Course Course Packet Packet 03 CM01-NGEC LM01-PRED 0423 1413 Course Module Mathematics in the Modern World Course Packet 03 Problem Solving and Reasoning Knowledge Area Code Course Code Learning Module Code Course Packet Code : : : : NGEC NGEC0423 CM01-NGEC0423 CM01-NGEC0423-03 Course Packet CM01-NGEC 03 Course Packet 03 Problem Solving and Reasoning Introduction This course packet includes the discussion of the two fundamental types of reasoning - the inductive and deductive reasoning. Mastery of such will help the students take different approaches in solving problems including the application of Polya’s four step approach. Worksheet, assessment, and assignment will be provided as well to help the students develop the skills they will be needing in understanding this course packet. Course Packet 03 Objectives After finishing this course packet you are expected to: differentiate inductive from deductive reasoning and be able to apply them in justifying statements and arguments about mathematics and its concepts, write clear and logical proofs, and employ Polya’s four steps approach in solving different types of mathematical problems. Learning Management System Google Classroom Links Duration • Course Packet 03: Problem Solving and Reasoning = 7.5 hours (6.5 hours self-directed learning with practical exercises and 1 hour assessment) Delivery Mode This course packet will be delivered online (synchronous and asynchronous). Assessment with Rubrics A multiple choice type of assessment will be given at the end of the course packet. It will be uploaded in Google Forms. Requirement with Rubrics Content analysis will serve as the requirement at the end of this course packet. The file of the problem set about Problems Solving and Reasoning is given. Course Module: Mathematics in the Modern World Page 1 of 16 0423 CM01-NGEC 0423 Standards/Basis for Grading to Use for Problem Solving Course Packet 03 0 The student did not make any attempt to solve any of the problems in the problem set or prove any of the statements in the quiz. 1 The student attempted to solve 50% of the problems in the problem set or displayed logical reasoning 50% of the time in attempting to prove the statement/s in the quiz. 2 The students attempted to solve all the problems in the problem set or displayed logical reasoning 75% of the time in attempting to prove the statement/s in the quiz. 3 The student is able to completely solve 50% of the problems in the problem set or completed 75% of the proof/s in the quiz. 4 The student is able to completely solve 75% of the problem set or completed all the proof/s in the quiz. Readings To have prior ideas about the topics in this course packet, you are recommended to have an advance reading of the following materials: Quintos Jr, R. et. al. (2020). Revised Edition Mathematics in the Modern World. Plaridel, Bulacan: St. Andrew Publishing House. pp. 67-101. Manuel-Guillermo, R. et. al. (2018). Mathematics in the Modern World A Woktext. Cubao, Quezon City: Nieme Publishing House Co. Ltd, pp. 69-98 Reyes, J.. Mathematics in the Modern World. Intramuros, Manila : Library Services & Publishing Inc., 2018, pp. 75-95. Sirug, W.. Mathematics in the Modern World CHED Curriculum Compliant. Intramuros, Manila : Mindshapers Co., Inc., 2018, pp. 44-26. Introduction Solving mathematical problems has always been a challenge to many students. One of the causes of this is poor reasoning skills. Reasoning is used to justify certain decisions. The development of this skill may lead to easier and diverse approaches to solving problems. Pre-Assessment Each of the letters in Figure 1 represents a distinct digit from 0 to 9. What does each letter represent? What strategy will you employ to solve the problem? What kind of reasoning will you use for to come up with a solution? (See page 15 to check you answer after you have tried it.) Course Module: Mathematics in the Modern World Page 2 of 16 CM01-NGEC 0423 Figure 1 Course Packet 03 Lesson Proper Reasoning is drawing of inferences/conclusions by means of justification (www.merriam-CM01-NGEC webster.com). It is a kind of thinking enlightened by logic (https://www.encyclopedia.com). It is an important part of problem solving. There are fundamentally two forms of reasoning – inductive and deductive reasoning. 