Mathematics in the Modern World Learning Modules

 DISCLAIMER These unpolished learning modules were compiled and prepared for personal use of students in GEC 05: Mathematics in the Modern World of Southern Luzon State University (SLSU) ONLY, and not as a reference material. Unauthorized distribution of the...

 DISCLAIMER These unpolished learning modules were compiled and prepared for personal use of students in GEC 05: Mathematics in the Modern World of Southern Luzon State University (SLSU) ONLY, and not as a reference material. Unauthorized distribution of the modules is not allowed. The topics included are given in summary form and does not claim to be complete. The instructors do not claim ownership of all the contents since it was taken from several resources including books, journals, and the internet.  Module 3 PROBLEM SOLVING AND REASONING “Every problem has a solution. You just have to be creative enough to find it.” - Travis Kalanick INTRODUCTION Problems are inevitable. And because of that, one must learn the necessary skills to arrive at solutions to different types of problems encountered. It is imperative that we study different approaches and strategies in problem solving. This module presents the fundamental principles of problem solving along with some specific processes that can be applied to certain types of problems. Its main goal is to develop students’ skills in pattern recognition, critical and logical thinking, and creativity. This also aims to help students become better problem solvers, develop interest in problem solving and consider it as an enjoyable experience. LEARNING OUTCOMES At the end of this module, students should be able to: 1. Explain Polya’s four-step problem solving strategy; 2. Apply Polya’s four-step problem solving strategy in solving certain problems; 3. Distinguish inductive reasoning from deductive reasoning; 4. Give examples of inductive and deductive processes of inference; and 5. Use either the inductive or deductive reasoning to solve practical problems; and 6. Solve problems using different approaches and strategies. DISCUSSION I. Polya’s Strategy George Polya, known as the father of modern problem solving, created his famous four-step process for problem solving. The steps are: Have a clear understanding of the problem. Understand the Problem Enumerate the steps to be taken to solve the problem: List down the given information, use variables to represent unknowns, translate to mathematical sentences, draw, create a table, Device a etc. Plan Carry out the plan carefully and accurately, taking note all the attempts. Carry out the Plan Ensure the solution is consistent with the facts of the problem, interpret the solution in the Review the context of the problem, review the details, and check the validity of the final answer. Solution Example 1: A baseball team won two out of their last four games. In how many different orders could they have two wins and two losses in four games? Solution: Understand the Problem – There are many different orders. The team may have won two straight games and lost the last two (𝑊𝑊𝐿𝐿) or they may have lost in the first two games and won the last two (𝐿𝐿𝑊𝑊). Of course there are other possibilities, such as 𝑊𝐿𝑊𝐿. Devise a Plan – We will make an organized list of all the possible orders. An organized list is a list that is produced using a system that ensures that each of the different orders will be listed once and only once. Carry Out the Plan – Each entry in our list must contain two 𝑊s and two 𝐿s. We will use a strategy that makes sure each order is considered, with no duplications. One such strategy is to always write a 𝑊 unless doing so will produce too many 𝑊s or a duplicate of one of the previous orders. If it is not possible to write a 𝑊, then and only then do we write an 𝐿. This strategy produces the six different orders shown below. 1. 𝑊𝑊𝐿𝐿 (Start with two wins) 2. 𝑊𝐿𝑊𝐿 (Start with one win) 3. 𝑊𝐿𝐿𝑊 4. 𝐿𝑊𝑊𝐿 (Start with one loss) 5. 𝐿𝑊𝐿𝑊 6. 