Math Logical Reasoning PDF

West Visayas State University | 2020 Unit 3: PROBLEM SOLVING AND REASONING Learning Outcomes 1. Distinguished inductive reasoning from deductive reasoning. 2. Used different types of reasoning to justify statements and arguments made about mathematics and mathematical conce...

West Visayas State University | 2020 Unit 3: PROBLEM SOLVING AND REASONING Learning Outcomes 1. Distinguished inductive reasoning from deductive reasoning. 2. Used different types of reasoning to justify statements and arguments made about mathematics and mathematical concepts. 3. Solved problems involving patterns and recreational problems, following Polya’s four steps. 4. Organized his/her methods and approaches for proving and solving problems. Introduction By Barbara Davidson https://www.cashnetusa.com/blog/how-good-are-your-problem-solving-skills/ Most of us are constantly exposed to opportunities and possibilities in life. These may be in our work, at school and at home or in every aspects of our existence. Problem comes in different ways. It involves setting out to achieve some goals or objectives. Problem solving involves a set of factors and tasks to achieve a defined goal and solve real life problems and apply mathematics in real life situations. A big part of being an adult is making decisions on your own-every day is full 112  West Visayas State University | 2020 of them, from simple, like what to eat for breakfast, to the critical, like a choice of major, career, job or mate. How far do you think you’ll get in life if you flip a coin to make every decision? I’m going to say, “Not every’. Instead, it’s important to be able to analyze a situation based on logical thinking. What are the possible outcomes of making that decision? How likely it is that each choice will have positive or negative consequences? We call the process of logical thinking reasoning. It doesn’t take a lot of imagination to understand how important reasoning is in everyone’s life. You make not realize it, but every day in your life, you use two types of reasoning to make decisions and solve problems: inductive reasoning and deductive reasoning. Mathematical problem solving is one of the important topics to learn and also one of the most complex to teach. The main objective in teaching problem solving is that students develop generic ability to solve real life problems and apply mathematics in real situations. 113  West Visayas State University | 2020 Pre-test Direction: Among the four choices, select the letter of the correct answer. 1. What conjecture can be formulated / drawn? Pick an even number. Multiply it by 4. Add 8 to the product. Divide the answer by 2. Subtract, 2 times the original number. A. The new number is 4 times the original number. B. The original number is increased by 4. C. The original number is decreased by 4. D. The original number is equal to the new number. 2. What is the next number in the given list? 1, 5, 12, 22, 35, _____ A. 50 B. 51 C. 52 D. 53 3. What are the next two terms in the sequence? 20, A, 18, C, 15, F, 1, J, ___, ___ A. 5, P B. 5, O C. 6, Q D. 6, O 4. The first five terms of Fibonacci numbers are 1, 1, 2, 3, and 5. Find the 11 th and 12th term of the sequence. A. 21 and 34 B. 34 and 55 C. 55 and 144 D. 89 and 144 5. Find the nth term formula of a sequence 1, 5, 11, 19, 29, … A. 𝑎𝑛 = 𝑛2 + 𝑛 C. 𝑎𝑛 = 𝑛2 + 𝑛 + 1 B. 𝑎𝑛 = 𝑛2 − 𝑛 D. 𝑎𝑛 = 𝑛2 + 𝑛 − 1 6. How many 3 – letter words can be made using the letters a, o, r, s, and t? A. 6 B. 7 C. 8 D. 9 7. Find the third, fourth and fifth terms of the sequence defined by 𝑎1 = 3 , 𝑎2 = 5 , and 𝑎𝑛 = 2𝑎𝑛−1 − 𝑎𝑛−2 for 𝑛 ≥ 3. A. 6, 7, 9 B. 7, 8, 9 C. 7, 8, 10 D. 7, 9, 11 8. Thrisha likes these numbers: 222, 900, 144, and 121. Which of the following does she like? A. 1 B. 2 C. 3 D. 8 9. How many 2-digit numbers can be formed using 3, 4, 6, and 8 without repetition of the digit, if the number formed is higher than 60? A. 4 B. 5 C. 6 D. 7 10. In how many different ways can change be made for a piso coin using 25 centavo coin and / or 5 centavo coin? A. 2 B. 3 C. 4 D. 5 11. One person works for 3 hours and another person works for 2 hours. They are given a total of P 600. How it should be divided up so that each person 114  West Visayas State University | 2020 receives a fair share? A. A person who works for 3 hours should receive P340 and P 260 for the person who works for 2 hours. B. A person who works for 3 hours should receive P360 and P240 for the person who works for 2 hours. C. The amount should be divided equally between the two persons. D. The person who works longer should receive larger amount than the person who works shorter. 12. The tens digit of a two-digit number is one more than twice the units digit. Find the number if the sum of the digits is ten. A. 64 B. 73 C. 82 D. 91 13. A diagonal of a polygon is a line segment that connects nonadjacent vertices corners of the polygon. How many diagonals can be made using a heptagon? A. 9 B. 11 C. 14 D. 17 14. One-fourth of a certain student’s age six years ago equals one-eight of his age twenty years hence. How old is he now? A. 22 B. 32 C. 42 D. 45 15. Mario made a deposit of P250. If this deposit consisted of 30 bills, some ten- peso bills and the rest twenty-peso bill. How many twenty-peso bill are there? A. 40 B. 30 C. 20 D. 10 16. The total number of ducks and pigs in the field is 35 and the total number of legs is 98. How many ducks are there? A. 14 B. 17 C. 21 D. 23 17. On three grading periods, John obtained grades of 84, 90, and 78. What grade does John need on the fourth grading to raise his average to 87? A. 98 B. 96 C. 94 D. 90 18. At what time between 8 and 9 o’clock do the hands of a clock coincide? 2 2 7 7 A. 38 𝑚𝑖𝑛. B. 40 𝑚𝑖𝑛. C. 42 𝑚𝑖𝑛. D. 43 𝑚𝑖𝑛. 11 11 11 11 19. One pipe can fill the water container in 15 minutes. Another pipe can fill it in 9 minutes. How long would it take both pipes to fill the container? A. 5.256 min B. 5.526 min C. 5.625 min D. 5.265 min 20. How many kilograms of refine sugar must be added to 20 kilograms of a 10% sweetened juice concentrate to prepare instead a light syrup 18% sweet? A. 1.59 kgs B. 1.65 kgs C. 1. 