Saturday, February 19, 2011
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Friday, February 4, 2011
Polya Steps According to Stewart
``There are no hard and fast rules that will ensure success in solving problems. However, it is
possible to outline some general steps in the problem-solving process and to give some principles
that may be useful in the solution of certain problems. These steps and principles are
just common sense made explicit. They have been adapted from George Polya’s book How
To Solve It.
1. Understand the Problem
For many problems it is useful to
Usually it is necessary to
In choosing symbols for the unknown quantities we often use letters such as $a, b, c, m, n, x,$
and $y$, but in some cases it helps to use initials as suggestive symbols; for instance, $V$ for
volume or $t$ for time.
2. Think of a Plan
at the unknown and try to recall a more familiar problem that has a similar unknown.
Try to Recognize Patterns Some problems are solved by recognizing that some kind of pattern
is occurring. The pattern could be geometric, or numerical, or algebraic. If you can see
regularity or repetition in a problem, you might be able to guess what the continuing pattern
is and then prove it.
Use Analogy Try to think of an analogous problem, that is, a similar problem, a related
problem, but one that is easier than the original problem. If you can solve the similar, simpler
problem, then it might give you the clues you need to solve the original, more difficult
problem. For instance, if a problem involves very large numbers, you could first try a similar
problem with smaller numbers. Or if the problem involves three-dimensional geometry,
you could look for a similar problem in two-dimensional geometry. Or if the problem you
start with is a general one, you could first try a special case.
Introduce Something Extra It may sometimes be necessary to introduce something new, an
auxiliary aid, to help make the connection between the given and the unknown. For instance,
in a problem where a diagram is useful the auxiliary aid could be a new line drawn in a diagram.
In a more algebraic problem it could be a new unknown that is related to the original
unknown.
Take Cases We may sometimes have to split a problem into several cases and give a different
argument for each of the cases. For instance, we often have to use this strategy in dealing
with absolute value.
Work Backward Sometimes it is useful to imagine that your problem is solved and work
backward, step by step, until you arrive at the given data. Then you may be able to reverse
your steps and thereby construct a solution to the original problem. This procedure is commonly
used in solving equations. For instance, in solving the equation $3x - 5 = 7$ , we suppose
that is a number that satisfies and work backward. We add $5$ to each side
of the equation and then divide each side by $3$ to get $x = 4$ . Since each of these steps can
be reversed, we have solved the problem.
Establish Subgoals In a complex problem it is often useful to set subgoals (in which the
desired situation is only partially fulfilled). If we can first reach these subgoals, then we may
be able to build on them to reach our final goal.
Indirect Reasoning Sometimes it is appropriate to attack a problem indirectly. In using
proof by contradiction to prove that $P$ implies $Q$, we assume that $P$ is true and $Q$ is false and
try to see why this can’t happen. Somehow we have to use this information and arrive at a
contradiction to what we absolutely know is true.
Mathematical Induction In proving statements that involve a positive integer , it is frequently
helpful to use the following principle.
3. Carry Out the Plan
possible to outline some general steps in the problem-solving process and to give some principles
that may be useful in the solution of certain problems. These steps and principles are
just common sense made explicit. They have been adapted from George Polya’s book How
To Solve It.
1. Understand the Problem
The first step is to read the problem and make sure that you understand it clearly. Ask yourself the following questions:
What is the unknown?
What are the given quantities?
What are the given conditions?
For many problems it is useful to
draw a diagramand identify the given and required quantities on the diagram.
Usually it is necessary to
introduce suitable notation
In choosing symbols for the unknown quantities we often use letters such as $a, b, c, m, n, x,$
and $y$, but in some cases it helps to use initials as suggestive symbols; for instance, $V$ for
volume or $t$ for time.
2. Think of a Plan
Find a connection between the given information and the unknown that will enable you toTry to Recognize Something Familiar Relate the given situation to previous knowledge. Look
calculate the unknown. It often helps to ask yourself explicitly: “How can I relate the given
to the unknown?” If you don’t see a connection immediately, the following ideas may be
helpful in devising a plan.
at the unknown and try to recall a more familiar problem that has a similar unknown.
Try to Recognize Patterns Some problems are solved by recognizing that some kind of pattern
is occurring. The pattern could be geometric, or numerical, or algebraic. If you can see
regularity or repetition in a problem, you might be able to guess what the continuing pattern
is and then prove it.
Use Analogy Try to think of an analogous problem, that is, a similar problem, a related
problem, but one that is easier than the original problem. If you can solve the similar, simpler
problem, then it might give you the clues you need to solve the original, more difficult
problem. For instance, if a problem involves very large numbers, you could first try a similar
problem with smaller numbers. Or if the problem involves three-dimensional geometry,
you could look for a similar problem in two-dimensional geometry. Or if the problem you
start with is a general one, you could first try a special case.
Introduce Something Extra It may sometimes be necessary to introduce something new, an
auxiliary aid, to help make the connection between the given and the unknown. For instance,
in a problem where a diagram is useful the auxiliary aid could be a new line drawn in a diagram.
In a more algebraic problem it could be a new unknown that is related to the original
unknown.
Take Cases We may sometimes have to split a problem into several cases and give a different
argument for each of the cases. For instance, we often have to use this strategy in dealing
with absolute value.
Work Backward Sometimes it is useful to imagine that your problem is solved and work
backward, step by step, until you arrive at the given data. Then you may be able to reverse
your steps and thereby construct a solution to the original problem. This procedure is commonly
used in solving equations. For instance, in solving the equation $3x - 5 = 7$ , we suppose
that is a number that satisfies and work backward. We add $5$ to each side
of the equation and then divide each side by $3$ to get $x = 4$ . Since each of these steps can
be reversed, we have solved the problem.
Establish Subgoals In a complex problem it is often useful to set subgoals (in which the
desired situation is only partially fulfilled). If we can first reach these subgoals, then we may
be able to build on them to reach our final goal.
Indirect Reasoning Sometimes it is appropriate to attack a problem indirectly. In using
proof by contradiction to prove that $P$ implies $Q$, we assume that $P$ is true and $Q$ is false and
try to see why this can’t happen. Somehow we have to use this information and arrive at a
contradiction to what we absolutely know is true.
Mathematical Induction In proving statements that involve a positive integer , it is frequently
helpful to use the following principle.
3. Carry Out the Plan
In Step 2 a plan was devised. In carrying out that plan we have to check each stage of the4. Look Back
plan and write the details that prove that each stage is correct.
Having completed our solution, it is wise to look back over it, partly to see if we have made
errors in the solution and partly to see if we can think of an easier way to solve the problem.
Another reason for looking back is that it will familiarize us with the method of solution and
this may be useful for solving a future problem. Descartes said, “Every problem that I solved
became a rule which served afterwards to solve other problems.”
These principles of problem solving are illustrated in the following examples. Before you
look at the solutions, try to solve these problems yourself, referring to these Principles of
Problem Solving if you get stuck. You may find it useful to refer to this section from time
to time as you solve the exercises in the remaining chapters of this book.
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