𝐿𝐿𝑊𝑊 (Start with two losses) Review the Solution – We have made an organized list. The list has no duplicates and the list considers all possibilities, so we are confident that there are six different orders in which a baseball team can win exactly two out of four games. Example 2: The product of the ages, in years, of three teenagers is 4590. None of the teens are the same age. What are the ages of the teenagers? Solution: Understand the Problem – We need to determine three distinct counting numbers, from the list 13, 14, 15, 16, 17, 18, and 19, that have a product of 4590. Devise a Plan – If we represent the ages by 𝑥, 𝑦, and 𝑧, then 𝑥𝑦𝑧 = 4590. We are unable to solve this equation, but we notice that 4590 ends in a zero. Hence, 4590 has a factor of 2 and a factor of 5, which means that at least one of the numbers we seek must be an even number and at least one number must have 5 as a factor. The only number in our list that has 5 as a factor is 15. Thus 15 is one of the numbers, and at least one of the other numbers must be an even number. At this point we try to solve by guessing and checking. Carry Out the Plan – 15 ∗ 16 ∗ 18 = 4320 → No, this product is too small. 15 ∗ 16 ∗ 19 = 4560 → No, this product is too small. 15 ∗ 17 ∗ 18 = 4590 → Yes, this is the correct product. The ages of the teenagers are 15, 17, and 18. Review the Solution – Because 15 ∗ 17 ∗ 18 = 4590, and each of the ages represents the age of a teenager, we know our solution is correct. None of the numbers 13, 14, 16, and 19 is a factor (divisor) of 4590, so there are no other solutions. II. Problem Solving Strategies 1. Guess and test (Trial and error) – this is the most common method that students tend to use when solving a given problem. This includes trying different possible answers to the question and checking if it is correct and if not, then try another guess. Example: Place the digits 8, 9, 10, 12, & 13 in the circles so that the sum across and vertically equal to 31. It is possible to systematically answer this problem, but since we are dealing with just 5 numbers and addition is the only operation, it will be faster and easier to just simply guess it. Few trials will be enough to correctly answer the problem. (Try it!) 2. Picture/Diagram/Experiment – this method is used by most people because it is much easier to solve a problem if you can draw or picture it, or try it out yourselves, instead of just imagining things. Example: Using only a 5L and an 11L can, how can you have exactly 7L of water? This is one of the problems that were presented in class during our group activities. 5L 11L And solving this easily would involve actually experimenting on it, or drawing scenarios or steps. (Try it!) 3. Working backwards – this is when we are given a problem wherein a final output is given, and we are asked to find something at the beginning or somewhere at the middle. Since the final output is provided, the best way is to start from there, then work backwards, reversing all actions. Example: Mary is thinking of a number. If you double it, and subtract 7 you obtain 11. What is the number? Solution: Since we are given the final answer which is 11, it is best to start from there, and then go backwards. First, we see that 11 is obtained after subtracting 7 to the previous number. Since we work backwards, instead of subtracting, we add. Thus, the previous number is 11 + 7 = 18. From 18 18, we see that it is obtained by doubling the previous number. Thus, the previous number is = 2 9. Therefore, 9 is Mary’s number. We can check it by substituting 9 to the problem. We have 9, if we double it, we get 18, and subtracting 7 from 18, we get 11. So our answer is correct. 4. Looking for patterns – this is used when we can find a pattern in a given problem or sequence. (More examples are presented in Section IV.) Example: Fill the missing the number in the sequence 1, 4, 9, 16, ___, 36, 49, 64, 81, 100, … Solution: In this sequence, we can observe that the given numbers are actually perfect squares, 1 = 12 , 4 = 22 , 9 = 32 , 16 = 42 and so as the others. Thus, we can conclude that the missing number is actually 52 which is equal to 25. 