75 kgs. D. 1.95 115  West Visayas State University | 2020 Lesson 3.1: MATHEMATICAL REASONING Mathematical Reasoning Is the critical thinking skill that enables a student to make use of all other mathematical skills like evaluating situations, selecting problem solving strategies, drawing logical conclusions, developing and describing solutions, and recognizing how these solutions can be applied. It is a part of mathematics where the truth values of the given statements are determined. The process of thinking about something in a logical way in order to form a conclusion/ judgment/ or conjecture. Two types of Mathematical Reasoning 1. Inductive Reasoning 2. Deductive Reasoning Inductive Reasoning is the process of reaching a general conclusion by examining the specific examples. The conclusion formed by using inductive reasoning is a conjecture, since it may or may not be correct. Since the conjecture may or may not be correct, you can give a counterexample. If for instance in one case a statement is not true, then the statement becomes false statement. It is only a true statement if it is true in all cases Example 1 1 is an odd number. 11 is an odd number. 21 is an odd number. Therefore, all numbers ending with 1 are odd numbers. Example 2 Essay test is difficult. Problem solving test is difficult. Therefore, all tests are difficult. 116  West Visayas State University | 2020 Uses of Inductive Reasoning 1. To predict a number Example 3 a. In the sequence 1, 3, 6, 10, 15, __. What is the next number? b. In the sequence 5, 10, 15, 20, ____, Find the missing number. Solution a. The first two numbers differ by 2. The second and the third differ by 3. It appears that the difference between any two numbers is always 1 more than the preceding difference. Since 10 and 15 differ by 5, we predict that the next number in the list will be 6 larger than 15, which is 21. b. Each successive number is 5 larger than the preceding number. Thus we predict that the next number in the list is 5 larger than 20, which is 25. Check your progress 1 Use inductive reasoning to predict the next number in each of the following lists. a. 3, 6, 9, 12, 15, __ b. 2, 5, 10, 17, 26, ___ 2. To make a conjecture Example 4 Pick a number multiply by 8, add 6 to the product, divide the sum by 2, and subtract 3. Complete the procedure for several different numbers and make a conjecture about the relationship between the size of the resulting number and the size of the original number. Solution Suppose we pick 5 as our original number. Then the procedure would produce the following results. Original number: 5 Multiply by 8 : 5 𝑥 8 = 40 Add by 6 : 40 + 6 = 46 Divide by 2 : 46 ÷ 2 = 23 Subtract by 3 : 20 − 3 = 20 ∴ When 5 is the original number, the number that will result is 4 times the given number 117  West Visayas State University | 2020 Check your progress 2: a. Given the table below, Number (n): 5 8 15 100 Result: 20 32 60 ? What is the result when a number (n) =100? What is the conjecture? b. Consider the following procedure. 1. Pick a number 2. Multiply by 9. 3. Add 15 to the product 4. Divide the sum by 3 5. and subtract by 5. Use the above procedure for any three numbers and make a conjecture. 3. To solve an application Galileo Galilei (1564 – 1642) used inductive reasoning to discover that the time required for a pendulum to complete one swing, called the period of the pendulum, depends on the length of the pendulum. Galileo did not have a clock, so he measured the period of pendulums in “heartbeats”. Example 5 Length of pendulum Period of pendulum (in units) (in heartbeats) 1 1 4 2 9 3 16 4 25 5∙ 36 6 The period of the pendulum is the time it takes for the pendulum to swing from left to right and back to its original position. a. If a pendulum has a length of 49 units, what is its period? b. if the length is quadrupled, what happens of its period? Solution a. 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. 118  West Visayas State University | 2020 b. A pendulum with a length of 4 units has a period that is twice that of a pendulum with a length of 1 unit. A pendulum with a length of 16 units has a period that is twice that of a pendulum with a length of 4 units. It appears that quadrupling the length of a pendulum doubles its period. Check your progress 3 A tsunami is a sea waver produced by an underwater earthquake. The height of a tsunami as it approaches land depends on the velocity of a tsunami. See the table below. Velocity of Tsunami Height of Tsunami (in feet / sec) (in feet) 6 4 9 9 12 16 15 25 18 36 21 49 24 64 a. What happens to the height of a tsunami when its velocity is doubled? b. What should be the height of a tsunami, if its velocity is 45ft. per second? 4.To find a counterexample A statement is a true statement, if it is true in all cases. If one case for which a statement is not true, called a counterexample, then the statement becomes false statement. Example 6 Verify that each of the following statement is a false statement by finding a counterexample. For all number x; a.) l x l> 0 b.) x 2 > 𝑥 c. √𝑥 2 = 𝑥 Solution A statement may have many counterexamples, but we need only one a. Let x = 0. Then |0| = 0. Because 0 is not greater than 0, we have found a counterexample. Thus “ for all numbers x, lxl> 0 “ is a false statement. b. Let x = 1 we have 12 = 1. Since 1 is not greater than 1, we have found a counterexample. Thus “for all numbers x, x2 = x “is a false statement. 