MMW-Part-4 Notes about Mathematical Problem Solving.pptx
- 1. MATHEMATICS IN THE MODERN WORLD Instructor: Joris N. Buloron, MS
- 2. Topic Outline I. Nature of Mathematics II. Speaking Mathematically III. Problem-Solving IV. Statistics V. Logic VI. Graphs VII. Mathematical Systems Textbook: Aufmann, R., Lockwood, J., Nation, R., Clegg, D., Epp, S., Abad, E. Jr. Mathematics in the Modern World. (Rex Book Store, Inc., Manila, Philippines). 2018.
- 3. III. Problem-Solving • Exercise 1: A regular polygon with n (n≥3) sides is a closed figure whose sides are of equal length such as the equilateral triangle, square, and so on. It is also called an n-gon. A diagonal of an n-gon is a line segment that connects two non-adjacent vertices. Observe the following pattern of the numbers of diagonals in several n-gons for small values of n and try to generalize. ?
- 4. Inductive Reasoning • Inductive reasoning is the process of reaching a general conclusion by examining specific examples. • Exercise 2: Use inductive reasoning to predict the next number in each list. a.) 5, 10, 15, 20, 25, ? b.) 2, 5, 10, 17, 26, ? • Exercise 3: Use inductive reasoning to make a conjecture. a.) Consider the following procedure: Pick a number. Multiply the number by 9, add 15 to the product, divide the sum by 3, and subtract 5. b.) Consider the following procedure: Pick a number. Multiply the number by 8, add 6 to the product, divide the sum by 3, and subtract 2.
- 5. • Exercise 4: A tsunami is a sea wave produced by an underwater earthquake. The height of a tsunami as it approaches land depends on the velocity of the tsunami. Use the table below and inductive reasoning to answer each of the following questions. a.) What happens to the height of a tsunami when its velocity is doubled? b.) What should be the height of a tsunami if its velocity is 30 feet per second? Velocity of Tsunami (feet per second) Height of Tsunami (feet) 6 4 9 9 12 16 15 25 18 36 21 49 24 64
- 6. • Exercise 5: Observe the number of regions made by drawing a diameter in each circle. Count the number of regions for the first few numbers of diameters. Make a conjecture on the number of regions. No. of Diameters 1 2 3 4 No. of regions 2 4 6 ?
- 7. Counterexamples • Exercise 6: A prime number is an integer greater than 1 such that its only positive integer divisors are 1 and itself. Consider the first few prime numbers 2, 3, 5, 7 and the following table: Compute the corresponding values for each of the remaining listed prime numbers. Observe what type of numbers you'd obtain using the expression. Make a conjecture. n 2 3 5 7 2n -1 22 -1=3 ? ? ?
- 8. • Exercise 7: For each circle, count the number of regions formed by the line segments that connect the dots on the circle. Using small numbers of dots, complete the list below. Make a guess on the maximum number of regions when there are six dots. Number of Dots 1 2 3 4 5 6 Maximum Number of Regions 1 2 ? ? ? ?
- 9. 1 x x • Exercise 8: Verify that each of the following statements is false by finding a counterexample. a.) For all real numbers x, |x|>0. b.) For all real numbers x, . c.) For all real numbers x, . N.B. Using inductive reasoning would not guarantee that your conclusion is true. x x 2
- 10. Deductive Reasoning • Deductive reasoning is the process of reaching a conclusion by applying general assumptions, procedures, or principles. • Exercise 9: Use deductive reasoning to derive the general form of the expression. a.) Consider the following procedure: Pick a number. Multiply the number by 9, add 15 to the product, divide the sum by 3, and subtract 5. b.) Consider the following procedure: Pick a number. Multiply the number by 8, add 6 to the product, divide the sum by 3, and subtract 2. • Exercise 10: Determine whether each of the following arguments is inductive or deductive. a.) 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. b.) All home improvements cost more than the estimate. The contractor estimated that my home improvement will cost $35,000. Thus my home improvement will cost more than $35,000.
- 11. • Exercise 11 (Logic Puzzle): Each of four neighbors, Sean, Maria, Sarah, and Brian, has a different occupation (editor, banker, chef, or dentist). From the following clues, determine the occupation of each neighbor. a.) Maria gets home from work after the banker but before the dentist. b.) Sarah, who is the last to get home from work, is not the editor. c.) The dentist and Sarah leave for work at the same time. d.) The banker lives next door to Brian. Note: In mathematics, when we say x and y are real numbers, we consider the possibility that x = y. In logic puzzle, when we say “dentist and Sarah”, we mean they are different persons.