PROBLEM SOLVING USING INDUCTIVE AND DEDUCTIVE REASONING Compiled

PROBLEM SOLVING USING INDUCTIVE AND DEDUCTIVE REASONING Compiled: Still John F. Reyes

INDUCTIVE REASONING Examples: 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, ? 2. Pick a number. Multiply the number by 8, add 6 to the product, divide the sum by 2, and subtract 3. Use inductive reasoning to make a conjecture.

DEDUCTIVE REASONING Examples: 1. Pick a number. Multiply the number by 8, add 6 to the product, divide the sum by 2, and subtract 3. Use deductive reasoning to establish a conjecture.

DEDUCTIVE REASONING 2. Each of four neighbors, Sean, Maria, Sarah, and Brian, has a different occupation (editor, banker, chef, or dentist). From the ff. 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.

PROBLEM-SOLVING STRATEGIES Polya’s Problem Solving Strategy

POLYA: “THE FATHER OF PROBLEM SOLVING” • George Pólya was a Hungarian mathematician. • He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory. He is also noted for his work in heuristics and mathematics education.

POLYA'S FOUR-STEP PROBLEM SOLVING STRATEGY • In 1945 George Polya published the book How To Solve It which quickly became his most prized publication. • It sold over one million copies and has been translated into 17 languages. • In this book he identifies four basic principles of problem solving.

1. UNDERSTAND THE PROBLEM • Do you understand all the words used in stating the problem? • What are you asked to find or show? • Can you restate the problem in your own words? • Can you think of a picture or diagram that might help you understand the problem? • Is there enough information to enable you to find a solution?

2. DEVISE A PLAN • There are many reasonable ways to solve problems. • The skill lies in choosing an appropriate strategy. • This best learned by solving many problems. You will find choosing a strategy increasingly easy.

3. CARRY OUT THE PLAN • This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. • Persist with the plan that you have chosen. • If it continues not to work discard it and choose another. Don't be misled, this is how mathematics is done, even by professionals.

4. LOOK BACK/REVIEW THE SOLUTION • Much can be gained by taking the time to reflect and look back at what you have done, what worked, and what didn't. • Doing this will enable you to predict what strategy to use to solve future problems.

PROBLEM SOLVING STRATEGIES • Guess and check • Use a model • Look for a pattern • Consider special cases • Make an orderly list • Draw a picture • Eliminate possibilities • Solve a simpler problem • Use symmetry • Work backwards • Use direct reasoning • Use a formula • Solve an equation

EXAMPLES 1. A baseball team won two out of their last four games. In how many different orders could they have two wins and two losses in four games? 2. The product of the ages, in years, of three teenagers is 4590. None of the teens are the same age. What are the ages of the teenagers? 3. Mang Tomas owns goats and ducks. Counting heads there are 39. Counting legs there are 110. How many goats and how many ducks has Mang Tomas?

4. Each one – Ann, Enya, Alvin, and Johnny have different favorite color among red, blue, green, and orange. No person’s name contains the same number of letters as his/her favorite color. Alvin and the boy who likes blue live in different parts of town. Red is the favorite color of one of the girls. What is each person’s favorite color?