119  West Visayas State University | 2020 c. Let x = -2 then √(−2)2 = 2. Since -2 is not equal to 2, we have found a counter example. Thus “for all numbers x, √𝑥 2 = 𝑥 is a false statement.” Check your progress 4 Verify that each of the following statements is a false by finding a counterexample for each. For all numbers x: 𝑥 𝑥+3 a. =1 b. = x+1 c. √(𝑥 2 + 16) = 𝑥 + 4 𝑥 3 Deductive Reasoning is the process of reaching a conclusion by applying assumptions, procedures, or principles. Deductive reasoning is used to establish a conjecture (an opinion or idea formed without proof or sufficient evidence / inference / supposition / guess). Example 7 All birds have feathers. Ducks are birds. Therefore, ducks have feathers. Example 8 Christopher is sick. If Christopher is sick, he won’t be able to go to work. Therefore, Christopher won’t be able to go to work. Example 9 Let us go back to the idea of previous example 4 in inductive reasoning. Pick a number multiply by 8, add 6 to the product, divide the sum by 2, and subtract by 3. Solution Let n = original number Multiply the number by 8; 8n Add 6 to the product; 8n + 6 Divide the sum by 2; (8n + 6) / 2 = 4n + 3 Subtract 3; 4n + 3 – 3 = 4n 120  West Visayas State University | 2020 Conclusion: We started with n and ended with 4n. The new number is 4 times the original number. Check your progress 5 Use deductive reasoning to show that the following procedure produces a number that is three times the original number. Procedure: Pick a number. Multiply the number by 6, add 10 to the product, divide the sum by 2, and subtract 5. Hint: Let n represent the original number. Self- Assessment Question 1 Direction: Answer the following as indicated. A. Determine whether each of the following arguments is an example of inductive reasoning or deductive reasoning. 1. All students in ADMU are smart. Mary Ann is a ADMU student. Therefore, Mary Ann is smart. 2. All passenger Jeepneys in Metro Manila are at least 10 years old. This jeepney is from Metro Manila. Therefore, this is at least 10 years old. 3. Squares have four sides. Rectangles have four sides. Therefore, squares are rectangles. 4. Obtuse angles are greater than 1800. This angle is 1900. Therefore, this angle is obtuse. 5. All men are good-looking. Some teachers are good looking. Therefore, some teachers are men. 6. Mark is a Science teacher. Mark is bald. Therefore, all science teachers are bald. 7. Marcelo did not win the game 2 days ago. Marcelo did not win the game yesterday. Thus, Marcelo will win the game today. 8. It is usually hot during summer season in the Philippines. It is summer now in the Philippines. Thus, it is hot now in the Philippines. 9. 25 is divisible by 5. 30 is divisible by 5. Therefore, numbers ending in 0 or 5 are divisible by 5. 10. All prime numbers are odd. 2 is a prime number. Therefore, 2 is an odd number. 121  West Visayas State University | 2020 B. Find a number that provides a counterexample to show that a given statement is false. (𝑥+ 1)(𝑥−1) 1. 𝐹𝑜𝑟 𝑎𝑙𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑥, 𝑥 3 ≥ 𝑥. 2. 𝐹𝑜𝑟 𝑎𝑙𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑥 , = 𝑥+1 (𝑥−1) C. Use deductive reasoning to show that the following procedure always produces a number 5. Procedure: Pick a number. Add 4 to the number and multiply the sum by 3. Subtract 7 and then decrease this difference by the triple of the original number. D. Make a conjecture for each of the following Fibonacci sums, where 𝐹𝑛 represents the 𝑛𝑡ℎ Fibonacci number. a. 𝐹𝑛 + 2𝐹𝑛+1 + 𝐹𝑛+2 = ? b. 𝐹𝑛 + 𝐹𝑛+1 + 𝐹𝑛+3 = ? E. Answer the following Logical Puzzle questions and Solve a Logic Puzzle 1. Meena’s mother has five daughters; Reene, Teena, Sheena, and Sheela. Who is the fifth daughter? 2. A woman had two girls who were born on the same hour of the same day of the same year. But they were not twins. How could this be so? 3. Each of the four friends Donna, Sarah, Nikkie and Shanelle has a four different pet (fish, cat, dog, and snake). From the following clues, determine the pet of each individual. a. Sarah is older than her friend who owns the cat and younger than her friend who owns the dog. b. Nikkie and her friend who owns the snake are both of the same age and the youngest members of their group. c. Donna is older than her friend who owns the fish. 4. “How many children do you have, and what are their ages?” ask the census taker. The mother answers, “I have three children. The product of their ages is 36, and the sum of their ages is the same as my house number.” The census taker looks at the house number, thinks for a moment, and responds, “I’m sorry, but I need more information.” “My oldest child likes chocolate ice cream,” says the mother. “Thank you”, replies the census taker. “I have all the information I require.” How old are the children? 122  West Visayas State University | 2020 Lesson 3.2: PROBLEM SOLVING Some basic terminologies in problem solving Problem – an inquiry of a given conditions to investigate or demonstrate a fact, result or law. It is also a question to be answered or solved, especially by reasoning or calculating. Problem solving – is a mathematical process of identifying, prioritizing, and selecting alternatives for a solution, and implementing a solution. It enables us to use the skills in a wide variety of situations. Math Patterns – are sequences that repeats based on a rule or rules. A rule is a set way to calculate or solve a problem. Sequence in mathematics – is an enumerated collection of objects in which repetitions are allowed and order does matter. It is an ordered list of numbers. Each number in a sequence is called a term of the sequence. 𝒏𝒕𝒉 − 𝒕𝒆𝒓𝒎 formula is formula that can be used to generate all the terms of a sequence Difference table is a table which shows the differences between successive terms of the sequence and in some cases, it can be used to predict the next term in a sequence. Problem Solving Involving Patterns Terms of sequence An ordered list of numbers such as 5, 14, 27, 44, 65, is called a sequence. The number in sequence that are separated by commas are called terms, a1 = 5, a2 = 14, a3 = 27, a4 = 44, a5 = 65,… The 3 dots mean the sequence continues beyond 65. It is customary to use the subscript notation an to designated the nth term of a sequence. a1 represents the first term of a sequence a2 represent the second term of a sequence a3 represents the third term of a sequence... an represents the nth term of a sequence. 