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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.
From Yahoo Answers
Answers:1. a.) Let the set of all who play basketball be designated B; the set of those who sing in the Choir is designated C, and the set of those who do Drama D. From the information given, B = 99. Since B C = 18, B D = 34, and 16 only do basketball, then B C D = 99 - (18 + 34 + 16) = 99 - 68 = 31. This also implies that 31 of the students who do Choir also do Basketball and Drama, and 31 of those who do Drama also do Choir and Basketball, because B C D = C B D = D B C. Therefore, x = 31. 1. b.) To determine the total number of students in the school, we must add up all the students in each group, and subtract the intersections of any two groups once from that total, because not to do so would be to double count those in the intersections. Also, we must subtract the intersection of all three groups twice, because not to do so would be to count that intersection three times. The total number of students in the school is given by this equation: T = B U C U D - [(B C) + (B D) + (C D) + 2 (B C D)]. Plugging in our given and calculated data, we get this: T = 99 + 88 + 110 - [18 + 34 + 18 + 2 (31)] T = 297 - (70 + 62) T = 297 - 132 T = 165. There are a total of 165 students in the school. 2. Read the following taken from purplemath.com. It should give you a good idea how to go about solving this problem. It even shows pictures of calculator screens setting up the problems. Good luck. http://www.purplemath.com/modules/scattreg.htm
Answers:Each student is in one of the following sets h = history only m = math only e = english only HM = history and math, but not english HE = history and english, but not math ME = math and english, but not history A = All three, history, math and english We are given: m + h + e + HM + HE + ME + A = 200 m + HM + ME + A = 80 h + HM + HE + A = 60 e + HE + ME + A = 140 HM + A = 40 HE + A = 30 ME + A = 20 Rearranging: HM = 40 - A HE = 30 - A ME = 20 - A Substituting and simplifing: m + h + e + (40 - A) + (30 - A) + (20 - A) + A = 200 m + (40 - A) + (20 - A) + A = 80 h + (40 - A) + (30 - A) + A = 60 e + (30 - A) + (20 - A) + A = 140 m + h + e - 2A = 110 m - A = 20 h - A = -10 e - A = 90 Rearrange m = 20 + A h = -10 + A e = 90 + A Substitute and simplify (20 + A) + (-10 + A) + (90 + A) - 2A = 110 100 + A = 110 A = 10
Answers:4) Although the weather was perfect for the beach party, 17 of the 30 people attending got a sunburn and 25 people were bitten by mosquitos. If 12 people were both bitten and sunburned, how many had neither affliction? s+m+t+x= 30 17 = s+t 25=m+t 12=t --> s= 5; m= 13; t= 12; x= 0; answer = 0 5) Of the 52 teachers at Roosevelt high school, 27 said they liked to go sailing, 25 said they like to go fishing, and 12 said they don't enjoy either recreation. how many enjoy fishing but not sailing? s+f+t+x=52 s+t=27 f+t=25 x=12 --> s= 15; f= 13; t= 12; x= 12; answer = 13
Answers:I can give you a hint there is a rule n(A B) = n(A) + n(B) n(A B) we have n(A B) = 73 n(A) = 25 n(B) = we don't know n(A B) = 21 ( this is same as N(AandB) = 21) did you get the idea now? if you need more help just write to me at firstname.lastname@example.org