Lesson 4 (Problem Solving) Concept Notes.pdf

Aklan Catholic College Arch. G.M. Reyes St. 5600 Kalibo, Aklan, Philippines Tel Nos.: (036) 26...

Aklan Catholic College Arch. G.M. Reyes St. 5600 Kalibo, Aklan, Philippines Tel Nos.: (036) 268-4010 Website: http://www.acc.edu.ph E-mail Add: [email protected] 1 SOLVING PROBLEMS USING POLYA’S STRATEGY AND REASONING ABILITIES Lesson 4 I. Intended Learning Outcomes At the end of this lesson, the learner will have 1. identified terms and concepts in problem solving; 2. described inductive and deductive reasoning in problem solving; and 3. solved problems using the four steps of Polya. II. General Instructions 1. Read the guide questions 2. Study the concept notes. Refer to the materials cited for further understanding. 3. Study supplementary materials for additional information III. Guide questions Use this guide questions to navigate through the keynotes and additional readings and media. Keep them in mind while studying. You can use a separate note to pick up answers from the materials as move along with them. 1. As a problem solver, what do you do when you get stuck while solving a problem? 2. Why problem solving important to learn? 3. As problem solving is one of the 21st century skills, how can you make yourself be a good problem solver? IV. Concept Notes Every day in our life, we always face so many problems that need solving. Whether the problem is big, or small, we all set objectives for ourselves. In this lesson, we will discuss the definition of problem and problem solving using different perspectives, the functions of problem solving, four steps in problem solving that was created by George Polya (1945), which used all over to aid people in Problem Solving and lastly, we will know how to reason out inductively and deductively. Let’s open our discussion by defining problem and problem solving. Watch the video “What is Problem Solving? 3 Key Points to Remember” by The Peak Learner (August 2015). This video is stored in the material section of GMathMod Google Classroom. What is Problem? The word problem may have different meanings depending on context. In English, a problem is any question or matter involving doubt, uncertainty or difficulty or a question proposed for solution or discussion. In business, problem is a perceived gap between the existing state and a desired state, or a deviation from a norm, a standard or status quo. In Mathematics, a problem is a statement requiring a solution, usually by means of a mathematical operation/geometric construction. Our focus here will be problems in Mathematics. What is Problem Solving? Prior to the detailed discussion of Problem Solving, it is important for us to understand first these three words "method", "answer" and "solution". The word "method" means the ways or techniques used to get an answer which will usually involve one or more problem solving strategies. On the other hand, the word "answer" means a number, quantity or some other entity that the problem is asking for. Finally, the word "solution" is the whole process of solving a problem, including the method of obtaining an answer and the answer itself. So, we can summarize the discussion above by this very simple equation which can be applied in problem solving: method + answer= solution Problem solving is about resolving problems. It is finding solutions and not just answers to problems. It is a mathematical process where one uses his skills creatively in new situations. Mathematics in the Modern World Aklan Catholic College HED Semester: 1st Academic Year: 2024 – 2025 Instructor: Alextaire C. Villanueva, MEd  2 Many mathematics skills were involved in problem-solving. Problem-solving is a process- an ongoing activity in which we take what we know to discover what we don't know. It involves overcoming obstacles by generating hypotheses, testing those predictions, and arriving at satisfactory solutions. Problem-solving involves three basic functions: 1. Seeking information 2. Generating new knowledge 3. Making decisions Problem-solving is the ability to identify and solve problems by applying appropriate skills systematically. FOUR STEPS IN PROBLEM SOLVING How do we perform Problem Solving? To be able to solve problems systematically, we follow the four basic steps enunciated by George Pólya in 1945 through all these steps were known already and used well before then. The Ancient Greek mathematicians like Euclid and Pythagoras certainly knew how it was done. Four Steps in Problem Solving 1. Understand the Problem First, you must understand the problem. Study the essential mathematical concepts by considering the terminology and notation used in the problem. Rephrase the problem in your own words, if needed. Then, write down specific examples of the conditions given in the problem. Ask also yourself these questions: a. What kind of a problem is it? b. What is the unknown? c. What information is given? d. What do the terms mean? e. Is there enough information or is more information needed? f. What is or are the conditions in the problem? Is it possible to satisfy the condition/s? Is/Are the condition/s sufficient to determine the unknown? 