add the balls from the top row to the balls from the bottom row. It is useful to make a sketch of the separate rows in the triangle. 3=1+2 Three rows: We add the balls from the top triangle to the balls from the bottom row. 6=3+3 Now we can fill in the first three rows of a table. Number of Rows Number of Balls 1 1 2 3 3 6 We can see a pattern. To create the next triangle, we add a new bottom row to the existing triangle. The new bottom row has the same number of balls as there are rows. (For example, a triangle with 3 rows has 3 balls in the bottom row.) To get the total number of balls for the new triangle, we add the number of balls in the old triangle to the number of balls in the new bottom row. Apply strategy/solve: We can complete the table by following the pattern we discovered. Number of balls = number of balls in previous triangle + number of rows in the new triangle 140  MATHEMATICS IN THE MODERN WORLD Number of Rows Number of Balls 1 1 2 3 3 6 4 6+4=10 5 10+5=15 6 15+6=21 7 21+7=28 8 28+8=36 Answer There are 36 balls in a triangle arrangement with 8 rows. Check Each row of the triangle has one more ball than the previous one. In a triangle with 8 rows, row 1 has 1 ball, row 2 has 2 balls, row 3 has 3 balls, row 4 has 4 balls, row 5 has 5 balls, row 6 has 6 balls, row 7 has 7 balls, row 8 has 8 balls. When we add these we get: 1+2+3+4+5+6+7+8=36 balls 4. Andrew cashes a $180 check and wants the money in $10 and $20 bills. The bank teller gives him 12 bills. How many of each kind of bill does he receive? Solution Method 1: Making a Table Understand Andrew gives the bank teller a $180 check. 141  MATHEMATICS IN THE MODERN WORLD The bank teller gives Andrew 12 bills. These bills are a mix of $10 bills and $20 bills. We want to know how many of each kind of bill Andrew receives. Strategy Let’s start by making a table of the different ways Andrew can have twelve bills in tens and twenties. Andrew could have twelve $10 bills and zero $20 bills, or eleven $10 bills and one $20 bill, and so on. We can calculate the total amount of money for each case. Apply strategy/solve $10 bills $ 20 bills Total amount 12 0 $10(12)+$20(0)=$120 11 1 $10(11)+$20(1)=$130 10 2 $10(10)+$20(2)=$140 9 3 $10(9)+$20(3)=$150 8 4 $10(8)+$20(4)=$160 7 5 $10(7)+$20(5)=$170 6 6 $10(6)+$20(6)=$180 5 7 $10(5)+$20(7)=$190 4 8 $10(4)+$20(8)=$200 3 9 $10(3)+$20(9)=$210 2 10 $10(2)+$20(10)=$220 1 11 $10(1)+$20(11)=$230 0 12 $10(0)+$20(12)=$240 142  MATHEMATICS IN THE MODERN WORLD In the table we listed all the possible ways you can get twelve $10 bills and $20 bills and the total amount of money for each possibility. The correct amount is given when Andrew has six $10 bills and six $20 bills. Answer: Andrew gets six $10 bills and six $20 bills. Check Six $10 bills and six $20 bills →6($10)+6($20)=$60+$120=$180 The answer checks out. Let’s solve the same problem using the method “Look for a Pattern.” Method 2: Looking for a Pattern Understand Andrew gives the bank teller a $180 check. The bank teller gives Andrew 12 bills. These bills are a mix of $10 bills and $20 bills. We want to know how many of each kind of bill Andrew receives. Strategy Let’s start by making a table just as we did above. However, this time we will look for patterns in the table that can be used to find the solution. Apply strategy/solve Let’s fill in the rows of the table until we see a pattern. $10 bills $20 bills Total amount 12 0 $10(12)+$20(0)=$120 11 1 $10(11)+$20(1)=$130 10 2 $10(10)+$20(2)=$140 143  MATHEMATICS IN THE MODERN WORLD We see that every time we reduce the number of $10 bills by one and increase the number of $20 bills by one, the total amount increases by $10. The last entry in the table gives a total amount of $140, so we have $40 to go until we reach our goal. This means that we should reduce the number of $10 bills by four and increase the number of $20 bills by four. That would give us six $10 bills and six $20 bills. 6($10)+6($20)=$60+120=$180 Answer: Andrew gets six $10 bills and six $20 bills. Learning Activity 1. A pattern of squares is put together as shown. How many squares are in the 12th diagram? 