0423 I. INDUCTIVE AND DEDUCTIVE REASONING INDUCTIVE REASONING (Image courtesy: https://www.scmp.com) Imagine yourself as a child creating an airplane, a house, or an animal using the Lego pieces. This is an illustration of the inductive reasoning. Inductive reasoning is the type of logical thinking where we form a generalization (the final output – an airplane, a house, an animal) from specifics (each Lego piece). Notice that putting up the pieces together does not necessarily yield only one product. Let us say that when you were a child, your mother would always feed you with a yellow mango that was sweet. In your mind, you may have concluded that all yellow mangoes are sweet. Is this conclusion necessarily true? Definitely, not. However, it is a valid guess because of your previous experiences. Course Module: Mathematics in the Modern World Page 3 of 16 CM01-NGEC 0423 This is a called a conjecture. It is a conclusion formed from an inductive reasoning that may or may not be correct (Quintos, R. et. al., 2019). Inductive reasoning may also be helpful to “predict” based on observed patterns. For example, what should be next figure in the pattern (Figure 2)? Should it be A, B, C, or D? CM01-NGEC Course Packet 03 0423 Figure 2 If you have carefully observed the pattern, the shaded region started form the leftmost part of the above row and moved towards right. The shaded region shifted to the next row below starting again form the leftmost part moving towards right. Following the observed pattern, the next figure should be C. Here are some illustrative examples of the application of the inductive reasoning: Illustrative Example 1. 14 is an even number. 24 is and even number. 54 is an even number. Therefore, all numbers that end with the digit 4 are even numbers. (True) Illustrative Example 2. 3 is a prime number 13 is a prime number. 23 is a prime number.. Therefore, all numbers that end with the digit 3 are prime numbers. (False. For example, 33) Illustrative Example 3. What should be the next number in the following sequences? a. 7, 11, 15, 19, ? b. 3, 6, 12, 24, ? CM01-NGEC Answer: a. 23. The succeeding numbers in the sequence are always 4 more than the previous. b. 48. The succeeding numbers in the sequence are always twice the previous. Illustrative Example 4. Course Module: Mathematics in the Modern World Page 4 of 16 0423 Think of a number. Multiply it by 6. Then, add 20. Divide the result by 2. Subtract 10 from the results. Compare the last result from the original number. What can you conclude? Does it hold true for all numbers? Solution: Let us say, you thought of 4. Multiplying it by 6 will give 24. Adding 20 to it will result to 44. Dividing the result by 2 will give 22. Subtracting 10 from 22 will give 12. What about if you have thought of 7? Multiplying it by 6 will give 42. Adding 20 to it will result to 62. Dividing the result by 2 will give 31. Subtracting 10 from it will give 21. Course Packet 03 In both cases, the results are thrice the original numbers. Find out for yourself if this isCM01-NGEC generally true. Illustrative Example 5. Mr. X, a Mathematics teacher gave a long quiz at the end of the second week of the month. He gave another long quiz at the end of the last week of the month? Is there a possibility of him giving another long quiz at the end of the second week of the next month even if no announcement was made? 0423 Answer: There is a possibility that this will happen based on the previous instances. Illustrative Example 6. Verify whether the mathematical statement b2 > 0 is true for all values of b under the set of real numbers. Otherwise, give a counter example. (A counter example is an example in which the given statement is NOT true.) Solution: If b = 4, then (4)2 > 0. Simplifying the inequality we have 16 > 0. (True) If b = -3, then (-3)2 > 0. Simplifying the inequality we have 9 > 0. (True) However, if b = 0, then (0)2 > 0. Simplifying the inequality we have 0 > 0. (False, 0 = 0) DEDUCTIVE REASONING The State requires that students who have finished the Education Program should pass the Licensure Examination for Teachers (LET). If a student who has graduated with the said program decided not to take it of failed to pass it, then he/she will not be able practice the profession of being a teacher. This is an example of a practical application of another type of reasoning - the deductive reasoning. Deductive reasoning is the opposite of the inductive reasoning. It is a “top-down” process of telling whether or not an assumption is true. This assumption, however, should be based on logic and experimentation (Heckmann, 2020). It starts with a generalization (or hypothesis) to reach a specificCM01-NGEC situation (conclusion). 