123  West Visayas State University | 2020 In the sequence 2, 6, 12, 20, 30,…, n2 + n a1 = 2, a2 = 6, a3 = 12, a4 = 20, a5 = 30, an = n 2 + n When we examine the sequence, it is natural to ask; (1) what is the next term? (2) what formula or rule can be used to generate the terms? Difference table Another way of finding the next term in the sequence is by constructing a difference table. A difference table shows the differences between successive terms of the sequence and in some cases it can be used to predict the next term in a sequence. Example 1 What is the next term for the sequence 2, 5, 8, 11, 14, ___ using difference table? Solution Sequence First difference The first differences of the sequence are all the same. If we predict the next number in the sequence, 14 + 3 = 17 is the next term of the sequence. Example 2 Use a difference table for the sequence: 5, 14, 27, 44, 65,___ Solution In this table, the 1st differences are not all the same. It is helpful to compute the successive difference of the first differences. These are shown in row 2. The differences of the first differences are called second differences, the differences of the second are called third differences. Stop finding the differences if the last row of the differences are all the same. 124  West Visayas State University | 2020 In the example, the last row has a common difference of 4 this means that if you add 21 and 4 you get 25. Then to get the next term in the sequence you will add 65 and 25 yielding 90. Therefore, the next term is 90. Check your progress 6 Use a difference table to predict the next term of a sequence. a. 2, 7, 24, 59, 118, 207, ___ b. 1, 14, 51, 124, 245, 426, ___ c. 10, 10, 12, 16, 22, 30, ___ d. 15, 36, 77, 144, 243, ___ e. 1, 7, 17, 31, 49, 71, ___ 𝒏𝒕𝒉 – term formula for a sequence The 𝒏𝒕𝒉 term formula is used to predict the terms of the sequence. Example 3 In the formula 𝑎𝑛 = 3𝑛2 + 𝑛 , If you replace n with 1, 2, 3, 4, 5, and 6 in the formula then it generates the sequence; 4, 14, 30, 52, 80, 114. To find the 40th term of the sequence, replace n with 40 𝑎40 = 3(40)2 + 40 = 4840 Example 4 Assume the pattern shown by the square tiles in the given figures continues. Answer the following a. Find an nth term formula for the number of tiles in the nth figure of the sequence. b. How many tiles are there in the seventh figure of the sequence? c. Which figure will consist of exactly 80 tiles? Solution a. Observe each figure starting from the first to the last given figure. 125  West Visayas State University | 2020 First figure consists of two tiles, second has 5 tiles and third has 8 tiles. From the observations, first figure, 2 tiles; second 1 tile plus 2 tiles on each horizontal section; third figure from 2 tiles plus 3 tiles on each horizontal section and 2 tiles between the horizontal sections, and the next figure has 4 tiles on each horizontal section and 3 tiles between the horizontal sections. Thus, the number of tiles in the 𝒏𝒕𝒉 figure is determined by two groups of n plus a group of ( n – 1). In symbols, 𝑎𝑛 = 2𝑛 + (𝑛 − 1) or 𝑎𝑛 = 3𝑛 − 1 𝑎1 = 2(1) + (1 − 1) a2 = 5 a3 = 8 a4 = 11 =2+0 𝑎1 = 2 b. The number of tiles in the seventh figure is; 𝑎7 = 3𝑛 − 1 𝑎7 = 3 (7) − 1 𝑎7 = 21 − 1 𝑎7 = 20 c. To determine which figure in the sequence with 80 tiles, we use the equation 3𝑛 − 1 = 80. Finding the value of n; 3n = 80 + 1 (add 1 to both sides of the equation) 3n = 81 n = 27 (dividing each side by 3) Thus, the 27th figure is composed of 80 tiles. Check your progress 7 Assume that the pattern shown by the square tiles in the following figure continues. Answer the following: a. What is the nth term formula for the number of tiles in the nth figure of 126  West Visayas State University | 2020 the sequence? b. How many tiles are there in the tenth figure of the sequence? c. Which figure will have exactly 419 tiles? The Fibonacci Sequence Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before! Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy. "Fibonacci" was his nickname, which roughly means "Son of Bonacci" He helped spread Hindu-Arabic Numerals (like our present numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). © Bettman/Corbis https://www.mathsisfun.com/numbers/fibonacci-sequence.html https://www.mathsisfun.com/numbers/fibo nacci-sequence.html Romanesque broccoli spirals resemble the Fibonacci sequence. (Image: © Shutterstock) The Fibonacci sequence is one of the most famous formulas in mathematics. Each number in the sequence is the sum of the two numbers that precede it. So, the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The mathematical equation describing it is 𝑥𝑛+2 = 𝑥𝑛+1 + 𝑥𝑛 127  West Visayas State University | 2020 Fibonacci’s Rabbit Problem Leonardo of Pisa introduces the sequence with a problem involving rabbits. The problem goes as follows: At the beginning of a month let say December you are given a pair of new born male and a female rabbit. After a month the rabbits have produce no offspring; however, every month thereafter, the pair of rabbits produces another pair of rabbits. The offspring reproduce in exactly the same manner. If none of the rabbits dies, after a year, how many rabbits would you have? The answer, it turns out, is 144. http://oldeuropeanculture.blogspot.com/2018/02/fibonacci_24.html Fibonacci discovered that the number of pairs of rabbits for any month after the first two months can be determined by adding the numbers of pairs of rabbits in each of the two previous months and the formula used to get to that answer is what's now known as the Fibonacci sequence The Fibonacci Numbers 𝐹1 = 1 , 𝐹2 = 1 , 𝑎𝑛𝑑 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 𝑓𝑜𝑟 𝑛 ≥ 3 128  West Visayas State University | 2020 Polygonal Numbers The ancient greek mathematicians were interested in the geometric shapes associated with numbers. The 𝑛𝑡ℎ term formula for the different polygons were introduced. 