2. Oswald is trying to cut down on drinking coffee. His goal is to cut down to 6 cups per week. If he starts with 24 cups the first week, then cuts down to 21 cups the second week and 18 cups the third week, how many weeks will it take him to reach his goal? 3. A new theme park opens in Milford. On opening day, the park has 120 visitors; on each of the next three days, the park has 10 more visitors than the day before; and on each of the three days after that, the park has 20 more visitors than the day before. How many visitors does the park have on the seventh day? How many total visitors does the park have all week? 4. Mark is three years older than Janet, and the sum of their ages is 15. How old are Mark and Janet? 144  MATHEMATICS IN THE MODERN WORLD 5. A pattern of squares is put together as shown. How many squares are in the 10th figure? Lesson 3.5 Recreational Problems Using Mathematics Specific Objective At the end of this lesson, the student should be able to: 1. Demonstrate appreciation of recreational games using the concept of mathematics. 2. Show student’s interest in a mathematical games by solving mathematical games. 3. Develop a sense of correct thinking to finish the game successfully. Introduction Puzzle, number games and mathematical riddles are some exciting games that we can solve or play. There are very essential most especially for the students in order to develop their critical thinking, enhance students’ computational work, deepen understanding with numbers and use different strategies and style of techniques through recreational games. In this new modern day, there are a lot of games that you may encounter not only in social media but also in different internet 145  MATHEMATICS IN THE MODERN WORLD site. Games are now easily downloaded and we can play this game on our palm hand. At the same time, person or persons that has high skills and knowledge in mathematics are those persons who could solve problems or games in mathematics easily. Discussion Now-a-days, there are a lot of recreational games that we could play whether it is an online or an offline games. Sometimes, we call this game as brain booster since our mind needs a lot of correct thinking on how to deal with the games and finish the game successfully. In this lesson, there would be an illustrative example that the students may look into on how to deal with a games using mathematical concept. 1. With the use of pencil or pen, connect by means of a line the nine dots (see figure below) without lifting a pen and re-tracing the line. To begin, you may think that it is easy for you to connect these nine dots by means of a line with the use of your pen or pencil with lifting the pen and re-tracing the line. On your first attempt, it could be like this. But as you can see, there is one dot left which is disconnected. 146  MATHEMATICS IN THE MODERN WORLD Trying for the second time, perhaps your presentation may looks like this. But still, there are dots which are disconnected. Let us use the technique called “think outside the box strategy” and for sure we can solve the puzzle. With this, we could extend the line or lines to connect the dots just like this one below. 2. How many squares could you find at the picture below? Some students would count manually the number of squares on the figure above. It is very tedious on the part of the student and it is prone to error. You cannot get the answer correctly at once if manually counting would be done. Not unless if you are very lucky to get the correct number of squares. But if we use the concept of mathematics here, you could be able to get accurately the total number 147  MATHEMATICS IN THE MODERN WORLD of squares in the figure. Remember that a square has an equal sides. Let us say an “n by n” is a square. The question is, how many squares are really on the figure above? First, let us always think that the square has an equal side. Let us ignore first the squares within the big square. On this figure, there are different “n x n” size of a square. The size of a big square is a 1 x 1 (12). But there is also a 2 x 2 (22) square on it as well as 3 x 3 and a 4 x 4 square. So, how many squares are there. If we look at the mathematical concept and we want to know the number of square, we need to add the different sizes of the squares such as; 12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30 squares Now, let us take a look on the squares within the big square. If you want to know the number of squares are there, let us use the upper selected square. The number of squares on the upper part would be; 12 + 22 = 1 + 4 = 5 squares The same as the selected square below, that is; 148  MATHEMATICS IN THE MODERN WORLD 12 + 22 = 1 + 4 = 5 squares Hence the total number of squares on the given figure is 30 + 5 + 5 = 40 squares. 