Chap6a.-Inductive-and-Deductive-Reasoning (1).pdf
- 1. PROBLEM SOLVING AND REASONING– Inductive and Deductive Reasoning Core Idea “Mathematics is not just about numbers; much of it is problem solving and reasoning.” CHAPTER 6a.
- 2. learning objectives 1. become familiar of the problem solving heuristics and be able to identify the strategy most appropriate for the problem; 2. use problem-solving strategies to investigate and understand mathematical content; 3. acquire skill in developing and applying a variety of strategies to solve problems; and 4. verify and generalize results with respect to the original problem situation.
- 3. challenge… 1 2 3 q Suppose you choose Door 1. q Monty Hall reveals a goat behind Door 3. q You can stay with Door 1 or switch to Door 2. Which choice will give a better chance of winning? The Monty Hall Problem
- 4. C B A
- 6. INDUCTIVE REASONING It is the process of reaching a general conclusion by examining specific examples. In this types of reasoning, conclusion is formed based on the examination of specific examples. Here, the conclusion formed by inductive reasoning is conjecture, which might be true or not.
- 7. INDUCTIVE REASONING Example. Predicting a Number Use inductive reasoning to predict the next number in each of the following lists. a. 3, 6, 9, 12, 15, __, __, __,…? b. 1, 3, 6, 10, 15, __, __, __,…? Answers: 18, 21, 24,… Answers: 21, 28, 36,…
- 8. INDUCTIVE REASONING Example. Making a Conjecture Consider the procedure: Pick a number. Multiply the number by 8, add 6 to the product, divide the sum by 2, and subtract 3. Conjecture: Such procedure generates a number that is four times the original number.
- 9. INDUCTIVE REASONING Example. Solving Applied Problems The table shows some results obtained for pendulum of various lengths. Galileo Galilei (1564- 1642) used inductive reasoning to discover that the time required for a pendulum to complete one swing, called the period of the pendulum, depends on the length of a pendulum . Historical background Length of pendulum, in units Period of pendulum, In heartbeats 1 1 4 2 9 3 16 4 25 5 36 6 a. If a pendulum has a length of 49 units, what is it period? b. If the length of a pendulum is quadrupled, what happens to its period?
- 10. INDUCTIVE REASONING How many regions can be formed with 6 dots on the circle? No. of dots No. of regions 1 1 2 2 3 4 4 8 5 16 6 ?
- 11. INDUCTIVE REASONING ü Conclusions based on inductive reasoning may be incorrect. ü Inductive reasoning does not guarantee that a conclusion is correct. Just because a pattern holds true for a few cases, it does not mean the pattern will continue.
- 12. Counterexample A statement is true provided that it is true in all cases. A counterexample proves that a statement is false for at least one case. Example. Finding a Counterexample Verify that each is false by finding a counterexample. For all numbers ! : 1. " > 0 2. "% > " 3. "% = "
- 13. DEDUCTIVE REASONING It is the process of reaching a general conclusion by applying general assumptions, procedures, or principles. Example 5. Establish a Conjecture Consider the procedure: Pick a number. Multiply the number by 8, add 6 to the product, divide the sum by 2, and subtract 3. Page 59, Example 5
- 14. Example. Inductive vs. Deductive Determine which argument 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. DEDUCTIVE REASONING Page 60, Example 6
- 15. Example. Solving a Logic 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. 1. Maria gets home from work after the banker but before the dentist. 2. Sarah, who is the last to get home from work, is not the editor. 3. The dentist and Sarah leave for work at the same time. 4. The banker lives next door to Brian. DEDUCTIVE REASONING Logic puzzles can be solved by deductive reasoning and a chart that displays the information in a visual manner. Page 61, Example 7
- 16. Practice. 1. Use deductive reasoning to construct an expression that represent the procedure: “Pick a number. Multiply the number by 6, add 10 to the product, divide the sum by 2, and subtract 5.” 2. Determine which argument is inductive and deductive. a. All Gillian Flynn novels are worth reading. The novel Gone Girl is a Gillian Flynn novel. Thus, Gone Girl is worth reading. b. I know I will win a jackpot on this slot machine in the next 10 tries, because it has not paid out any money during the last 45 tries. DEDUCTIVE REASONING Page 60, Check your progress 6
- 17. Practice. Solving a Logic Brianna, Ryan, Tyler, and Ashley were recently elected as the class officers (president, vice president, secretary, treasurer) of the sophomore class at Summit College. Determine which position each holds. 1. Ashley is younger than the president but older than the treasurer. 2. Brianna and the secretary are both the same age, and they are the youngest members of the group. 3. Tyler and the secretary are next neighbors. DEDUCTIVE REASONING Page 62, Check your progress 7