Problem Solving Lecture Notes PDF

Chapter 3 Problem Solving 1. Inductive and Deductive Reasoning Inductive Reasoning – is the process of reaching a general conclusion by examining specific examples. A conclusion based on inductive reasoning is called a conjecture. A conj...

Chapter 3 Problem Solving 1. Inductive and Deductive Reasoning Inductive Reasoning – is the process of reaching a general conclusion by examining specific examples. A conclusion based on inductive reasoning is called a conjecture. A conjecture may or may not be correct. 2 PROBLEM SOLVING 1. Inductive and Deductive Reasoning EXAMPLE 1: 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, ? 3 PROBLEM SOLVING 1. Inductive and Deductive Reasoning EXAMPLE 2: Consider the following procedure: Pick a number. Multiply the number by 8, add 6 to the product, divide the sum by 2, and subtract 3. Complete the above procedure for several different numbers. Use inductive reasoning to make a conjecture about the relationship between the size of the resulting number and the size of the original number. 4 PROBLEM SOLVING  1. Inductive and Deductive Reasoning EXAMPLE 3: Length of Period of pendulum, pendulum, in units in heartbeats 1 1 4 2 9 3 16 4 25 5 36 6 The period of a pendulum is the time Use the data in the table and inductive reasoning to answer each of it takes for the the following questions. pendulum to swing a) If a pendulum has a length of 49 units, what is its period? from left to right and b) If the length of a pendulum is quadrupled, what happens to its back to its original period? position. 5 PROBLEM SOLVING 1. Inductive and Deductive Reasoning Deductive Reasoning – is the process of reaching a conclusion by applying general assumptions, procedures, or principles. 6 PROBLEM SOLVING 1. Inductive and Deductive Reasoning EXAMPLE 4: Solve a Logic Puzzle Each of the four friends, Donna, Sarah, Nikkie, and Xhanelle, has a different pet (fish, cat, dog, and snake). From the following clues, determine the pet of each individual. 1) Sarah is older than her friend who owns the cat and younger than her friend who owns the dog. 2) Nikkie and her friend who owns the snake are both of the same age and are the youngest members of their group. 3) Donna is older than her friend who owns the fish. 7 PROBLEM SOLVING  SOLUTION: From Clue 1, Sarah does not own a cat nor a dog. In the following table, write X1(which stands for rule out by clue1) in the cat and dog column for Sarah Fish Cat Dog Snake Donna Sarah X1 X1 Nikkie Xhanell e From Clue 2, Nikkie does not own a snake and a dog being the youngest. And since Sarah is not the youngest from Clue 1, then Sarah does not own a snake as well.Write X2 n snake column for Nikkie And X1 in snake column for Sarah.There are now Xs in the 3 pets in Sarah’s row, therefore Sarah owns the fish.Put a √ which means Sarah’s pet is a fish.So, Donna, Nikkie and Xhanelle do not own the fish.  Fish Cat Dog Snake Donna Sarah √ X1 X1 X1 Nikkie X2 X2 Xhanell e From Clue 3, Donna is older than Sarah, hence Donna owns the dog. Write X3 in cat snake columns for Donna.There are now Xs in snake column for Donna, Sarah, and Nikkie; therefore, Xhanelle owns the snake. Put a check in that box. Write X3 the cat column for Xhannele, hence Nikkie owns the cat. Put a check in that box.  Fish Cat Dog Snake Donna X2 X3 √ X3 Sarah √ X1 X1 X1 Nikkie X2 √ X2 X2 Xhanelle X2 X3 X3 √ Thus, Sarah owns the fish, Donna owns the dog, Xhanelle owns the snake and Nikkie owns the cat. Try this: Each of the four siblings (Edmund, Genalyn, Madelyn, and Sonia) bought four different cars. One chooses a Honda, a Mitsubishi, a Toyota, and a Suzuki car. From the following clues, determine which sibling bought which car. 1.) Edmund, living alone stays next door to his sister who bought the Honda car and very far from his sister who bought the Suzuki car. 2.) Genalyn, also living alone, is younger than the one who bought the Mitsubishi car and older than her sibling who bought the Toyota car. 3.) Madelyn did not like Toyota and Suzuki cars.But she and her sibling, who bought the Toyota car,live in the same house. 1. Inductive and Deductive Reasoning A statement is a true statement provided it is true in all cases. If you can find one case in which a statement is not true, called a counterexample, then the statement is a false statement. 