3. A 3 x 3 grid table is given below. Filled out each cell of a digit from 1 – 9 except 5 since it is already given and without repetition where the sum of horizontal, vertical and diagonal are all equal to 15. 5 There are several ways to present the 3 x 3 grid table magic square number. First thing that you’re going to do is just to add all digits from 1 to 9 giving a sum of 45. In a 3 x 3 square number, you have to add three numbers again and again hence it will give an average that the sum of three number is 15, i.e. 45/3 = 15. This number 15 is what we called the magic number of a 3 x 3 square number where when you add three numbers horizontally, vertically and diagonally will give us a sum of 15. To achieve this, the number 5 should be placed in the middle part of a 3 x 3 square number just like in the given figure above. You can also achieve 15, if you add the middle number 5 three times. You can reduce 15 in a sum of three summands eight times: 15=1+5+9 15=2+4+9 15=2+6+7 15=3+5+7 15=1+6+8 15=2+5+8 15=3+4+8 15=4+5+6 The odd numbers 1,3,7, and 9 occur twice in the reductions, the even numbers 2,4,6,8 three times and the number 5 once. Therefore you have to place number 5 in the middle of the magic 3x3 square. The remaining odd numbers have to be in the middles of a side and the even numbers at the corners. Under these circumstances there are eight possibilities building a square and two of these are presented below. 2 9 4 2 7 6 7 5 3 9 5 1 6 1 8 4 3 8 149  MATHEMATICS IN THE MODERN WORLD 4. The figure below is arranged using 16 matchsticks to form 5 squares. Rearrange exactly 2 of the matchsticks to form 4 squares of the same size, without leaving any stray matchsticks. Solution: Learning Activity Directions: Do as indicated. 1. Place 10 coins in five straight lines so that each line contains exactly four coins. Can you arrange four coins so that if you choose any three of them (i.e., no matter which three of the four you pick), the three coins you chose form the corners of an equilateral triangle? 2. If you add the square of Tom's age to the age of Mary, the sum is 62; but if you add the square of Mary's age to the age of Tom, the result is 176. Can you say what the ages of Tom and Mary are? 3. A man and his wife had three children, John, Ben, and Mary, and the difference between their parents' ages was the same as between John and Ben and between Ben and Mary. The ages of John and Ben, multiplied together, equalled the age of the father, and the ages of Ben and Mary multiplied together equalled the age of the mother. The combined ages of the family amounted to ninety years. What was the age of each person? 150  MATHEMATICS IN THE MODERN WORLD 4. Refer to example number 3. Look for another remaining six (6) 3 x 3 magic square number. 5. Complete the grid table. Filled out the table (a) by digit from 1 to 4, (b) by digit from 1 to 9 and (c) by digit from 1 to 12; some digits are already identified whose sum is 8, 23 and 25 respectively. a. b. c. 151  MATHEMATICS IN THE MODERN WORLD Chapter Test 3 Test 1. TRUE OR FALSE Directions: Read the following statement carefully. Write T if the statement is true, otherwise write F on the space provided before each item. _______1. Deductive reasoning uses a set of specific observations to reach an overarching conclusion or it is the process of recognizing or observing patterns and drawing a conclusion. _______2. The conclusion formed by using inductive reasoning is often called a conjecture. _______3. Inductive reasoning is the process of reaching conclusion by applying general assumptions, procedure or principle or it is a process of reasoning logically from given statement to a conclusion. _______4. Conjecture is a form of deductive reasoning where you arrive at a specific conclusion by examining two other premises or ideas. _______5. In deductive reasoning, the two premises are the major and the minor premises and these are called an argument also known as syllogism. ______6. A categorical syllogisms follow the statement that "If A is part of C, then B is part of C". _____7. Intuition is an immediate understanding or knowing something without reasoning. _____8 A certainty is an inferential argument for a mathematical statement while proofs are an example of mathematical logical proof. ____9. An indirect proof is also known as contrapositive proof. ____10. Greg Polya is known as the “Father of Problem Solving”. Test 2. MULTIPLE CHOICE 152