5. Listing/tabular – this is often used when we are asked to answer problems where we need to list down possibilities so we can look at it properly. Of course we can just use a scratch paper and write things randomly but this is more organized and easier to look at. Example: Let 𝐹(𝑛) denote the 𝑛𝑡ℎ term in the Fibonacci Sequence where 𝐹(1) = 1, 𝐹(2) = 1, 𝐹(3) = 2 and so on. Find the least value of 𝑛 such that 𝐹(𝑛) > 500. Solution: In this problem, we are asked to find the smallest 𝑛 so that 𝐹(𝑛) is greater than 500. One way to do it is to make a table and list down the terms of the Fibonacci sequence with the corresponding 𝑛. 𝒏 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 𝑭(𝒏) 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 In this table, all we do is to find the smallest 𝑛 so that 𝐹(𝑛) > 500. We see that 610, 987, 1597 are all greater than 500. Therefore, the possible values of 𝑛 that will give us these values are 15, 16, 17, and even greater. But, we are only asked to find the smallest. And the smallest here is 15. So the answer is 15. 6. Algebraic equations – this method is used when it is not enough to simply look at the problems, draw it, or guess. This happens when the given problem is too complex and there are way too many possible answers, or the values are too big. Example: The sum of the two digits of a 2-digit number is 11. Reversing the digits increase the number by 45. What is the number? Solution: We use Polya’s strategy in solving this problem We enumerate the steps as follows: Understand the problem – In this problem, we are asked to find a two digit number. It says that if we add the two digits of this number, we get 11. (For example, 65 is possible because 6 + 5 = 11, but that’s not the only possible combination.) Next, it tells us that if reverse the numbers, meaning if we interchange their positions, then the resulting number is 45 more than the original (Well, if we reversed 65, it will be 56, so definitely it’s not the answer). Translate – After understanding the problem, we now try to translate it to mathematical sentences. First, how do we translate “The sum of the two digits of a 2-digit number is 11”? Since we are dealing with two unknown values, we can use two variables to represent them, say 𝑥 is the first and 𝑦 is the second. Therefore, 𝑥 + 𝑦 = 11. (Equation 1) But what is the number? Is it 𝑥𝑦? No. Because if it is 𝑥𝑦, you will be multiplying the two digits, and that is not right (It is different when we are dealing with variables already). For example, if you have 65, we know that to get 65 out of the digits 6 and 5, we have to multiply the tens digit with 10 and add the ones digit. So 10(6) + 5 = 60 + 5 = 65. Now, if we have the 𝑥 and 𝑦 as variables, then the number that we are looking for is  10𝑥 + 𝑦 (The unknown number) But since we have two variables to solve, it is not enough that we only have one equation (The second one is not an equation). Thus, we need another one. Going back to the problem, it tells us that if we reversed the digits, we get a number that is 45 more than the original one. So we are saying that if we have 10𝑦 + 𝑥 (we interchanged 𝑥 and 𝑦), then this will be equal to 45 + (10𝑥 + 𝑦). Thus we have the second equation 10𝑦 + 𝑥 = 45 + 10𝑥 + 𝑦 (Equation 2) Solve – Since we have enough equations to solve for the unknown values, we can already solve it. 𝑥 + 𝑦 = 11 → 𝑦 = 11 − 𝑥 Rewrite Equation 1. 10𝑦 + 𝑥 = 45 + 10𝑥 + 𝑦 Simplify Equation 2 by combining like terms 10𝑦 − 𝑦 + 𝑥 − 10𝑥 = 45 and leaving the variables in one side and the 9𝑦 − 9𝑥 = 45 constant in the other side of the equation. 9(11 − 𝑥) − 9𝑥 = 45 Since 𝑦 = 11 − 𝑥 from Equation 1, substitute 99 − 9𝑥 − 9𝑥 = 45 11 − 𝑥 to 𝑦 in Equation 2. Solve for 𝑥. 99 − 18𝑥 = 45 −18𝑥 = 45 − 99 −18𝑥 = −54 −18𝑥 −54 = −18 −18 𝒙=𝟑 𝑦 = 11 − 3 Solve for 𝑦 using the computed value of 𝑥 and 𝒚=𝟖 Equation 1. Therefore, we have 𝑥 = 3 and 𝑦 = 8, so our number is 38. Check – We now try to check if our answer is correct. If we have 38, then the sum of its digits, 3 and 8 is obviously 11. Now, reversing the digits, we have 83. We can check that 83 = 38 + 45. Therefore, 38 is correct. 