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An Introduction to Problem solving

An Introduction to Problem solving. Fruit Problem.

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An Introduction to Problem solving

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  1. An Introduction to Problem solving

  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?

  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).

  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.

  5. Some problems to consider

  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

  7. Four-step problem-solving process • 1. Understand the problem • 2. Devise a plan • 3. Carry out the plan • 4. Look back

  8. Step oneUnderstanding 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?

  9. Step TwoDevising 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.

  10. Step threeCarrying out the plan • Implement the strategy or strategies. • Check each step of the plan as you proceed. • Keep an accurate record of your work.

  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.

  12. The Great problem solver“The prince of Mathematics”

  13. Gauss’s 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!

  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.)

  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?

  16. Round Robin Problem“The Solution” • Sixteen people in a round-robin handball tournament played every person once. How many games were played? • Let’s look at some patterns that develop when we look at some simpler problems. • Let’s label the participants as: A, B, C, D, . . .

  17. Round RobinSimpler Problems

  18. Round RobinObservation of pattern • So for 16 players it would be: 15 + 14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 That is 120! • Remember the Gauss problem 15 + 14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11+ 12 +13 +14 + 15 16 + 16 + 16 + 16 + 16 + 16 + 16 +16 + 16 + 16 +16 +16 +16 +16 + 16 That is = 120

  19. Round RobinGeneral formula • If there were n players, how many tournaments would have to be played?

  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

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