129  West Visayas State University | 2020 Example 5 Use the definition of Fibonacci numbers to find the seventh and eighth Fibonacci numbers. Solution The first six Fibonacci numbers are 1, 1, 2, 3, 5, and 8. The seventh Fibonacci number is the sum of the two previous Fibonacci numbers. Thus, 𝐹7 = 𝐹6 + 𝐹5 𝐹8 = 𝐹7 + 𝐹6 = 8+5 = 13 + 8 = 13 = 21 Check your progress 8 Use the definition of Fibonacci numbers to find the ninth Fibonacci number. Determining the Properties of Fibonacci Numbers Example 6 Determine whether each of the following statements about Fibonacci numbers is true or false. Note: The first 10 terms of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, and 55. a. If n is even, then 𝐹𝑛 is an odd number. b. 2𝐹𝑛 − 𝐹𝑛−2 = 𝐹𝑛+1 𝑓𝑜𝑟 𝑛 ≥ 3 Solution a. An experiment of Fibonacci numbers shows that the second Fibonacci number, 1, is odd and the fourth Fibonacci number, 3, is odd, but the sixth Fibonacci number, 8, is even. Thus, the statement “If n is even, then 𝐹𝑛 is an odd number “is false. b. Experiment to see whether 2𝐹𝑛 − 𝐹𝑛−2 = 𝐹𝑛+1 for several values of n. For instance, for n = 7, we get 2𝐹𝑛 − 𝐹𝑛−2 = 𝐹𝑛+1 2𝐹7 − 𝐹7−2 = 𝐹7+1 2𝐹7 − 𝐹5 = 𝐹8 130  West Visayas State University | 2020 2(13) − 5 = 21 26 − 5 = 21 21 = 21 which is true. Evaluating 2𝐹𝑛 − 𝐹𝑛−2 for several additional values of , 𝑛 ≥ 3 , we find that in each case 2𝐹𝑛 − 𝐹𝑛−2 = 𝐹𝑛+1. Thus, by inductive reasoning, we conjecture 𝐹𝑛 − 𝐹𝑛−2 = 𝐹𝑛+1 𝑓𝑜𝑟 𝑛 ≥ 3 is a true statement. Note: This property of Fibonacci numbers can also be established using deductive reasoning. Check your progress 9 Determine whether each of the following statements about Fibonacci numbers is true or false. a. 2𝐹𝑛 > 𝐹𝑛+1 , 𝑓𝑜𝑟 𝑛 ≥ 3 b. 2𝐹𝑛 + 𝐹𝑛+1 = 𝐹𝑛+3 𝑓𝑜𝑟 𝑛 ≥ 3 131  West Visayas State University | 2020 Lesson 3.3: PROBLEM-SOLVING STRATEGIES Polya’s Problem -Solving Strategy Ancient mathematicians such as Euclid and Pappus were interested in solving mathematical problems, as well as, heuristics – the study of the methods and rules of discovery and invention. Rene Descartes (1596- 1650), a mathematician and a philosopher contributed to the field of heuristics. Descartes developed a universal problem-solving method. Gottfried Wilhelm Leibnitz (1646-1716) wrote a book on heuristics entitled “Art of invention” One of the foremost recent mathematicians to make a study of problem solving was George Polya (1887-1985). He was born in Hungary and moved to US.in 1940. The basic problem-solving strategy that Polya advocated consisted of the following four steps. Polya’s Four-Step Problem - Solving Strategy Polya’s four steps are deceptively simple. To become a good problem solver, it helps to examine each of these and steps and determine what is involved. 1. Understand the Problem Sometimes the problem lies in understanding the problem itself. To help us understand the problem, we might consider the following:  What is the goal? 132  West Visayas State University | 2020  What is being asked?  What is the condition?  What sort of a problem is it?  What is known or unknown?  Is there enough information?  Can you draw a figure to illustrate the problem?  Can you restate the problem in your own words? 2. Devise a Plan Devising a plan (translating) is a way to solve the problem by picturing how we are going to attack the problem.  Make a list of the known information.  Draw a diagram.  Make an organized list that shows all the possibilities.  Act it out.  Work backwards.  Make a table or chart.  Try to solve a similar but simpler problem.  Look for a pattern.  Use a variable, such as x.  Look for a formula / formulas.  Write an equation. If necessary, define what each variable represents.  Perform an experiment.  Guess at a solution and then check your result. 3. Carry Out the Plan In carrying out the plan ( solve ), we need to execute the equation we came up in step 2. The main key is to be patient and careful, even if we have necessary skills.  Be patient.  Work carefully.  Modify the plan or try a new plan.  Keep trying until something works.  Implement the strategy and strategies in step 2.  Try another strategy if the one isn’t working.  Keep a complete and accurate record of your work. 133  West Visayas State University | 2020  Be determined and don’t get discouraged if the plan does not work immediately. 4. Look back This step helps in identifying if there is a mistake in the solution. It is time to reflect and look back at what is done, what worked, and what didn’t.  Look for an easier solution.  Does the answer make sense?  Check the results in the original problem.  Interpret the solution with facts of the problem.  Recheck any computations involved in the solution.  Can the solution be extended to a more general case?  Ensure that all the conditions related to the problem are met.  Determine whether there is another method of finding the solution.  Ensure the consistency of the solution in the context of the problem. Examples 1 Use Polya’s 4 steps to solve the problems: 1. A baseball team won two out of their last four games. In how many different orders could they have two wins and two losses in four games? Solution a. Understand the Problem There are many different orders. The team may have won two straight games and lost the last two or maybe they lost the first two games and won the last two games. Of course, there are other possibilities such as WLWL. b. Devise a Plan Make an organized list of all possible orders and ensure that each of the different orders is accounted for only once. c. Carry Out the Plan Each entry in our list must contain two Ws and two Ls. Use a strategy to each order. One strategy is to start to write Ws, then write L if it is not possible to write W. This strategy produces six different orders. WWLL LWWL WLWL LWLW WLLW LLWW 134  West Visayas State University | 2020 d. Look Back The list has no duplications. So we are confident that there are 6 different orders in which a baseball team can win exactly two out of four games. 2. Two times the sum of a number and 3 is equal to thrice the number plus 4. Find the number. Solution a. Understand the Problem We need to make sure that we have read the question carefully several times. Since we are looking for a number, we will let x be a number. b. Devise a Plan We will translate the problem mathematically. Two times the sum of a number and 3 is equal to thrice the number plus 4. 2(x + 3) = 3x + 4 c. Carry Out the Plan We solve the value of x, algebraically. 