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How to Solve It suggests the following steps when solving a mathematical problem:
- First, you have to understand the problem.
- After understanding, then make a plan.
- Carry out the plan.
- Look backon your work.How could it be better?
If this technique fails, PÃ³lya advises: "If you can't solve a problem, then there is an easier problem you can solve: find it." Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"
First principle: Understand the problem
"Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, PÃ³lya taught teachers how to prompt each student with appropriate questions, depending on the situation, such as:
- What are you asked to find or show?
- Can you restate the problem in your own words?
- Can you think of a picture or a diagram that might help you understand the problem?
- Is there enough information to enable you to find a solution?
- Do you understand all the words used in stating the problem?
- Do you need to ask a question to get the answer?
The teacher is to select the question with the appropriate level of difficulty for each student to ascertain if each student understands at their own level, moving up or down the list to prompt each student, until each one can respond with something constructive.
Second principle: Devise a plan
PÃ³lya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:
- Guess and check
- Make an orderly list
- Eliminate possibilities
- Use symmetry
- Consider special cases
- Use direct reasoning
- Solve an equation
- Look for a pattern
- Draw a picture
- Solve a simpler problem
- Use a model
- Work backward
- Use a formula
- Be creative
- Use your head/noggin
Third principle: Carry out the plan
This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals.
Fourth principle: Review/extend
PÃ³lya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't. Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.
The book contains a dictionary-style set of heuristics, many of which have to do with generating a more accessible problem. For example:
The technique "have I used everything" is perhaps most applicable to formal educational examinations (e.g., n men digging m ditches) problems.
The book has achieved "classic" status because of its considerable influence (see the next section).
- It has been translated into several languages and has sold over a million copies, and has been continuously in print since its first publication.
- Marvin Minsky said in his influential paper Steps Toward Artificial Intelligence that "everyone should know the work of George PÃ³lya on how to solve problems."
- PÃ³lya's book has had a large influence on mathematics textbooks as evidenced by the bibliographies for mathematics education.
- Russian physicistZhores I. Alfyorov, (Nobel laureate in 2000) praised it, saying he was very pleased with PÃ³lya's famous book.
- Russian inventor Genrich Altshuller developed an elaborate set of methods for problem solving known as TRIZ, which in many aspects reproduces or parallels PÃ³lya's work.
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Answers:It's important to know that vertical angles are congruent. Therefore angles 2 and 4 are =. That means: 2x + 3 = 4x - 7 solve for x: -2x = -10 x = 5 that's it! ;)
Answers:the measure of vertical angles are equal so you have an equation: 5x - 20 = 3x - 2 solve for x: 2x = 18 x = 9 plug in the 9 for x: 5(9) - 20 = 45 - 20 = 25 or 3(9) - 2 = 27 - 2 = 25 The measure of each angle is 25 that's it! ;)
Answers:The vertical angle theorem tells us that opposite angles are equal. So, 6x - 16 = 4x + 6 so 2x = 22 => x = 11. Once you plug it in, you get that the angle is 50 degrees. Then you can say that 4x + 6 + (8x + 7y) = 180 (since they form a straight line). Since you know that 4x + 6 is 50 degrees you can plug that in and solve for y. Also plug in x = 11 for x. 50 + 88 + 7y = 180 138 + 7y = 180 7y = 42 y = 6 So you get that 8x + 7y = 130. 8x + 7y = 130 6x - 16 = 4x + 6 = 50
Answers:Since adjacent angles can never be vertical angles, the two circles representing then would not overlap.