0423 Illustrative Example 1. (Sound deductive reasoning) All dogs are animals; poodles are dogs, therefore poodles are animals. All even numbers are divisible by 2. Course Module: Mathematics in the Modern World Page 5 of 16 78 is an even number. Therefore, 78 is divisible by 2. Obtuse angles measure between 90° and 180°. Angle A measures 100°. Therefore angle A is an obtuse angle. Course Packet 03 Illustrative Example 2. (Logically unsound deductive reasoning) Cheetahs have spots; Dalmatians have spots, therefore Dalmatians are cheetahs. CM01-NGEC Illustrative Example 3. (Predictive deductive reasoning) Most people who join reality shows become actors/actresses. Camille joined a reality show. Therefore, she will become an actress. 0423 Illustrative Example 4. (Deductive reasoning used in logic puzzles) In a class of 40 students, 31 love the color red, 28 love the color blue, and 25 love the color yellow. 18 of them love blue and yellow, 22 love red and yellow, and 19 love red and blue. 15 of them love the three colors. How many of them love red, blue, and yellow only? Solution: This kind of puzzle can be solved using a Venn diagram. Venn diagram normally makes use of circles (each representing a set or a group) that overlap to show the logical relationships between them. The overlapping parts of the circles denote the commonality in the members of the set or group. Draw three circles that overlap as shown in Figure 3. Course Module: Mathematics in the Modern World Page 6 of 16 CM01-NGEC Course Packet 03 0423 Figure 3 Start with the overlapping region of the three circles. This region represents what is common with the three colors. From the given, there are 15 students that love the three colors. Next, consider the overlapping regions of two colors. Let us start with blue and yellow. From the given, there are 18. However, part of this region is the region where the three circles overlap. This means that 15 should be subtracted from 18. Therefore, there should be 3 in this region. Same thing should be done with the regions where other two colors overlap: CM01-NGEC Red and Yellow: 22 – 15 = 7 Red and Blue: 19 – 15 = 4 To answer the question, subtract from the total number of students who love a specific color the ones that you have for the overlapping parts: Red only: 31 – (15 + 7 +4) = 5 Blue only: 28 – (15 +4 +3) = 6 Yellow only: 25 – (15 + 7 +3) = 0 Therefore, 5 students love the color red only, 6 love the color blue only, and none love the color yellow only. 0423 Activity 3.1: Video Watching Click the link below to learn more about inductive and deductive reasoning. Don’t Memorise (July 31, 2019). Introduction to Inductive and Deductive Reasoning / Don’t Memorise. https://www.youtube.com/watch?v=yAjkQ1YqLEE Let’s Get Logical (May 26, 2020). What Is Deductive vs Inductive Reasoning / Deductive vs Inductive Arguments. https://www.youtube.com/watch?v=MKRH03msgMg Course Module: Mathematics in the Modern World Page 7 of 16 CM01-NGEC 0423 Assignment 3.1: Assignment 3.1 will be posted in the Google Classroom via Google forms. Work Sheet 3.1: Work Sheet 3.1 will be posted in the Google Classroom via Google forms. Course Packet 03 II. PROBLEM SOLVING WITH PATTERNS Generally speaking, there is no one size fits all approach to solving problems. One student might consider one approach as right while another student one might have a different one. One prolific Mathematician has shown how a different approach to solving problems may lead to a much more convenient way. When he was in his primary school, Carl Friedrich Gauss, was punished by his teacher for displaying misbehavior. As a “punishment”, he was asked to get the sum of the numbers from 1 to 100. In just seconds, he gave the answer – 5,050 (Math and Multimedia)! How did he do it? Here’s how: (Image courtesy: https://www.britannica.com/) Figure 4 (Image courtesy: Math and Multimedia) He