2(x + 3) = 3x + 4 2x + 6 = 3x + 4 3x – 2x = 6–4 x = 2 d. Look Back If we take two times the sum of 2 and 3, that is the same as thrice the number2 plus 4 which is 10, so this does check, Thus, the number is 2. 3. Three siblings Sofia, Achaiah and Riana. Sofia gave Achaiah and Riana as much money as each had. Then Achaiah gave Sofia and Riana as much money as each had. Then Riana gave Sofia and Achaiah as much money as each had. Then each of the three had P128. How much money did each have originally? Solution a. Understand the Problem The problem is a little bit confusing and needs to be carefully analyzed. b. Devise a Plan We will be working backwards. c. Carry Out the Plan 135  West Visayas State University | 2020 There are four stages to this problem. We will number it from fourth to first. Fourth: Each has P128. Third: Riana gave Sofia and Achaiah as much money as each has. Second: Achaiah gave Sofia and Riana as much money as each has. First: Sofia gave Achaiah and Riana as much money as each has. Stages Sofia Achaiah Riana Fourth P 128 P128 P128 Third 64 64 256 Second 32 224 128 First 208 112 64 Initially Sofia had P208, Achaiah had P112, and Riana had P64. d. Look Back We check the result. Stages Sofia Achaiah Riana First 208 112 64 Second 208 – 112 – 64 = 32 112 + 112 = 224 64 + 64 = 128 Third 32 + 32 = 64 224 – 32 – 128 = 64 128 + 128 = 256 Fourth 64 + 64 = 128 64 + 64 = 128 256 – 64 – 64 = 128 Thus, Sofia, Achaiah, and Riana’s initial money are P208, P112, and P64, respectively. 4. If the length of the top of a rectangle is 15 inches more than its width and the area is 1,350 square inches. Find the dimension of the table. Solution a. Understand the Problem We are looking for the length and width of the rectangular table; and we will let l be the length and w be the width. It is indicated in the problem that the length is 15 inches longer than the width (l = 15 + w). 136  West Visayas State University | 2020 b. Devise a Plan We will apply the area of the rectangle formula Area = lw. 1,350 = lw = (15 + w) w c. Carry Out a Plan 1,350 = (15 + w) 1,350 = 15w + w2 w2 – 15w = 1,350 (w + 45) (w – 30) = 0 w + 45 = 0 w – 30 = 0 w = - 45 w = 30 Since, measurement cannot be negative, the width of the rectangle is 30. The length is, l = 15 + w = 15 + 30 = 45 inches. d. Look Back If the width of the rectangle is 30 inches and the length is 15 inches longer than the width which is 45 inches. The area of a rectangle is Area = lw = 30(45) = 1,350 square inches. Thus, the width is 30 and the length is 45 inches. 137  West Visayas State University | 2020 Lesson 3.4: RECREATIONAL PROBLEM SOLVING Recreational Mathematics Is a carried out by mathematical activity which involves mathematical puzzles and games. Most topics in recreational mathematics require no deeper knowledge in advanced mathematics. Some of the topics are the magic square, logic puzzles, aesthetics, culture mathematics and others. Problem solving through recreational mathematics is a collection of puzzles and other mathematical problems that are solved using things like graph theory and other math techniques Magic Square of Order n An arrangement of numbers in a square such that the sum of the n numbers in each row, column, and diagonal is the same number. Logic Puzzle Can be solved by using deductive reasoning, and a chart that enables us to display the given information in a visual manner. Kenken Puzzle An arithmetic-based logic puzzle that was invented by the Japanese mathematics teacher Tetsuya Miyamoto in 2004. Kenken are similar to Sudoko puzzles, but they also require you to perform arithmetic to solve puzzle. Magic Square of Order n An arrangement of numbers in a square such that the sum of the n numbers in each row, column, and diagonal is the same number. The magic square in the 1 st figure below has order 3, and the sum of the numbers is each row, column, and diagonal is 15; and the magic square in the 2nd figure below has order 4, and the sum of the number in each row, column, and diagonal is 34. 138  West Visayas State University | 2020 Logic Puzzle Can be solved by using deductive reasoning, and a chart that enables us to display the given information in a visual manner. Example 1 Each of four neighbors, Sean, Maria, Sarah and Brian has a different occupation; (Editor, banker, chef, or dentist). From the following clues, determine the occupation of each neighbors. 1. Maria gets home from work after the banker but before the dentist. 2. Sarah, who is the last to get home from work, is not an editor. 3. The dentist and Sarah leave for work at the same time. 4. The banker lives next door to Brian. Solution From Clue 1, Maria is not the banker or dentist. In the following chart, write X1 in the Banker and the Dentist columns of Maria’s row. Editor Banker Chef Dentist Sean Maria X1 X1 Sarah Brian From clue 2, Sarah is not the editor. Write X2 in the editor column of Sarah’s row. We know from clue 1 that the banker is not the last to get home, and we know from clue 2 that Sarah is the last to get home; Therefore, Sarah is not the Banker. Write X2 in the Banker column of Sarah’s row. Editor Banker Chef Dentist Sean Maria X1 X1 Sarah X2 X2 Brian From clue 3, Sarah is not the dentist. Write X3 for this condition. There are now Xs for three of the four occupation in Sarah’s row; therefore, Sarah must 139  West Visayas State University | 2020 be the chef. Place a  in the box. Since Sarah is the chef, none of the other three people can be the chef. Write X3 for these conditions. There are Xs for the three of the four occupations in Maria’s row; therefore, Maria must be the editor. Insert a  to indicate that Maria is the editor, and write X3 twice to indicate that neither Sean nor Brian is the editor. Editor Banker Chef Dentist Sean X3 X3 Maria  X1 X3 X1 Sarah X2 X2  X3 Brian X3 X3 From clue 4, Brian is not the banker. Write X4 for this condition. Since there are three Xs in the Banker column, Sean must be the banker. Place a  in that box. Thus, Sean cannot be the dentist. Write X4 in the box. Since there are 3 Xs in the Dentist column, Brian must be the dentist. Place  in that box. Editor Banker Chef Dentist Sean X3  X3 X4 Maria  X1 X3 X1 Sarah X2 X2  X3 Brian X3 X4 X3  Sean is the banker, Maria is the editor, Sarah is the chef, and Brian is the dentist. Kenken Puzzles An arithmetic-based logic puzzle that was invented by the Japanese mathematics teacher Tetsuya Miyamoto in 2004. The noun “ken” has “knowledge” and “awareness” as synonyms. Hence, Kenken translates as knowledge squared, or awareness squared. Kenken are similar to Sudoko puzzles, but they also require you to perform arithmetic to solve puzzle. 