AN INTRODUCTION TO PROBLEM SOLVING. FRUIT PROBLEM There are three bags of fruit in front of you....
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AN INTRODUCTION TO PROBLEM SOLVING
AN INTRODUCTION TO PROBLEM SOLVING. FRUIT PROBLEM There are three bags of fruit in front of you. One bag contains all apples, one bag contains all oranges,
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?
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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).
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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.
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SOME PROBLEMS TO CONSIDER
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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
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FOUR-STEP PROBLEM-SOLVING PROCESS 1. Understand the problem
2.Devise a plan 3.Carry out the plan 4.Look back
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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?
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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.
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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.
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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.
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THE GREAT PROBLEM SOLVER THE PRINCE OF MATHEMATICS
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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!
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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.)
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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?
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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,...
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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
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ROUND ROBIN OBSERVATION OF PATTERN
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ROUND ROBIN GENERAL FORMULA
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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