GEC6 Module 3: Problem Solving 2024-2025 PDF

MATHEMATICS IN THE MODERN WORLD  -- -- -- -- -- -- LEARNING OUTCOMES ----------------- **Topic Learning Outcomes \[TLO\]**   ### What is a Problem-Solving? ###...

MATHEMATICS IN THE MODERN WORLD  -- -- -- -- -- -- LEARNING OUTCOMES ----------------- **Topic Learning Outcomes \[TLO\]**   ### What is a Problem-Solving? ### [Two (2) Types of Reasoning]   a. 5, 10, 15, 20, 25, [ ] ? b. 1, 4, 9, 16, 25, [ ] ? ### Solution: a. Each successive number is 5 units larger than the preceding number. Thus, it can be predicted that the next number in the list is 5 units larger than 25, which is 30. b. Observe that all the numbers are perfect squares, 1 = 12, 4 = 22, 9 = 32, 16 = 42, 25 = 52. Meaning, it can be predicted that the next number to be 36 = 62. 1. Pick a number. 2. Multiply the number by 10. 3. Add 8 to the product. 4. Divide the sum by 2. 5. And subtract 4. ### Solution: ----- ---- 7.5 5 7.6 9 7.7 13 7.8 17 7.9 21 8.0 25 8.1 29 8.2 33 8.3 37 ----- ---- a. If the earthquake magnitude is 8.5, how high (in meters) can the tsunami be? b. Can a tsunami occur when the earthquake magnitude is less than 7? Explain your answer. ### Solution: a. In the table, for every 0.1 increase in earthquake magnitude, the maximum tsunami height increases by 4 meters. So, it is conjectured that the maximum tsunami height for the earthquake magnitude of 8.5 is ***45 meters.*** b. The answer is ***No***, because when the earthquake magnitude is 7.4 the maximum tsunami height is only 1 meter. Hence, a tsunami does not occur when the earthquake magnitude is less than 7. - ***Conclusions based on inductive reasoning may not always be true. In other words, a conjecture formed by using inductive reasoning may be incorrect as stated about the meaning of conjecture*.** To illustrate this, let us consider the results below,  ### Let's have some examples on Counterexamples a. x/x = 1 b. √x2 + 4 = x + 2 ### Solution: a. Let x = 0. Then 0/0 ≠ 1. This implies that x/x = 1 is not true for all x. b. For x = 1, √12 + 4 = √5. But x + 2 = 3 when x = 1. Then, √5 ≠ 3, implies ### Solution: ### Example 6: solve a logic puzzle 1. Joy is older than her friend who owns the cat and younger than her friend who owns the dog. 2. Janice and her friend who owns the snake are both of the same age and are the youngest members of the group. 3. Jenny is older than her friend who owns the fish.  ### Solution: -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- Problem Solving with Patterns ============================= ### [Take these examples on Fibonacci sequence]  Solution: --------- ### Binet's Formula ### Solution: F20 = 20 𝐹50 = 50 Polya's Problem-Solving Strategy ======================================================= Polya's Four-Step Problem-Solving Strategy ------------------------------------------ 1. Understand the problem. 2. Devise a plan. 3. Carry out the plan. 4. Review the solution.  ### Let us elaborate further on the different steps for better understanding: [Understand the Problem] ### [Devise a Plan] ### [Carry out the Plan] ### [Review the Solution] ### Solution: 1. WWWLLL 9. LLLWWW 2. WWLWLL 10. LLWLWW 3. WWLLWL 11. LLWWLW 4. WWLLLW 12. LLWWWL 5. WLLLWW 13. LWWWLL 6. WLLWWL 14. LWWLLW 7. WLWWLL 15. LWLLWW 8. WLWLWL 16. LWLWLW ### Solution:  ### Carry out the Plan. ### Solution: a. Solve for y in equation **(1)** b. Substitute y = 10 -- x in equation **(2)** Modulo n -------- Solution: ---------