140  West Visayas State University | 2020 Rules for Solving a Kenken puzzle For a 3 by 3 puzzle, fill in each box (square) of the grid with one of the numbers 1, 2, or 3. For a 4 by 4 puzzle, fill in each square of the grid with one of the numbers 1, 2, 3, or 4. For a n by n puzzle, fill in each square of the grid with one of the numbers 1, 2, 3,…, n. Grids range in size from a 3 by 3 up to a 9 by 9.  Do not repeat a number in any row or column.  The numbers in each heavily outlined set of squares called cages, must combine (in some order) to produce the target number in the top left corner of the cage using mathematical operation indicated.  Cages with just one square should be filled in with the target number.  A number can be repeated within a cage as long as it is not in the same row or column. Example 2 4 x 4 Puzzle 6x 7+ 2 8x 4x 12x 1- 1 6x 7+ 2 1 3 4 2 8x 3 2 4 1 4x 12x 1- 1 4 2 3 1 4 3 1 2 141  West Visayas State University | 2020 Self- Assessment Question 2 Direction: Answer the following as indicated. A. Write the first five terms of the sequence whose nth term is given by the formula. 1. 𝑎𝑛 = 1 – 2𝑛 2. 𝑎𝑛 = 𝑛3 – 1 3. 𝑎𝑛 = (𝑛 + 1)2 4. 𝑎𝑛 = 3𝑛 5. 𝑎𝑛 = 𝑛2 + 2𝑛 B. Use the difference table to predict the next term in the sequence. 1. 3 , 10 , 17 , 24 , 31 ,... 2. 1 , 4 , 9 , 16 , 25 , 36 ,... 3. 3 , 6 , 11 , 18 ,... 4. 15 , 36 , 77 , 144 , 243 ,... 5. 12 , 45 , 98 , 171 , 264 ,... C. Find a pattern. 1.The figure shows a series of rectangles where each rectangle is bounded by 10 dots.. a) How many dots are required for 7 rectangles? b) If the figure has 73 dots, how many rectangles would there be? 2. Each triangle in the figure below has 3 dots. Study the pattern and find the number of dots for 6 layers of triangles. 142  West Visayas State University | 2020 3. The following figures were formed using matchsticks a. How many triangles are there if the figure in the series has 8 squares? b. How many matchsticks would be used in the figure in the series with 10 squares? D. Solve each of the following problems. Show pertinent solutions clearly. 1 1. Ruben has a bag containing 128 chicos. He sold of them to Misael. Next he sold 4 1 of was left to Andres. He then selected the biggest chico and gave it to his 4 mother. How many chicos are left to Ruben? 2. Karlo’s father is 26 years older than Karlo. The sum of their ages five years ago is 42. How old are they now? 3. A number has two digits whose sum is 10. The difference between the original number and the number resulting from an interchange of its digits is 54. What is the original number? 4. How soon after one o’clock will the hands of the clock extend in opposite directions? 5. An athlete jogs a certain distance at the rate of 8 kph and returns over the same 1 track running at the rate of 24 kph. If it took him a total time of 2 hours, what is 2 the total distance he covered? 6. A consumer goods manufacturer wanted to blend two grades of coffee, one costing P1.50 per kg. the other 200 kgs costing P1.90 per kg. to be sold at P1.60 per kg. How much coffee costing P1.50 per kgs would be needed? 7. A sum of money amounting to P5.15 consists of 10 centavo coins (dimes) and 25 centavo coins (quarters). If there are 32 coins in all, how many 25 centavo coins (quarters) are there? 143  West Visayas State University | 2020 8. A term paper consists of 10,000 words. Martha can type 60 words per minute, while Jess can type 40 words per minute. In how much time can they finish the job if they started together? 9. On three examinations, Myra received scores of 82, 91, and 76. What score does Myra need on the fourth examination to raise her average to 85? 10. A cube whose side measures 8 units is painted outside. After painting, the cube is disassembled into 512-unit cubes. How many of the 512 units cubes have no paint on its surface? 11. Five cousins met at the family reunion. Each cousin kisses each of the other cousins just once. How many kisses were given in all? 12. Sharon wrote down a number and added 2 to it, she multiplied by 6, then subtracted 7 and the answer was 23. What was the original number that she wrote down? SUMMARY In mathematics, we are exposed to opportunities and possibilities in attaining our goals and objectives. Making decisions is important in achieving a certain goals. The decisions may or may not be correct but we need to have a conclusions or judgment. Reaching a conclusion needs our logical way of thinking, or reasoning. Reasoning may it be inductive or deductive way of reaching a conclusion /judgment or conjecture. Inductive reasoning is the process of reaching a general conclusion by examining the specific examples. The conclusion formed by using inductive reasoning is conjecture, since it may or may not be correct. The process of reaching conclusion by applying assumptions, procedures, or principles is deductive reasoning. It is used to establish a conjecture ( an o pinion or idea formed without proof or sufficient evidence / inference / supposition / guess ). A statement is a true statement provided it is true in all cases. If you can find one case in which a statement is not true, called counterexample , then a statement is a false statement. Mathematical problem is one of the important topics to learn and also one of the most complex topics to teach. Problem is an inquiry of a given conditions to investigate or demonstrate a fact, result or law. It is also a question to be answered or solved, especially by reasoning or calculating. A mathematical process of 144  West Visayas State University | 2020 identifying, prioritizing, and selecting alternatives