GEC 104 - Mathematics in the Modern World Problem Solving and Reasoning PDF

GEC 104 – MATHEMATICS IN THE MODERN WORLD Problem Solving & Reasoning INTRODUCTION Alfred A. Montapert once warned, “Expect Problems and eat them for breakfast.” Indeed men face problems of varying complexity from time to time. Problems have b...

GEC 104 – MATHEMATICS IN THE MODERN WORLD Problem Solving & Reasoning INTRODUCTION Alfred A. Montapert once warned, “Expect Problems and eat them for breakfast.” Indeed men face problems of varying complexity from time to time. Problems have become integral in one’s daily affairs and form the spectrum by which a person’s character is sometimes defined. This module will start with kinds of reasoning like inductive and deductive reasoning, intuition and proof, followed by Polya’s 4-steps in problem solving that will help you solve problems through organized methods and approaches specially in solving problems involving finances. Topics: I. Reasoning A. Kinds of Reasoning o Deductive and Inductive Reasoning o Intuition o Proof II. Polya’s 4-Steps in Problem Solving A. Word Problems Involving Finances III. Recreational Mathematics Schedule: 4th Week Instructional Materials: Module Calculator Learning Outcomes: At the end of this module, students are expected to: 1. use different types of reasoning to justify statements and arguments made about mathematics and mathematical concept; 2. solve mathematical problems involving finances following Polya’s 4 steps in problem solving; 3. solve recreational problems involving mathematics 1 GEC 104 – MATHEMATICS IN THE MODERN WORLD MOTIVATION SURVIVAL SCENARIO Say you are stranded on an island with your group of friends, knowing that help will not arrive in three days. The problem is that the group has a limited amount of food and water and must create shelter from items around the island. How will you make it through the three days as safely and comfortably as possible? CONTENT Problems arise inevitably and so they must be dealt with objectively and successfully. Mathematics is not just about numbers; much of it is problem solving and reasoning. Problem solving and reasoning are basically inseparable. The art of reasoning is the skill needed in exemplifying the critical thinking and problem solving ability. Reasoning is the practice of stating ideas clearly and precisely to arrive at a certain conclusion. In our life, we often make judgement and conclusion based on facts and observation. These are not always true. Thus, we have to know the different ways of arriving at accurate conclusions. Kinds of Reasoning 1. The Deductive and Inductive Reasoning Deductive and inductive reasoning are methods of making inferences, used in developing statements about some mathematical concepts or phenomena. Below illustrates the distinction between the two processes: Deductive Reasoning Inductive Reasoning Moves from general principles Moves from specific observations to or statements to specific general claims. conclusions. The process of gathering specific Relies on the idea of true information, usually through observation statements having a and then making a conclusion based on consequence. the gathered information. 2 GEC 104 – MATHEMATICS IN THE MODERN WORLD Example 1. Consider the following arguments Argument 1: The sum of two odd integers is an even number. 3 and 5 are odd integers. Thus, the sum of 3 and 5 is even. Argument 2: 3 + 5 = 8, 7 + 11 = 18, -5 + 9 = 4. Therefore, the sum of two odd integers is even. In argument 1, the first is a general principle, while the second statement is an information based from the given principle. The third statement is the conclusion, which is a specific statement drawn from the general principle. Thus, Argument 1 uses the deductive form of reasoning. (rule then the example). In argument 2, the first statement is the premise while the second is a conclusion. Argument 2 uses the inductive form of reasoning. As can be observed, the premise consists of specific cases while the conclusion is the general principle of what is claimed in the premise (examples before the rule). Example 2. In a right triangle, the two acute angles are complementary. Triangle ABC has B = 90°. Therefore, A + C = 90° Here, the premise is a known property of right triangles. The second statement implies that triangle ABC is a right triangle with ∠𝐴 and ∠𝐵 being acute. The third is the conclusion which is a specific statement drawn from the general principle, hence, it uses deductive reasoning. Example 3. Triangle ABC is a right triangle with A = 30° and C = 60°. A + C = 90°. Therefore, in a right triangle, the two acute angles are complementary. The example uses inductive reasoning. As can be observed, the premise consists of specific cases while the conclusion is the general principle of what is claimed in the premise. Example 4. All numbers ending in “0” and “5” are divisible by 5. The number 35 ends with a”5”, hence it is divisible by 5. The given uses deductive reasoning. From a general rule to a specific case. Example 5. The Chair in the living room is red. The chair in the dining room is red. The chair in the bed room is red. Therefore, all chairs in the house are red. The given uses inductive reasoning. From observations or specific cases to a general conclusion based from the observations. Here are some additional examples of deductive and inductive reasoning. 