7. Logical Reasoning – this deals with the way we analyze things, and how we come up with solutions to our problems. This can go with every other method that was discussed earlier, whichever is appropriate. There are two types of reasoning which will be discussed in the next section. III. Inductive and Deductive Reasoning 1. Inductive Reasoning – the type of reasoning that forms a conclusion based on the examination of specific examples. Example 1: During the past 10 years, a tree has produced plums every other year. Last year, the tree did not produce plums, so this year, the tree will produce plums. We arrive at our answer based on our observation on what has been happening for the past years, thus, this might not be that accurate, but, it is what is most likely to happen based on experience. Example 2: Use inductive reasoning to predict the next number in the sequence 5, 10, 15, 20, ? Solution: 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.  The conclusion formed using inductive reasoning is often called a conjecture, since it may or may not be correct. Example 3: 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. The following table shows some results obtained for pendulums of various lengths. (For the sake of convenience, a length of 10 inches has been designated as 1 unit.) If a pendulum has a length of 49 units, what is its period? If the length of a pendulum is quadrupled, what happens to its period? Length of pendulum Period of pendulum, in units in heartbeats 1 1 4 2 9 3 16 4 25 5 36 6 Solution: 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. To answer the second question, note that 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.  A statement is a true statement provided that it is true in all cases. If you can find one case for which a statement is not true, called a counterexample, then the statement is a false statement. Example 4: Verify that each of the following statements is a false statement by finding a counterexample. For all numbers 𝑥: a) |𝑥 | > 0 b) 𝑥 2 > 𝑥 Solution: A statement may have many counterexamples, but we need only fi nd one counterexample to verify that the statement is false. For a), we can let 𝑥 = 0, so that |𝑥 | = 0 which is equal (not greater than) 0. Thus, this is a false statement because we have found a counterexample. For b), 1 1 we can let 𝑥 = 1, so that 𝑥 2 = 1, which is again, equal to 1. We can also take 𝑥 = , so 𝑥 2 = , and 2 4 1 1 < , which is a contradiction to the statement. 4 2 2. Deductive Reasoning – it is the process of reaching a conclusion by applying general principles and procedure. Example 1: All home improvements cost more than the estimate. The contractor estimated that my home improvement will cost P200,000.00. Thus, my home improvement will cost more than P200,000.00. Here, it is already given as a fact that ALL home improvements cost more than the estimate. Therefore, if we are given estimate, then we know that the actual cost will be bigger. It is not just based on observations, but with given facts. Example 2: 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 9, add 6 to the product, divide the sum by 3, and subtract 2. Solution: Let 𝑥 represent the original number. 9𝑥 Multiply the number by 9. 9𝑥 + 6 Add 6 to the product. 9𝑥 + 6 Divide the sum by 3. 3 9𝑥 + 6 −2 Subtract 2. 3 9𝑥 + 6 −2 Simplify. 3 3𝑥 + 2 − 2 𝟑𝒙 We started with 𝑥 and ended with 3x. The procedure given in this example produces a number that is three times the original number.  Logic puzzles can be solved by using deductive reasoning and a chart that helps us visualize the problem. Example 3: Each of four neighbors, Kristan, Michael, Luis, and Francis, has a different occupation (editor, banker, chef, or dentist). From the following clues, determine the occupation of each neighbor.  Michael gets home from work after the banker but before the dentist.  Luis, who is the last to get home from work, is not the editor.  