or a solution, and implementing a solution is problem solving. It enables us to use the skills in a wide variety of situations. Numbers are everywhere in our daily lives and mathematics is based on numbers. Mathematics is useful to predict, and number pattern is about prediction. Number pattern leads directly to the concept of functions in mathematics in relation to different quantities which is defined as a list of the same number following a particular sequence. Number pattern can also be applied to problem-solving whether a pattern is present. Math patterns are sequences that repeats based on a rule or rules. A rule is a set way to calculate or solve a problem. Sequence is an ordered list of numbers. Each number in a sequence is called a term. A formula that can be used to generate all terms of a sequence is called an nth-term formula. One way to determine the next term in a sequence is through the difference table. In problem solving, different techniques and strategies were being used to obtain the truth value (s) or solutions of the given problem or situations. To solve or answer the problem one of the famous strategy by Polya’s Four-Step problem solving strategy , understand the problem , devise a plan , carry out the plan and look back. Problem solving through recreational mathematics is carried out mathematical activity which involves puzzles, logic puzzles, and games and other mathematical problems that are solved using things like graph theory and other math techniques. 145  West Visayas State University | 2020 Post-test Direction: Among the four choices, select the letter of the correct answer. 1. What conjecture can be formulated / drawn? Pick an even number. Multiply it by 4. Add 8 to the product. Divide the answer by 2. Subtract, 2 times the original number. A. The new number is 4 times the original number. B. The original number is increased by 4. C. The original number is decreased by 4. D. The original number is equal to the new number. 2. What is the next number in the given list? 1, 5, 12, 22, 35, _____ A. 50 B. 51 C. 52 D. 53 3. What are the next two terms in the sequence? 20, A, 18, C, 15, F, 1, J, ___, ___ A. 5, P B. 5, O C. 6, Q D. 6, O 4. The first five terms of Fibonacci numbers are 1, 1, 2, 3, and 5. Find the 11 th and 12th term of the sequence. A. 21 and 34 B. 34 and 55 C. 55 and 144 D. 89 and 144 5. Find the nth term formula of a sequence 1, 5, 11, 19, 29, … A. 𝑎𝑛 = 𝑛2 + 𝑛 C. 𝑎𝑛 = 𝑛2 + 𝑛 + 1 B. 𝑎𝑛 = 𝑛2 − 𝑛 D. 𝑎𝑛 = 𝑛2 + 𝑛 − 1 6. How many 3 – letter words can be made using the letters a, o, r, s, and t? A. 6 B. 7 C. 8 D. 9 7. Find the third, fourth and fifth terms of the sequence defined by 𝑎1 = 3 , 𝑎2 = 5 , and 𝑎𝑛 = 2𝑎𝑛−1 − 𝑎𝑛−2 for 𝑛 ≥ 3. A. 6, 7, 9 B. 7, 8, 9 C. 7, 8, 10 D. 7, 9, 11 8. Thrisha likes these numbers: 222, 900, 144, and 121. Which of the following does she like? A. 1 B. 2 C. 3 D. 8 9. How many 2-digit numbers can be formed using 3, 4, 6, and 8 without repetition of the digit, if the number formed is higher than 60? A. 4 B. 5 C. 6 D. 7 10. In how many different ways can change be made for a piso coin using 25 centavo coin and / or 5 centavo coin? A. 2 B. 3 C. 4 D. 5 11. One person works for 3 hours and another person works for 2 hours. They are given a total of P 600. How it should be divided up so that each person 146  West Visayas State University | 2020 receives a fair share? A. A person who works for 3 hours should receive P340 and P 260 for the person who works for 2 hours. B. A person who works for 3 hours should receive P360 and P240 for the person who works for 2 hours. C. The amount should be divided equally between the two persons. D. The person who works longer should receive larger amount than the person who works shorter. 12. The tens digit of a two-digit number is one more than twice the units digit. Find the number if the sum of the digits is ten. A. 64 B. 73 C. 82 D. 91 13. A diagonal of a polygon is a line segment that connects nonadjacent vertices corners of the polygon. How many diagonals can be made using a heptagon? A. 9 B. 11 C. 14 D. 17 14. One-fourth of a certain student’s age six years ago equals one-eight of his age twenty years hence. How old is he now? A. 22 B. 32 C. 42 D. 45 15. Mario made a deposit of P250. If this deposit consisted of 30 bills, some ten- peso bills and the rest twenty-peso bill. How many twenty-peso bill are there? A. 40 B. 30 C. 20 D. 10 16. The total number of ducks and pigs in the field is 35 and the total number of legs is 98. How many ducks are there? A. 14 B. 17 C. 21 D. 23 17. On three grading periods, John obtained grades of 84, 90, and 78. What grade does John need on the fourth grading to raise his average to 87? A. 98 B. 96 C. 94 D. 90 18. At what time between 8 and 9 o’clock do the hands of a clock coincide? 2 2 7 7 A. 38 𝑚𝑖𝑛. B. 40 𝑚𝑖𝑛. C. 42 𝑚𝑖𝑛. D. 43 𝑚𝑖𝑛. 11 11 11 11 19. One pipe can fill the water container in 15 minutes. Another pipe can fill it in 9 minutes. How long would it take both pipes to fill the container? A. 5.256 min B. 5.526 min C. 5.625 min D. 5.265 min 20. How many kilograms of refine sugar must be added to 20 kilograms of a 10% sweetened juice concentrate to prepare instead a light syrup 18% sweet? A. 1.59 kgs B. 1.65 kgs C. 1. 75 kgs. D. 1.95 147  West Visayas State University | 2020 References Aufmann, R., et al. (2017) Mathematics in the Modern World. Philippine ed. Rex Bookstore ( Cengage) (Chapter 3) Aufmann, R., et al. (2013) Mathematical Excursions. 3rd ed. Belmont, CA: Cengage Learning (Chapter 1) Medallon, Merlita C., et. al. (2018) Mathematics in the Modern World (Worktext) Minshapers Co, Inc. (Chapter 3) Sirug, Winston S. (2018) Mathematics in the Modern World. Philippine ed., Mindshapers Co. Inc. (Chapter 3) Young, Johnny c. (2003) Problem Solving Reviewer in Algebra ( For High School and College). O’dell , Isabelita c. , et. al. NUMLOCK 1. https://www.onlinemathlearning.com › http://oldeuropeanculture.blogspot.com/2018/02/fibonacci_24.html https://www.mathsisfun.com/numbers/fibonacci-sequence.html https://www.cashnetusa.com/blog/how-good-are-your-problem-solving-skills/ 148