2. Devise a Plan First, find the link between the data and the unknown. You must start somewhere, so try something. But if an immediate connection cannot be found, then it would be necessary to consider more problems. You should obtain eventually a plan of the solution. Think of ways on how you are going to attack the problem, that is, try using strategies that could help you solve the problem. Here are some of the possible strategies that you can use: a. ldentifying a Subgoal b. Making a Table c. Make an Organized List (Tree Diagram, Venn Diagram) d. Making an illustration/Drawing e. Eliminating Possibilities f. Writing an Equation/ Using a Variable g. Solving a simpler version of the problem h. Trial and Error / Guess and Check i. Work Backwards j. Look for a Pattern/s 3. Carry Out the Plan As soon as you have an idea for the solution of the problem, write it down instantly then carry out your plan of the solution. Just make sure that each step in the solution is logically correct. However, if the plan does not seem to be working well, then start over again and try another strategy. Sometimes, the first approach will not work. But do not worry because if a strategy does not work, it does not mean you did it wrong. It could be that there is a more appropriate strategy that you can use for that problem. Remember, the secret here is to keep trying until something works. 4. Look Back Once you have a potential solution, check to see if it works. Ask the following to yourself a. Did you answer the question? b. Is your result reasonable? Mathematics in the Modern World Aklan Catholic College HED Semester: 1st Academic Year: 2024 – 2025 Instructor: Alextaire C. Villanueva, MEd  3 Then, double check your solution to make sure that all the conditions related to the problem are satisfied. Make sure also that any computation involved in finding your solution is correct. If you find that your solution does not work or satisfy the problem, there may only be a simple mistake. Try to fix or modify your existing solution before disregarding it. Remember what you tried is likely that at least part of it will end up being useful. Another way of checking your solution is to make use of other concepts or formulas or even strategies to solve the problem. Example: Apply Polya’s Strategy (Make an organized list) 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: 1. Understand the Problem. There are many different orders. The teams may have won straight games and lost the last two (WWLL). Or maybe they lost the first two games and won the last two (LLWW). Of course, there are possibilities, such as WLWL 2. 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. 3. 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 L. This strategy produces six different orders shown below. a. WWLL (Start with two wins) b. WLWL (Start with one win) c. WLLW d. LWWL (Start with one loss) e. LWLW f. LLWW (Start with two losses) 4. 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. Inductive and Deductive Reasoning What is Inductive Reasoning? Inductive reasoning is a process that uses our knowledge in making a general inference about unfamiliar occurrences based on observation and patterns. It is using specific examples to make a general rule. Examples: 1. Use inductive reasoning to find the next two terms. a. 5, 50, 500, 5000, 50000 b. a, 6, c, 12, e, 18, Solution: a. Looking at the terms in the given sequence, notice that the succeeding terms are multiplied by a power of 10. Hence, we can deduce that the next term will also be a product of a power of 10. Since we are looking for 4th and 5th terms, then we multiply the 4th term by 104 and the 5thterm by 105 so the next two terms in the sequence are 50000 and 500000. b. Examining the terms in the given sequence, observe that letters and numbers alternate. The letters are those that are in the odd position in the alphabet while the numbers are multiples of 6. Thus, the next two terms in the sequence are the letter g and the number 24 (6 x 4). What is Deductive Reasoning? Deductive reasoning is the process by which conclusions are made based on previously known facts or by employing general assumptions, procedures, or principles. It is applying a general rule to specific examples. Deductive reasoning is also the way of showing that certain statements follow logically from agreed-upon assumptions and proven facts and there is a need to justify every step with a reason. Mathematics in the Modern World Aklan Catholic College HED Semester: 1st Academic Year: 2024 – 2025 Instructor: Alextaire C. Villanueva, MEd  4 Examples: 1. Use