Chapter 6a - Inductive and Deductive Reasoning (PDF)

CHAPTER 6a. PROBLEM SOLVING AND REASONING– Inductive and Deductive Reasoning Core Idea “Mathematics is not just about numbers; much of it is problem solving and reasoning.” learning objectives 1. become familiar of the problem solving he...

CHAPTER 6a. PROBLEM SOLVING AND REASONING– Inductive and Deductive Reasoning Core Idea “Mathematics is not just about numbers; much of it is problem solving and reasoning.” 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. challenge… The Monty Hall Problem 1 2 3 ❑ Suppose you choose Door 1. ❑ Monty Hall reveals a goat behind Door 3. ❑ You can stay with Door 1 or switch to Door 2. Which choice will give a better chance of winning? A B C A B C 1/3 2/3 2/3 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. 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, __, __, __,…? Answers: 18, 21, 24,… b. 1, 3, 6, 10, 15, __, __, __,…? Answers: 21, 28, 36,… 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. INDUCTIVE REASONING Example. Solving Applied Problems The table shows some results obtained for pendulum of various lengths. Length of pendulum, Period of pendulum, a. If a pendulum has in units In heartbeats a length of 49 Historical 1 1 units, what is it background 4 2 period? Galileo Galilei (1564- 1642) used inductive 9 3 b. If the length of a reasoning to discover that the time required 16 4 pendulum is for a pendulum to 25 5 quadrupled, what complete one swing, happens to its called the period of the pendulum, depends on 36 6 period? the length of a pendulum. Practice. On Inductive Reasoning a. What happens to b. What should be c. Pick a number. the height of the the height of a Multiply the tsunami when its tsunami if its number by 9, add velocity is velocity is 30 feet 15 to the product, doubled? per second? divide the sum by 3, and subtract 5. Velocity of tsunami, Height of tsunami, in feet per second in feet 6 4 d. Predict the next 9 9 numbers in the 12 16 sequence: 15 25 2, 5, 7, 10, 17, 26 18 36 21 49 INDUCTIVE REASONING How many regions can be formed with 6 dots on the circle? No. of No. of dots regions 1 1 2 2 3 4 4 8 5 16 6 ? INDUCTIVE REASONING ✓ Conclusions based on inductive reasoning may be incorrect. ✓ 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. 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 𝒙 : 1. 𝑥 > 0 2. 𝑥 2 > 𝑥 3. 𝑥 2 = 𝑥 DEDUCTIVE REASONING Page 59, Example 5 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. DEDUCTIVE REASONING Page 60, Example 6 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 61, Example 7 Logic puzzles can be solved by deductive reasoning and a chart that displays the information in a visual manner. 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 Page 60, Check your progress 6 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 62, Check your progress 7 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. CHAPTER 6b. PROBLEM SOLVING AND REASONING– Mathematical Patterns Core Idea “Mathematics is not just about numbers; much of it is problem solving and reasoning.” 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. TERMS OF A SEQUENCE sequence - an ordered list of numbers 5, 14, 27, 44, 65,… term - a number in a sequence TERMS OF A SEQUENCE ❑ difference table - shows the differences between successive terms of the sequence. 5 14 27 44 65 … TERMS OF A SEQUENCE Example. Predict the Next Term (Book: Page 70) Use a difference table to predict the next term in the sequence a. 2, 7, 24, 59, 118, 207, ___ b. 1, 14, 51, 124, 245, 426, ___ THE FIBONACCI SEQUENCE Month 1 Newborn 1 pair 2 Mature 1 pair 3 2 pairs 4 3 pairs THE FIBONACCI SEQUENCE Month 4 3 pairs 5 pairs 5 8 pairs 6 THE FIBONACCI SEQUENCE THE FIBONACCI SEQUENCE A recursive definition for a sequence is one in which each successive term of the sequence is defined by using some of the preceding terms. 