noticed that certain pairs of numbers always have sums of 101 (the first and the last, the second and the second to the last, etc.) and that there are 50 pairs of them (Figure 4). He simply multiplied the common sum of 101 and the number of pairs (50). Voila, he got the sum – 5,050! This proves how looking at patterns can help solve problems. Here are some illustrative examples: Illustrative Example 1. Identify the formula for the general term an of the given sequence and use this to identify the 25th term: 3, 8, 13, 18, 23,… Solution: This particular example can be solved using the formula for arithmetic sequence. However, this can also be solved using the concept of reasoning. First, observe the pattern. You can get the succeeding term by adding 5 to the previous terms. Use this observation to make a generalization: Course Module: Mathematics in the Modern World Page 8 of 16 CM01-NGEC 0423 3 8=3+5 13 = 8 + 5 = 3 + 5 + 5 18 = 13 + 5 +15 = 3 + 5 + 5 + 5 23 = 18 + 5 = 3 + 5 + 5 + 5 + 5 Or Course Packet 03 3 = 3 + 5(0) 8 = 3 + 5(1) 13 = 3 + 5(2) 18 = 3 + 5(3) 23 = 3 + 5(4) We know that the first (a1) is 3, the second term (a2) is 8, the third (a3) term is 13, the fourth (a4) term is 18, and the fifth (a5) term is 23. Notice that the multipliers of 5 in each mathematical expression is always 1 less than the order of the term (i.e. in the expression equal to 3, the multiplier of 5 is zero which is 1 less than its order which is first. The same can be observed in expressions equal to 8, 13, 18, and 23.). Therefore, it can be concluded that if the order of the term is n, the multiplier of 5 for each expression should be 1 less than n (or n – 1). Following such conclusion, we have: an = 3+5(n -1) or an = 3+5n – 5 = 5n – 2 Therefore, the formula for the general term is an = 5n – 2. To find the 25th term, simply replace n with 25: a25 = 5(25) – 2 = 125 – 2 = 123. Hence, the 25th term is 123. Illustrative Example 2. Identify the formula for the general term an of the given sequence and use this to identify the 9th term: 5, 10, 20, 40, 80,… Solution: This particular example can be solved using the formula for geometric sequence. However, this can also be solved using the concept of reasoning. First, observe the pattern. You can get the succeeding term by multiplying the previous terms by 2. Use this observation to make a generalization: 5 10 = 5(2) 20 = 10(2) = 5(2)(2) 40 = 20(2) = 5(2)(2)(2) Course Module: Mathematics in the Modern World Page 9 of 16 CM01-NGEC 0423 80 = 40(2) = 5(2)(2)(2)(2) Or Course Packet 03 5 = 5(2)0 10 = 5(2)1 20 = 5(2)2 40 = 5(2)3 80 = 5(2)4 Again, we know that the first (a1) is 5, the second term (a2) is 10, the third (a3) term is 20, the fourth (a4) term is 40, and the fifth (a5) term is 80. Notice that the exponents of 2 in each mathematical expression is always 1 less than the order of the term (i.e. in the expression equal to 5, the exponent of 2 is zero which is 1 less than its order which is first. The same can be observed in expressions equal to 10, 20, 40, and 80.). Therefore, it can be concluded that if the order of the term is n, the exponent of 2 for each expression should be 1 less than n (or n – 1). Following such conclusion, the formula for the general term of the given sequence is: an = 5(2)n-1 To find the 9th term, simply replace n with 9: a9 = 5(2)9-1 = 5(2)8 = 5(256) = 1,280. Hence, the 9th term of the sequence is 1,280. Illustrative Example 3. Identify the pattern suggested by the following sequence (Figure 5). What will be next in the sequence? Figure 5 Solution: Notice that every time, the image inside rotates 135° counter-clockwise. Therefore, the next figure should be: Course Module: Mathematics in the Modern World Page 10 of 16 CM01-NGEC 0423 3 0423 CM01-NGEC 3 Alternate solution: You might also observe that the image inside rotates 45° clockwise. However, the colors also alternate (from black to white or from white to black). Following this observation, you will also have the answer as the previous. Course Packet 03 Activity 3.2: Video Watching Click the links below to learn more about problem solving with patterns. Miacademy Learning Channel (March 31, 2017). How to Find a Pattern when Solving Problems. https://www.youtube.com/watch?v=w40E9RX6-XE LearnYouSomeMath (October 10, 2015). Problem