1. MICROSOFT OFFICE USER 3 GEC 104 – MATHEMATICS IN THE MODERN WORLD 2. Intuition - Intuition is similar to guessing or also called reasoning by common sense. - It requires less mental activity and highly subjective. - It is the ability to acquire knowledge without proof, evidence, or conscious reasoning. Examples: Given these two figure , you can identify which is bigger by using intuition and without actually measuring. Saying “I think I will like this subject” after your first meeting in class is an intuition. When you’re not sure with your final answer to an item in an exam but you have a gut feeling that it is the correct answer is a reasoning by intuition. 3. Proof - Shows a statement to be true using definitions, theorems & postulates, conclusion of some other previously accepted statements, algebraic properties. Example If x – 7 = 10, then x = 17. To prove that the statement is true, we have to solve for x. x – 7 = 10 x – 7 + 7 = 10 + 7 By the use of Addition Property of Equality, we can x = 17 show that the value of x is 17. Thus, the statement is true. In 1945, George Polya published the book How to Solve it: A New Aspect of Mathematical Method which later became widely accepted design of how students should proceed with problem-solving processes. Although the book generally revolves around mathematical problems, Polya emphasized that the same strategy can be applied when students encounter real-life problems. George Polya’s 4-steps in problem solving Understand the problem. This requires reading the problem slowly and recognizing the information given in the problem. This is achieved when all the words used in stating the problem is fully understood, it can be restated in one’s own words and what is being asked for SEE is identified. One has to have a clear grasp of the problem at hand, especially the relationship that exists between the given and the unknown quantities. Devise a plan. To devise a plan is to come up with a way to solve the problem. There are many different types of plans for solving problems which include: setting up an equation, drawing a diagram, making a chart, guess and check, look for a PLAN pattern, make an orderly list, draw a picture, eliminate the possibilities, use a model, work backwards. Carry out the plan. This step is where the identified plan is applied to solve the problem. If a plan does not work, it can be modified or changed. Common sense and DO natural thinking abilities can help this step. Look back. This step is where answers can be verified and checked and where mistakes can be identified. Verify and check if the answer makes sense or correct based on the situation posed in the problem. This is a time to review what you have C HECK MICROSOFT O FF I C E USER done, what worked and what didn’t. 4 GEC 00 – MATHEMATICS IN THE MODERN WORLD Now let us try to solve mathematical problems involving finances using Polya’s 4-steps in problem solving. Problem 1. Seph invested a certain amount of money at 5% interest rate. The interest earned in a year is Php15 000.00. How much money was invested? Note: Simple interest refers to an interest that is Step 1. See. To understand the problem, we have to identify the given and what is being asked. computed only once from the time the Given: a certain amount of money is invested at 5% interest rate amount is borrowed or Php15 000.00 interest was earned in a year invested until it is paid Note: Find: amount of money invested Use the decimal equivalent of the given percentage. Step 2. Plan. Devise a plan by representing the given and by forming equations using the information in step 1. Simple interest is determined by multiplying the principal by Represent the given: the interest rate by the time that Let P = amount of money invested elapse between payments. I = Prt I = 15 000 Where: Come up with an equation by substituting the values represented to r = 5% or 0.05 I = Simple interest the formula t = 1 year P = amount invested/borrowed or I = Prt principal If I = Prt, then 15 000 = P (0.05)(1) r = interest rate t = time Step 3. Do. Carry out the plan by solving the equation in step 2. 15 000 = P (0.05)(1) 15 000 0.05 𝑃 Simplify the right side of the = equation. 0.05 0.05 Then divide both sides by 0.05 to isolate P. P = 300 000 Simplify to get the value of P. If P = 300 000, then the amount of money invested was Php300 000.00. Step 4. Check. Look back and see if the values (answers) obtained are correct. Again, using the formula I = Prt, we can verify if Php300 000.00 was the amount invested from the lending company but this time we will substitute P with 300 000. I = (300,000) (0.05) (1) I = 15 000 Since Php300, 000.00 earned an interest amounting to Php15 000.00, then we did the problem correctly. 