Slide 1

AN INTRODUCTION TO PROBLEM SOLVING

Slide 2

FRUIT PROBLEM There are three bags of fruit in front of you. One bag contains all apples, one bag contains all oranges, and one bag contains apples and oranges. Each bag is labeled with one of the labels: Apples, Oranges, or Apples & Oranges. However each bag is incorrectly labeled. Your task is to select one bag and reach in and grab one piece of fruit. Having done this and using the information above can you label each bag correctly?

Slide 3

WHAT IS PROBLEM SOLVING? Problem solving has long been recognized as one of the hallmarks of mathematics. Solving a problem means finding a way out of difficulty, a way around an obstacle, attaining an aim which was not immediately attainable. George Polya (1887-1985).

Slide 4

GOOD MATHEMATICAL PROBLEM SOLVING OCCURS WHEN : Students are presented with a situation that they understand but do not know how to proceed directly to a solution. Students are interested in finding the solution and attempt to do so. Students are required to use mathematical ideas to solve the problem. Note: A reasonable amount of tension and discomfort improves problem-solving performance. Mathematical experience often determines whether situations are problems or exercises.

Slide 5

SOME PROBLEMS TO CONSIDER

Slide 6

GEORGE POLYA (1887 1995) Born in Hungary Received his Ph.D. from the University of Budapest Moved to the United States in 1940 After a brief stay at Brown University he joined the faculty at Stanford University He focused on the vital importance of mathematics education Published 10 books including How to Solve It (1945) Developed the four-step problem-solving process

Slide 7

FOUR-STEP PROBLEM-SOLVING PROCESS 1. Understand the problem 2.Devise a plan 3.Carry out the plan 4.Look back

Slide 8

STEP ONE UNDERSTANDING THE PROBLEM Can you state the problem in your own words? What are you trying to find or do? What are the unknowns? What information do you obtain from the problem? What information, if any, is missing or not needed?

Slide 9

STEP TWO DEVISING A PLAN (SOME STRATEGIES YOU MAY FIND USEFUL) Look for a pattern. Examine related problems and determine if the same technique can be used. Examine a simpler problem to gain insight into the solution of the original problem. Make a table or list. Make a diagram. Write an equation. Use guess and check. Work backward. Identify a subgoal. Use indirect reasoning. Use direct reasoning.

Slide 10

STEP THREE CARRYING OUT THE PLAN Implement the strategy or strategies. Check each step of the plan as you proceed. Keep an accurate record of your work.

Slide 11

LOOKING BACK Check the results in the original problem. Interpret the solution in terms of the original problem. Determine whether there is another method of finding the solution. If possible, determine other related or more general problems for which the techniques will work.

Slide 12

THE GREAT PROBLEM SOLVER THE PRINCE OF MATHEMATICS

Slide 13

GAUSSS PROBLEM When Carl Gauss was a child, his teacher required the students to find the sum of the first 100 natural numbers. The teacher expected this problem to keep the class occupied for some time. Gauss gave the correct answer almost immediately. With a partner solve this problem. Be prepared to explain how you arrived at your answer. The answer is 5050!

Slide 14

A MAGIC SQUARE Arrange the numbers 1 through 9 into a square subdivided into nine smaller squares like the one shown so that the sum of every row, column and main diagonal is the same. (The result is a magic square.)

Slide 15

ROUND-ROBIN Sixteen people in a round-robin handball tournament played every person once. How many games were played? Work with a partner to solve the problem. Be prepared to share your solution. What strategy did you use?

Slide 16

ROUND ROBIN PROBLEM THE SOLUTION Sixteen people in a round-robin handball tournament played every person once. How many games were played? Lets look at some patterns that develop when we look at some simpler problems. Lets label the participants as: A, B, C, D,...

Slide 17

ROUND ROBIN SIMPLER PROBLEMS Two Players Three Players Four Players Five Players Six Players ABAB ACAB AC ADAB AC AD AEAB AC AD AE AF BCBC BDBC BD BEBC BD BE BF CDCD CECD CE CF DEDE DF EF Total Number of Rounds 1361015

Slide 18

ROUND ROBIN OBSERVATION OF PATTERN

Slide 19

ROUND ROBIN GENERAL FORMULA

Slide 20

PROBLEMS?... "The problem is not that there are problems. The problem is expecting otherwise and thinking that having problems is a problem. Theodore Rubin The best way to escape from a problem is to solve it.--Brendan Francis Every problem contains within itself the seeds of its own solution.--Stanley Arnold It isn't that they can't see the solution. It's that they can't see the problem.--G. K. Chesterton Problems are to the mind what exercise is to the muscles, they toughen and make strong. - Norman Vincent Peale