https://www.youtube.com/watch?v=9cSTCfcrKNg Solving with Patterns. Assignment 3.2: Assignment 3.2 will be posted in the Google Classroom via Google forms. Work Sheet 3.2: Work Sheet 3.2 will be posted in the Google Classroom via Google forms. III. POLYA’S PROBLEM SOLVING STRATEGY George Polya (1887-1985) (Photo courtesy: https://www.deviantart.com/) Another Mathematician also had introduced a way to solve problem. George Polya (18871985) was a Hungarian educator in Mathematics who wrote the book “How to solve it”. He introduced what is now-called Polya’s Problem Solving Strategy or Polya’s FourStep Approach (www.ms.uky,edu). The book identifies four basic principles on how to solve problems: understand the Course Module: Mathematics in the Modern World Page 11 of 16 CM01-NGEC 0423 3 Course Packet 03 problem, devise a plan, carry out the plan, and look back (UCB Mathematics). 1. Understand the problem. Understanding the problem is arguably the part where the difficulty in solving problem lies. Polya suggested to have the following as guide questions to understand the problem: Do you understand the words in the problem? What is asked to be found or shown? Can you paraphrase the problem in your own words? Is it possible to use a picture or a diagram to illustrate the problem? Is the information given by the problem enough? 2. Devise a plan. Plan is always crucial to find solutions to problems. It serves as the blueprint to be followed. In devising a plan, one may use some or any of the following CM01-NGEC strategies: Guess and check (trial and error) Systematic/orderly listing Elimination of possibilities Considering special cases Using direct/indirect reasoning Writing and equation (or making a model) Looking for patterns Drawing pictures Solving a simpler version of the problem Working backwards Applying a known formula 0423 3 3. 4. Carry out the plan. Of course, once you have already devised a plan, the next thing to do is to put the plan into action. As much as possible, follow whatever plan you have made. However, keep the following in mind: Be patient when you carry out the plan. Giving up is always an option but don’t choose it. Carefully execute the plan. It is okay to modify the plan or try a new should you have realized something wrong is happening along the way. Be systematic and orderly. Sometimes you need to keep a complete and accurate record. Look back. Normally, when a student has already “done” answering the question, it is when the process ends. However, have you “really” answered the question? This part is very important as it verifies the answer to the problem. You may consider the following: Are you finding the solution difficult? Then consider looking for an easier one. Examine the solution. Does the answer make sense? Check if the answer agrees with the conditions stated in the problem Recheck any computation Here are some illustrative examples to help you understand it more: Illustrative Example 1. Course Module: Mathematics in the Modern World Page 12 of 16 CM01-NGEC 0423 3 Barangay Panalo has made the finals of Palarong Pambarangay 2021. They will need to face another for a best of five series game. In how many ways can they win the championship? Solution: Step 1: Understand the problem. To win the championship, they have to win three games. They may or may not win these games consecutively. Step 2: Devise a plan. Make a list all the possible orders of winning. Be systematic in doing the list. Step 3: Carry out the plan. Won all first three games: WWW Won the first two games: CM01-NGEC WWLW WWLLW Won the first game: WLWW WLLWW WLWLW Lost the first game: LWWW LWWLW LWLWW Lost the first two games: LLWWW Step 4: Look back. Check whether there is something missing in the list or a duplication of entries in the list. Since, we have listed ALL the possibilities, simply count them. There are 10 in the list. Therefore, there are 10 ways for them to win the championship. Course Packet 03 0423 3 Illustrative Example 2. Two times the sum of a number and 3 is equal to thrice the number plus 4. Find the number. Solution: Step 1: 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 the number. Step 2: Devise a plan. We will translate the problem mathematically (make an equation) and them solve it. Step 3: Carry out the plan. The equation is 2(x + 