𝑭𝟏 𝑭𝟐 𝑭𝟑 𝑭𝟒 𝑭𝟓 𝑭𝟔 1, 1, 2, 3, 5, 8, … FIBONACCI NUMBERS Let 𝐹1 = 1, 𝐹2 = 1, 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 for 𝑛 ≥ 3. THE FIBONACCI SEQUENCE Example. Properties of Fibonacci Numbers (Book: Page 74) a. If 𝑛 is even, then 𝐹𝑛 is an odd number. b. If 𝐹12 = 144 and 𝐹13 = 233, what is 𝐹14 ? c. If 𝐹16 = 987 and 𝐹18 = 2584, what is 𝐹17 ? NTH-TERM FORMULA FOR A SEQUENCE An nth-formula is constructed when there is a certain pattern in a sequence. It also generates the terms of a sequence. Example 2. Generate Terms of a Sequence Consider the formula 𝑎𝑛 = 3𝑛2 + 𝑛 where 𝑛 is the number of the term in the sequence. Generate the first 5 numbers of the sequence. NTH-TERM FORMULA FOR A SEQUENCE Example. Find an nth-Term Formula Assume the pattern shown by the square tiles. a. What is the nth-term formula for the number of tiles in the nth figure of the sequence? b. How many tiles are in the eighth figure of the sequence? c. Which figure will consist of exactly 320 tiles? NTH-TERM FORMULA FOR A SEQUENCE Practice. Find an nth-Term Formula Assume the pattern shown by the square tiles. a. What is the nth-term formula for the number of tiles in the 𝑛th figure of the sequence? b. How many tiles are in the thirteenth figure of the sequence? c. What figure will consist of exactly 419 tiles? CHAPTER 6c. PROBLEM SOLVING AND REASONING– Problem Solving Strategies Core Idea “Mathematics is not just about numbers; much of it is problem solving and reasoning.” 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. POLYA’S PROBLEM SOLVING STRATEGY 1. Understand the problem. 2. Devise a plan. 3. Carry out the plan. 4. Review the solution Historical background ❑George Polya (1887-1985): advocated basic problem- solving strategy which consists of four steps. ROUTE PROBLEM Example 1a. Consider the map shown. Allison wishes to walk along the streets from point A to point B. How many direct routes can Allison take? ROUTE PROBLEM Example 1a. Consider the map shown. Allison wishes to walk along the streets from point A to point B. How many direct routes can Allison take? A simple diagram of the street map. ROUTE PROBLEM Example 1a. Consider the map shown. Allison wishes to walk along the streets from point A to point B. How many direct routes can Allison take? A street diagram with the number of routes to each intersection labeled ROUTE PROBLEM Example 1b. Consider the map shown. Allison wishes to walk along the streets from point A to point B (No back tracking). How many direct routes can Allison take? ROUTE PROBLEM Example 1c. How many direct routes are there from A to B if Allison wants to pass by Starbucks? RIVER CROSSING PROBLEM Example 2. Four people on one side of the river need to cross the river in a boat that can carry a maximum load of 180 pounds. The weights of the people are 80, 100, 150, and 170 pounds. a. Explain how the people can use the boat to get everyone to the opposite side of the river. b. What is the minimum number of crossings that must be made by the boat? PUZZLE FROM A MOVIE Example 3. In the movie die hard: With a Vengeance, Bruce Willis and Samuel Jackson are given a 5-gallon jug and a 3- gallon jug and they must put exactly 4 gallons of water on a scale to keep a bomb from exploding. Explain how they could accomplish this feat. FAKE COIN PROBLEM Example 4. You have eight coins. They all look identical, but one is a fake and slightly lighter than the others. Explain how you can use the balance scale to determine which coin is the fake in exactly a. three weighings b. two weighings OTHER PROBLEMS Example 5. In a basketball league of 12 teams, each team plays each of the other teams exactly three times. How many league games will be played? Example 6. In consecutive turns of a Monopoly game, Stacy first paid $800 for a hotel. She then lost half her money when she landed on Boardwalk. Next, she collected $200 for passing GO. Shen then lost half of her remaining money when she landed on Illinois Avenue. Stacy now has $2500. How much did she have just before she purchased the hotel? End of Discussion…