1 4 PROBLEM SOLVING 2. Problem Solving with Patterns Sequences A sequence is an ordered list of numbers. Each number in a sequence is called a term of the sequence. The is used to designate the term of a sequence. A formula that can be used to generate all the terms of a sequence is called an formula. 1 5 PROBLEM SOLVING 2. Problem Solving with Patterns EXAMPLE 1: Predict the Next Term Use a difference table to predict the next term in the sequence. 2, 7, 24, 59, 118, 207, … 1 6 PROBLEM SOLVING What is the next term of the sequence, -1, 4, 21, 56, 115, 204?  What is a problem? It is a question that motivates a person to search for a solution. 1.It implies that one wants or needs to solve the problem. 2.One has to search for a way to find a solution. 3. Problem-Solving Strategies One of the foremost recent mathematicians to make a study of problem solving was George Polya (1877-1985). He was born in Hungary and moved to the United States in 1940. the basic problem-solving strategy that Polya advocated consisted of the following four steps. 2 0 PROBLEM SOLVING 3. Problem-Solving Strategies Polya’s Four-Step Problem- Solving Strategy 1) Understand the problem. 2) Devise a plan. 3) Carry out the plan. 4) Review the solution. 2 1 PROBLEM SOLVING 3. Problem-Solving Strategies Polya’s four steps are deceptively simple. To become a good problem solver, it helps to examine each of these steps and determine what is involved. 2 2 PROBLEM SOLVING  3. Problem-Solving Strategies You must have a clear understanding of the Once you have found a problem. solution, check the “Can you restate the solution. problem in your own Ensure that the words?” solution is consistent with the facts of the problem. Successful problem Work carefully. solvers use a variety of Keep an accurate techniques when they and neat record of attempt to solve a all your attempts. problem. 2 Realize that some of 3 your initial plans will notPROBLEM work and modify SOLVING  3. Problem-Solving Strategies EXAMPLE 5: Apply Polya’s Strategy 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. She then lost half her remaining money when she landed on Illionois Avenue. Stacy now has $2,500. How much did she have just before she purchased the hotel? 2 4 PROBLEM SOLVING 1.Three couples go picnic together. They must move across a river with only one boat carrying a maximum of 2 people. Help them move across the river, provided that husbands would not allow their wife to go with another man or left with another man without their presence. 2.Find the sum of the first 100 positive integers. 3.In how many ways can we make change a quarter, using only dimes, nickels and pennies? PROBLEM SOLVING 25 Modulo n Two integers a and b are said to be congruent modulo n, with n being a natural number, if is an integer. In this case, we write a≡bmod n.The the number n is called modulus. The statement a≡bmodn is called congruence. Example. Finding a day of a week. In 2017, Venus’ birthday fell on a Saturday, June 3. On what day of the week does Venus’ birthday fall in 2020? Note that the year 2020 is a leap year. Solution: The number of days in a year is 365 except when it is a leap year where there is one day added. How many days are there from June 3, 2017 to June 3, 2020? PROBLEM SOLVING 26 Number of days: After June 3, 2017 to June 3, 2018: 365 After June 3, 2017 to June 3, 2019: 365 After June 3, 2017 to June 3, 2020: 366(leap year) Total :1096 Beacause 1096/7 = 156 has a remainder of 4,then we write 1096 ≡4mod7. Since a week is a cycle then any multiple of 7 days past a given day will be the same day of the week. It means that on the 1092nd day, 1092 being a multiple of 7,after June 3, 2017 is also a Saturday. Furthermore, on the 1096th day, four days after, is a Wednesday. Thus June 3, 2020 will be Wednesday.  3. Problem-Solving Strategies EXAMPLE 7: Solve a Deceptive Problem A hat and a jacket together cost $100. the jacket costs $90 more than the hat. What are the cost of the hat and the cost of the jacket? 2 8 PROBLEM SOLVING Reference: Aufmann, R. N., Lockwood, J. S., Nation, R. D. & Clegg, D. K. (2013). Mathematical Excursions, Third Edition. CA: Brooks/Cole, Cengage Learning. 2 9 PROBLEM SOLVING