deductive reasoning to show that when a number is multiplied by 10, the product is decreased by 8, the difference is divided by 2, and 4 is added to quotient, then the number is five times the original number. Solution: Let n be the original number. Then, A number is multiplied by 10: 10n The product is decreased by 8: 10n − 8 10n−8 The difference is divided by 2: = 5n − 4 2 4 is added to quotient: 5n − 4 + 4 = 𝟓𝟓𝟓𝟓 Since from the original number, n, we got 5n, the statement therefore is proven. What is recreational mathematics? Recreational Mathematics consists of games, tricks or puzzles that are most of the time designed for entertainment, pleasure, amusement, or fun. Though its main purpose is for recreation, it cannot be denied that the amount of mathematical content, understanding, and purpose also requires some sort of rigor analysis and high degree of patience. Example: Three musicians appeared at a concert. Their last names were Benton, Lanier, and Rosario. Each play only one of the following instruments: guitar, piano, or saxophone. 1. Benton and the guitar player arrived at the concert together. 2. The saxophone player performed before Benton. 3. Rosario wished the guitar player good luck. Who played each instrument? Solution: The solution can be summarized using a chart. From Clue 1, Benton is not the guitarist. We mark X1 (this means “ruled out by clue 1), in the guitar column of Benton’s row. From clue 2, Benton does not play saxophone, hence he must be the pianist. From Clue 3, Rosario plays saxophone. This leaves Lanier as the guitar player. Guitar Piano Saxophone Benton X1 Yes X2 Lanier Yes No No Rosario X3 No Yes Benton plays piano, Lanier plays guitar, and Rosario plays the saxophone. A cryptogram is a mathematical puzzle where symbols are used to represent digits for a system that is true. Logical deductions and a series of tests are needed to solve such problem. It may have unique or multiple solutions. 2. Find the value of A if: 1A1 + A0A 8A8 Solution: Using the concept of expansion of number, the addend and sum may be expressed as follows, 1𝐴𝐴1 → 1(100) + 𝐴𝐴(10) + 1 + 𝐴𝐴0𝐴𝐴 → 𝐴𝐴(100) + 0(10) + 𝐴𝐴 8𝐴𝐴8 → 8(100) + 𝐴𝐴(10) + 8 The sum of the unit digits, 1 + 𝐴𝐴 = 8 implies 𝐴𝐴 = 8 − 1, 𝑨𝑨 = 𝟕𝟕. Want to know more? If you need to read or review, kindly open the file Polya’s Four Steps in Problem Solving, and watch the videos, Polya’s Problem Solving Processby Math Videos that Motivate (August 2018) and Introduction to Inductive and Deductive Reasoning by Don’t Memorise (July 2019). Mathematics in the Modern World Aklan Catholic College HED Semester: 1st Academic Year: 2024 – 2025 Instructor: Alextaire C. Villanueva, MEd  5 Reference list 1. Aufmann, R. N., Lookwood, J. S., Nation, R. D., & Clegg, D. K. (2013). Mathematical excursion. Brooks/ Cole. 2. Cengage Learning. (2018). Mathematics in the modern world. Rex Book Store, Inc. 3. Libretexts. (2020, July 27). Module 1: Problem solving strategies. Mathematics LibreTexts. https://math.libretexts.org/Courses/Monroe_Community_College/MTH_155_Mathematics_for_Elementary_T eachers_I_(placeholder)/Chapter_1%3A_Problem_Solving_Strategies/Module_1%3A_Problem_Solving_Str ategies 4. Mathematical reasoning and statements. (2019). BYJU’S. https://byjus.com/maths/statements-in- mathematical-reasoning/ 5. Rodriguez, M. J., Salvador, I. G., Ragma, F., Torres, E., Manalang, E., Oredina, N., & Ogoy, J. (2018). Mathematics in the modern world. Nieme Publishing House Co. Ltd. 6. Seward, K. (2011). Introduction to problem solving. WTAMU. https://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut8_probsol.htm 7. Tool for the field: Polya’s Problem-Solving method. (2018, March 12). OPEPP. https://www.opepp.org/lesson/hsdm-unit7-tool-for-field/ 8. von Renesse, C. (2014). Inductive and deductive reasoning | discovering the art of Mathematics. Art of Mathematics. https://www.artofmathematics.org/blogs/cvonrenesse/inductive-and-deductive-reasoning Supplementary Video 1. Don’t Memorise. (2019, July 30). Introduction to inductive and deductive reasoning | don’t memorise [Video]. YouTube. https://www.youtube.com/watch?v=yAjkQ1YqLEE&t=65s 2. Miacademy Learning Channel. (2017, April 20). 4 steps in solving problems [Video]. YouTube. https://www.youtube.com/watch?v=kn8frIzQupA&t=36s 3. Puzzle guy. (2018, May 30). Tower of Hanoi, 8 disks. Only 255 moves requires to solve it. [Video]. YouTube. https://www.youtube.com/watch?v=SceJ_kIQ7xE&t=62s 4. ThePeakLearner. (2015, July 31). What is problem solving? 3 key points to remember [Video]. YouTube. https://www.youtube.com/watch?v=rxn407cm6MM&t=54s 5. WOW MATH. (2020, December 2). Mathematics in the modern world [Video]. YouTube. https://www.youtube.com/playlist?list=PLPPsDIdbG32Auf61Nq_mFwIe7Xel3VfsW Mathematics in the Modern World Aklan Catholic College HED Semester: 1st Academic Year: 2024 – 2025 Instructor: Alextaire C. Villanueva, MEd