Guess is a wonderful method and plays an important role in many mathematical problems. When we can't solve the problem with conventional channels, we might as well use "guessing" to work out the answer to the problem slowly.
However, it needs to be emphasized that mathematical guessing is not a blind guess, a roll of dice, or a trick like "three short and one long to choose the longest", but a well-founded and relatively accurate guess based on mathematical logic and reasoning.
Unfortunately, many candidates don't understand this, and they are always at two extremes when facing multiple-choice questions.
So how to make a mathematical guess?
Estimate numerical results.
The estimated result is not a high-end skill. Most candidates have a sense that the results between candidates are usually "regular". If a particularly "wonderful" figure is calculated, it is likely to be miscalculated. This is the simplest result estimate.
Further, we can look at such an example.
If it can be estimated that A is between 0 and 1 and B is less than-1, then it can be estimated that a+b and ab are both negative numbers, options C and D must be wrong, and the correct answer will only be between A and B.
This estimate itself is not difficult, but candidates often can't think of an estimate. When they saw that it was the most difficult multiple-choice question, they gave up directly and chose 1 as one of the four choices, trying to bet on a "hit rate" of 25%. But even if you gamble, it is estimated that after gambling, is the "hit rate" of 50% higher than 25%?
Guess with a special case
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When Goldbach made a guess, he only tested a series of big even numbers, such as 4=2+2, 6=3+3, 8=3+5, 10=5+5, 12 = 5+7 ..., and then he got that "even numbers greater than 2 can be divisible into the sum of two prime numbers.
If a general conclusion is correct, then it must be correct for every special situation. According to limited special cases, we can also conclude and guess a possible correct universal conclusion.
This method is often seen in series. For example, if we can't directly find the general formula an of the series, we can guess one according to the first few items a 1, a 2, A 3, and then prove it (this was tested in the new curriculum standard 3 17( 1) in 2020). This method can also be used in other fields.
The general conclusion given in the title holds true for all triangles, so it should also hold true for any special triangle. Therefore, it can be specialized into an isosceles right triangle, and the results can be calculated directly by establishing a plane right-angled coordinate system.
This "guessing" method can be used not only for uncertain results, but also for testing. The reason why many students don't check their grades is that it is a waste of time to do it again, but if you use the method given above, you can check your grades quickly. For example, after finding the general formula a n, substitute n = 1 for testing. If a 1 is incorrect, can this an be correct?
It can be seen that although we are "guessing", our thinking is still "analyzing the problem first, then thinking logically" and still mathematical thinking. Mastering mathematical thinking is the key to improve the level of mathematics.
Mathematics is a logical subject and a logical language, so the process of reasoning is particularly important. In our atlas, it writes all the reasoning processes, all the thinking modes and combines them with practical applications in reality.