5 GEC 00 – MATHEMATICS IN THE MODERN WORLD Problem 2. A certain amount of money was borrowed from a certain lending company payable for 2 years at 8% simple interest. A Php6 000.00 was paid for the interest of the loan. How much was the amount of money borrowed from the lending company? Step 1. See. To understand the problem, we have to identify the given and what is being asked. Given: amount borrowed payable for 2 years at 8% simple interest Php6 000.00 was paid for the interest of the loan Find: amount of money borrowed. Step 2. Plan. Devise a plan by representing the given and by forming an equation using the information in step 1. Represent the given: Use simple interest Let P = principal or the amount borrowed formula I = 6 000 r = 8% or 0.08 t = 2 years If I = Prt, then 6 000 = P (0.08) (2) Step 3. Do. Carry out the plan by solving the equation in step 2. 6 000 = P (0.08) (2) Simplify the right side of the 6 000 0.16 𝑃 equation. = Then divide both sides by 0.16 to 0.16 0.16 isolate P. P = 37 500 Simplify to get the value of P. If P = 37 500, then the amount of money borrowed was Php37 500.00. Step 4. Check. Look back and see if the values (answers) obtained are correct. Again, using the formula I = Prt, we can verify if Php37 500.00 was the amount borrowed from the lending company but this time we will substitute P with 37, 500. I = (37,500) (0.08) (2) I = 6 000 Since Php37 500.00 gained a simple interest amounting to Php6 000.00, then we did the problem correctly. 6 GEC 00 – MATHEMATICS IN THE MODERN WORLD Problem 3. A Php12 000.00 was borrowed from a rural bank which is payable within 3 years with 12% simple interest. How much will be repaid after three years? Maturity Value is Step 1. See. To understand the problem, we have to identify the given and what is being asked. the sum of the principal amount and Given: amount borrowed was Php12, 000.00 the interest. payable within 3 years at 12% simple interest 𝐹 = 𝑃 +𝐼 Where: Find: amount of money to be repaid after three years or the maturity value F = maturity value P = principal Step 2. Plan. Devise a plan by representing the given and by forming an equation using the I = interest information in step 1. Using simple Represent the given: interest formula I = Prt, Let F = maturity value maturity value can P = 12 000 be computed as r = 12% or 0.12 Come up with an equation by 𝐹 = 𝑃 + 𝑃𝑟𝑡 substituting the values represented to t = 3 years the formula By factoring Therefore, F = 12 000[1 + (0.12) (3)] 𝑭 = 𝑷(𝟏 + 𝒓𝒕) Step 3. Do. Carry out the plan by solving the equation in step 2. F = 12 000[1 + (0.12) (3)] F = 12 000(1.36) F= 16 320 If F = 16 320, then the total amount of money to be repaid is Php16 320.00. Step 4. Check. Look back and see if the values (answers) obtained are correct. By adding the principal and the interest we can verify if the computed maturity value is 16, 320. I = (12000) (0.12) (3) I = 4 320 F = 12 000 + 4 320 F = 16 320 Since the amount computed in step 3 and step 4 are the same, then we did the problem correctly. Problem 4. An employee applied for a salary loan amounting to Php15, 000.00 from their association payable in 12 months with a monthly interest of 1.5%. How much will be deducted from his salary every month? Step 1. See. To understand the problem, we have to identify the given and what is being asked. Given: Php15, 000.00 amount of salary loan Payable in 12months 1.5% monthly interest Find: monthly salary deduction. 7 GEC 00 – MATHEMATICS IN THE MODERN WORLD Step 2. Plan. Devise a plan by representing the given and by forming an equation using the information in step 1. Represent the given: Let F = maturity value Use maturity value I = interest formula P = 15 000 r = 1.5% or 0.015 t = 12 months If F = P (1 + rt), then F = 15 000[1 + (0.015) (12)] Come up with an equation to solve for the monthly deduction Monthly salary deduction = 𝑭 𝟏𝟐 Step 3. Do. Carry out the plan by solving the equations in step 2. Simplify by performing F = 15 000[1 + (0.015) (12)] operations inside the grouping F = 15 000(1.18) symbols first. F= 17 700 If F = 17 700, then the total amount of money to be repaid is Php17 700.00. Monthly deduction = 𝑭 Now that the maturity value (F) is 𝟏𝟐 already known, we can now solve for the monthly salary deduction. 17 700 = 12 = 1 475 The employees’ monthly salary deduction will be Php1, 475.00. Step 4. Check. Look back and see if the values (answers) obtained are correct. Using I = Prt, we can verify if the monthly salary deduction is Php1 475.00 I = 15 000 (0.015) (12) = 2 700 Monthly salary deduction = 15,000+2,700 = 1,475 12 Since we still got the same value using different methods, then we did the problem right. Most people don’t do mathematics for recreation. You can go your whole life thinking of “mathematics” as calculations, a bit of algebra, and maybe the quadratic formula. (Oldridge, 2019). But have you ever heard of Recreational Mathematics? Recreational mathematics A mathematics carried out for recreation (entertainment) and is intended to be fun rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Many topics in this field require no knowledge of advanced mathematics. 