3) = 3x +4. Solving the equation we have: 2(x + 3) = 3x + 4 2x + 6 = 3x + 4 2x – 3x = 4 – 6 -x = -2 x=2 Step 4: Look back. If we take two times the sum of 2 and 3, that is the same as thrice the number 2 plus 4 which is 10, so this does check. Thus, the number is 2. Course Module: Mathematics in the Modern World Page 13 of 16 CM01-NGEC 0423 3 Illustrative Example 3. There are 7 sacks. Each sack has 7 cats. Each cat has 7 kittens. How many cat/kitten legs are there in all? Course Packet 03 Solution: Step 1: Understand the problem. We are to identify the total number of legs of all the cats and the kittens given the condition that there are 7 sacks, where there are 7 cats in each sack, and each cat has 7 kittens. Step 2: Devise a plan. Draw a table Step 3: Carry out the plan. Number of Cats/Kittens Number of Legs Cat: 7 x 7 = 49 49 x 4 = 196 Kitten: 7 x 7 x 7 = 343 343 x 4 = 1,372 Total 1,568 CM01-NGEC 0423 3 Step 4: Look back. All conditions of the problems are met. Therefore, there are 1,568 legs in all. Activity 3.3: Video Watching Click the links below to learn more about Polya’s problem solving strategy. Apologs Santos (September 29, 2020). Problem Solving / Polya’s Four Steps / Mathematics in the Modern World. https://www.youtube.com/watch?v=Q-LUqWhsSbo Math Videos that Motivate (August 10, 2018). Polya’s Problem Solving Process. https://www.youtube.com/watch?v=zhL3EMFSm6o Assignment 3.3: Assignment 3.3 will be posted in the Google Classroom via Google forms. Work Sheet 3.3: Work Sheet 3.3 will be posted in the Google Classroom via Google forms. Course Assessment 3: Course Assessment 3 will be posted in the Google Classroom via Google forms. CM01-NGEC 0423 3 Annexes Course Module: Mathematics in the Modern World Page 14 of 16 CM01-NGEC 0423 Answer Key - Course Packet Pre-Assessment Course Packet 03 M=1 O=0 S=9 R=8 D=7 E=5 N=6 Y=2 3 References Books Quintos Jr, R. et. al. (2020). Revised Edition Mathematics in the Modern World. Plaridel, Bulacan: St. Andrew Publishing House. pp. 67-101. Manuel-Guillermo, R. et. al. (2018). Mathematics in the Modern World A Woktext. Cubao, Quezon City: Nieme Publishing House Co. Ltd, pp. 69-98 Reyes, J.. Mathematics in the Modern World. Intramuros, Manila : Library Services & Publishing Inc., 2018, pp. 75-95. Sirug, W.. Mathematics in the Modern World CHED Curriculum Compliant. Intramuros, Manila : Mindshapers Co., Inc., 2018, pp. 44-26. On Line Sources https://www.merriam-webster.com/dictionary/reasoning Heckmann, C. (2020). What is deductive reasoning? Definition and examples. StudioBinder. https://www.studiobinder.com/blog/what-is-deductive-reasoning-definition/ Math and Multimedia (September 15, 2010). Young Gauss and the sum of the first n positive integers). http://mathandmultimedia.com/2010/09/15/sum-first-n-positiveintegers/#:~:text=Gauss%20displayed%20his%20genius%20at,in%20a%20matter%20of%20sec onds. Polya's four steps to solving a problem. Math Sciences Facility. https://www.ms.uky.edu/~carl/ma310/spring03/polya/Polya.htm UCB Mathematics | Department of Mathematics Berkeley. https://math.berkeley.edu/~gmelvin/polya.pdf at University Computing of California Videos: Don’t Memorise (July 31, 2019). Introduction to Inductive and Deductive Reasoning / Don’t Memorise. https://www.youtube.com/watch?v=yAjkQ1YqLEE Course Module: Mathematics in the Modern World Page 15 of 16 CM01-NGEC 0423 3 Let’s Get Logical (May 26, 2020). What Is Deductive vs Inductive Reasoning / Deductive vs Inductive Arguments. https://www.youtube.com/watch?v=MKRH03msgMg Miacademy Learning Channel (March 31, 2017). How to Find a Pattern when Solving Problems. https://www.youtube.com/watch?v=w40E9RX6-XE LearnYouSomeMath (October 10, 2015). Problem Solving with Patterns. https://www.youtube.com/watch?v=9cSTCfcrKNg Apologs Santos (September 29, 2020). Problem Solving / Polya’s Four Steps / Mathematics in the Modern World. https://www.youtube.com/watch?v=Q-LUqWhsSbo Math Videos that Motivate (August 10, 2018). Polya’s Problem Solving Process. https://www.youtube.com/watch?v=zhL3EMFSm6o Course Packet 03 Feedback Form To further improve the discussion of this course packet your feedback is very important. Please fill out the learner’s feedback form and submit it to your teacher. Course Module: Mathematics in the Modern World Page 16 of 16