The dentist and Luis leave for work at the same time.  The banker lives next door to Francis. Solution: From the first clue, we can already tell that Michael is not the banker or the dentist. Thus, we can already mark it x in our table. From the second clue, it is mentioned that Luis is not the editor, and the third clue is implying that he is also not the dentist, so we mark it x. Finally, Francis is living next to the banker, so clearly, he is not the banker. We mark it with x. editor banker chef dentist Kristan Michael x x Luis x x Francis x The table shows the initial information that we can get from the clues given. Now, to deduce more information, we analyze further the clues. It was mentioned from the second clue that Luis is the last to get home from work, therefore, from the first clue, we observe that he is not the banker, because clearly the banker is not the last one to go home because Michael and the dentist goes home after him. We can mark it with x. editor banker chef dentist Kristan Michael x x Luis x x x Francis x From the table, we can observe that the only one who can possibly be the banker is Kristan. Thus, we can already mark it with ❶. Obviously, Kristan can no longer have the other occupations so we can mark the entire row with x. editor banker chef dentist Kristan x ❶ x x Michael ❸ x x Luis x x ❹ x Francis x ❷ Continuing this fashion, we can conclude that Francis is the dentist, mark it ❷. Then Michael is the editor, mark it ❸. Finally, Luis, is the chef, marked ❹. editor banker chef dentist Kristan x ❶ x x Michael ❸ x x x Luis x x ❹ x Francis x x x ❷ IV. Mathematical Problems Involving Patterns  An ordered list of numbers such as 6, 12, 18, 24, 30, … is called a sequence. The numbers in a sequence are called terms of the sequence. In the given sequence, we call 6 as the first term, 12 the second term, and so on.  The 𝒏th term of a sequence is denoted by 𝒂𝒏.  That is, a sequence consisting of 𝑛 terms is represented by 𝑎1 , 𝑎2 , 𝑎3 , … , 𝑎𝑛. By examining the given terms in the sequence, we can analyse and try to find the pattern so that we can identify the next term. In the sequence above, we can observe that the next term is computed by adding 6 to the preceding term. Thus, we can conclude that the term after 30 is 36, followed by 42, 48, and so on.  In some cases, it is possible to predict or derive a formula, called the 𝒏𝒕𝒉 term formula, which can generate the terms of the given sequence. Example 1: Consider the sequence 2, 4, 6, 8, 10, … of even numbers. Observe that we can write the term 2 as 2(1), 4 as 2(2), 6 as 2(3), and so on. Generally, we can write the terms in the sequence in the form 2𝑛, where 𝑛 is a natural number. Thus, the formula for the 𝑛th term of this sequence is: 𝑎𝑛 = 2𝑛 Example 2: Consider the sequence 4, 14, 30, 52, 80, 114. To find the 𝑛th term of the sequence, we have the formula 𝑎𝑛 = 3𝑛2 + 𝑛. Thus, if we want to find the 20th term of the sequence, we simply substitute 20 to the formula. 𝑎𝑛 = 3𝑛2 + 𝑛 𝑎20 = 3(202 ) + 20 𝑎20 = 3(400) + 20 𝑎20 = 1200 + 20 𝑎20 = 𝟏𝟐𝟐𝟎 V. Recreational Problems using Mathematics KenKen® Puzzles KenKen® is an arithmetic-based logic puzzle that was invented by the Japanese mathematics teacher Tetsuya Miyamoto in 2004. The noun “ken” has “knowledge” and “awareness” as synonyms. Hence, KenKen translates as knowledge squared, or awareness squared. In recent years the popularity of KenKen has increased at a dramatic rate. More than a million KenKen puzzle books have been sold, and KenKen puzzles now appear in many popular newspapers, including the New York Times and the Boston Globe. KenKen puzzles are similar to Sudoku puzzles, but they also require you to perform arithmetic to solve the puzzle. 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 the mathematical operation indicated.  Cages with just one square should be fi lled 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. Here is a 4 by 4 puzzle and its solution. Properly constructed puzzles have a unique solution.  