8 GEC 104 – MATHEMATICS IN THE MODERN WORLD It involves mathematical puzzles and games, often appealing to children and untrained adults, inspiring their further study of the subject. It can be surprisingly popular, with millions of people at one time or another having enjoyed Rubik's Cube, Sudoku, Sungka or Philippine Mancala, logic problems, and many other diversions. Examples of Recreational Mathematics 1. The Magic Square – an arrangement of numbers in a square such that all rows, all columns, and both main diagonals sum to the same number, a number referred to as the magic constant. 4 Given a 3x3 square, place the numbers 1 to 9 so that each row, column and diagonal adds up 3 to the same number. Use each number exactly once. (Bold faced numbers are the given) 8 6 6. Since 9 is the only number not in the box, then the last cell would be 9 5. To fill this cell you may use any of the 1. In the given square, the sum of the numbers in 4 9 2 available info. Let’s try to use 15 – (7+6) = 2 first column is 15, hence, we can say that 15 is 3 5 7 the Magic constant. 8 1 6 4. You can now fill the middle row. 15 – (3+5) = 7 2. Look for a cell which you can easily fill next. In this case, 3. Another cell you can the bottom row. 15 – (8+6) = 1 easily fill in is this diagonal (you may also do this diagonal first before the bottom row) 15 – (4+6) = 5 2. Sungka or Philippine Mancala –a two-player turn-based Filipino local game wherein the players compete to obtain more stones than the other. The game unleashes the competitiveness of a person. Somehow, the Sungka game seems to be a mind game. It is also known to improve the mathematical thinking and observation skills of an individual. The game is played on a solid wooden block with two rows of seven circular holes and two large holes at both ends called "head". It begins with 49 game pieces (shells, marbles, pebbles or seeds) equally distributed to alternate holes - seven pieces in every other hole - except "heads" which remain empty. Each player controls the seven holes on his side of the board and owns the "head" to his right. The goal is to accumulate as many pieces in your own "head". https://kapampangan.org/sungka/ You may want to play the game for a break from this module. SUMMARY Reasoning is the practice of stating ideas clearly and precisely to arrive at a certain conclusion There are different kinds of reasoning, namely: Inductive reasoning – from specific cases to general statements. Deductive reasoning – from general principle to specific cases Intuition – guessing or similar to common sense Proof - shows a statement to be true using definitions, theorems & postulates, conclusion of some other previously accepted statements, algebraic properties. 9 GEC 104 – MATHEMATICS IN THE MODERN WORLD Problems can be solved methodically using Polya’s 4-steps in problem solving. First is to understand the problem. Second is to devise a plan to solve the problem. Third is to carry out the plan. And fourth is to look back and verify if you solved the problem correctly. Solving problems can also be fun and recreational through Recreational mathematics. This includes but are not limited to, mathematical puzzles and games, Rubik's Cube, Sudoku, Sungka or Philippine Mancala, logic problems, and many other diversions. Here are some additional examples of deductive and inductive reasoning. 1. All birds lay eggs. Pigeons are birds. Pigeons lay eggs. – Deductive 2. Every dog I meet is friendly. Most dogs are usually friendly. – Inductive 3. I tend to catch colds when people around me are sick. Colds are infectious. – Inductive 4. All mammals have backbones. Humans are mammals. Humans have backbones. – Deductive 5. All plants perform photosynthesis. A cactus is a plant. A cactus performs photosynthesis. - Deductive REFERENCES Romeo M. Daligdig, EdD (2019). Mathematics in the Modern World. Problem Solving and Reasoning. Lorimar Publishing,Inc. 40 – 49 Earnhart, R T., and Adina E M. (2018). Mathematics in the Modern World. Problem Solving and Reasoning. C & E Publishing,Inc. 78 – 88 Aduana, Nick L. (2012). Mathematics of Investment. Quezon City: C&E Publishing,Inc. Matthew Oldridge (July 16, 2019). Martin Gardner and Recreational Mathematics. Retrieved from https://medium.com/@MatthewOldridge/recreational-mathematics-d72dfbd68226 James Yolkowski (January 12, 2015) Recreational Mathematics. Retrieved from https://mathlair.allfunandgames.ca/recreational.php (nd) Retrieved from. https://kami.com.ph/112571-how-to-play-sungka.html https://www.hostos.cuny.edu/Hostos/media/Office-of-the-President/Institutional-Research- Assessment/Problem-Solving-Rubric.pdf 10