Cómo resolver problemas (de matemáticas)?.

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ÍNDICE

Problem Solving

1. UNDERSTAND THE PROBLEM.

2. THINK OF A PLAN.

3. CARRY OUT THE PLAN.

4. LOOK BACK.

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Problem Solving

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.

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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:
1.What is the unknown?
2.What are the given quantities?
3.What are the given conditions?

For many problems it is useful to draw a diagram and 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, y but in some cases it helps to use initials as suggestive symbols, for instance, V for volume, t for time.

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2. THINK OF A PLAN.

Find a connection between the given information and the unknown that will enable you to calculate the unknown. If you do not see the connection immediately, the following ideas may be helpful in devising a plan.

(a) Try to recognize something familiar. Relate the given situation to previous knowledge. Look at the unknown and try to recall a more familiar problem having a similar unknown.

(b) 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, then you might be able to guess what thecontinuing pattern is, and then prove it.

(c) Use analogy. Try to think of an analogous problem, that is, a similar and 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 one. For instance, if a problem involves very large numbers, you could first try a similar problem with smaller numbers. Or if the problem is in 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.

(d) 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 geometry the auxiliary aid could be a new line drawn in a diagram. In algebra it could be a new unknown that is related to the original unknown.

(e) Take cases. You may sometimes have to split a problem into several cases and give a different argument for each of the cases.

(f) Work backwards. Sometimes it is useful to imagine that your problem is solved and work backwards, step by step, till you arrive at the given data. Then you may be able to reverse your steps and thereby construct a solution to the original problem.

(g) Use indirect reasoning. Sometimes it is appropriate to attack a problem indirectly. For instance, in a counting argument it might be best to count the total number of objects and subtract the number of objects that do not have the required property. Another example of indirect reasoning is proof by contradiction in which we assume that the desired conclusion is false and eventually arrive at a contradiction.

(h) Use mathematical induction. In proving statements that involve a positive integer n, it is frequently helpful to use the Principle of Mathematical Induction, which is discussed in Section 4.7

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3. CARRY OUT THE PLAN.

In Step 2 a plan was devised. In carrying out that plan you have to check each stage of the plan and write the details that prove that each stage is correct.

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4. LOOK BACK.

Having completed your solution, it is wise to look back over it, partly to see if there are errors in the solution, and partly to see if there is an easier way to solve the problem. Another reason for looking back is that it will familiarize you 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 I then used in solving other problems."