column 1 column 2 column 3 column 4 ↓ ↓ ↓ ↓ 𝟔× 𝟕+ 𝟔× 𝟕+ row 1 → 2 1 3 4 𝟐 𝟖× 𝟐 𝟖× row 2 → 3 2 4 1 𝟒× 𝟏𝟐 × 𝟏− 𝟒× 𝟏𝟐 × 𝟏− row 3 → 1 4 2 3 𝟏 𝟏 row 4 → 4 3 1 2 A 4 x 4 puzzle with 8 cages The solution to the puzzles Basic Puzzle Solution Strategies Single-Square Cages – Fill cages that consist of a single square with the target number for that square. In the puzzle above, the cell in row 4, column 3 is filled with 1. Cages with Two Squares – Next examine the cages with exactly two squares. Many cages that cover two squares will only have two digits that can be used to fill the cage. For instance, a 5 × cage can only be filled with 1 and 5. Large or Small Target Numbers – Search for cages that have an unusually large or small target number. These cages generally have only a few combinations of numbers that can be used to fill the cage. For example, in a 5 by 5 puzzle, a 60 × cage with exactly 3 squares can only be filled with 3, 4, and 5. Duplicate Digit in a Cage – Consider the 4 + cage shown below. The digits 1, 1, and 2 produce a sum of 4; however, we cannot place the two 1s in the same row or the same column. Thus the only way to fill the squares is to place the 2 in the corner of the L-shaped cage as shown below. Remember: A digit can occur more than once in a cage, provided that it does not appear in the same row or in the same column. 𝟒+ 2 1 1 Remember the Following Rules – In an 𝑛 by 𝑛 puzzle, each row and column must contain every digit from 1 to 𝑛. In a two-square cage that involves subtraction or division, the order of the numbers in the cage is not important. For instance, a 3 − cage with two squares could be filled with 4 and 1 or with 1 and 4. A 3 ÷ cage with two squares could be filled with 3 and 1 or with 1 and 3. Make a List of Possible Digits – For each cage, make a list of digits, with no regard to order, that can be used to fill the cage. It’s like having a list of possibilities for later use. Guess and Check – In most puzzles you will reach a point where you will need to just guess, then try and see if it works. Assume that the possible digits in a particular cage are arranged in a particular manner and then see where your assumption takes you. If you find that the remaining part of a row or column cannot be filled in correctly, then you can eliminate your assumption and proceed to check out one of the remaining possible numerical arrangements for that particular cage. A Famous Puzzle The Tower of Hanoi is a puzzle invented by Edouard Lucas in 1883. The puzzle consists of three pegs and a number of disks of distinct diameters stacked on one of the pegs such that the largest disk is on the bottom, the next largest is placed on the largest disk, and so on as shown in the next figure. The object of the puzzle is to transfer the tower to one of the other pegs. The rules require that only one disk be moved at a time and that a larger disk may not be placed on a smaller disk. All pegs may be used. Determine the minimum number of moves required to transfer all of the disks to another peg for each of the following situations. There are many possible solutions of course. But what we need is to solve the problem with minimum number of moves. First, label the pegs from left to right with A, B, and C. We follow the following steps: 1. Transfer the yellow disk to peg A 2. Transfer the pink disk to peg C. 3. Transfer the yellow disk to peg C. 4. Transfer the blue disk to peg A. 5. Transfer the yellow disk to peg B. 6. Transfer the pink disk to peg A.  7. Finally, transfer the yellow disk to peg A. The puzzle can be played with any number of disks, although many toy revisions have around 7 to 9 of them. The minimal number of moves required to solve a Tower of Hanoi puzzle is 2𝑛 − 1, where 𝑛 is the number of disks. REFERENCES Aufmann, Richard N., et al. Mathematical Excursions, 3rd ed., Cengage Learning, 2013. Earnhart, Richard T., & Adina, Edgar M., Mathematics in the Modern World, Outcome-Based Module, C&E Publishing, Inc., 2018. Petkovic, Miodrag, Famous Puzzles of Great Mathematicians, AMS Bookstore, 2009. https://static2.bigstockphoto.com/0/3/1/large1500/130158797.jpg