Keywords: common and systematic transformation methods in solving mathematical problems
China Library ClassificationNo.: G633.6 Document ID: A DocumentNo.:1002-7661(2011) 09-0080-02.
Conversion is an important thinking method to solve mathematical problems, and the thought of conversion is an important basic idea to analyze and solve problems. Many mathematical ideas are the embodiment of transforming ideas. Any new knowledge is always the result of the development and transformation of the original knowledge. In mathematics teaching, especially in the process of problem-solving training, our teachers should gradually teach students some transformed thinking methods, so that they can learn new knowledge and analyze new problems with transformed views. Therefore, the transfer of students' learning mathematics is conducive to the realization of learning transfer, especially the transfer of principles and attitudes, thus rapidly improving the learning quality and mathematics ability. There are many methods of transformation, so we should master them flexibly, realize the "four modernizations" and complete the purpose of cultivating students' ability to analyze and solve problems.
First, be familiar with unfamiliar problems.
Turning unfamiliar problems into familiar ones is a common way of thinking in solving problems. Careful observation and application of the knowledge learned in the past can turn unfamiliar problems into familiar ones and avoid psychological obstacles to new problems, which can often get twice the result with half the effort.
1. Transform by analogy.
Example 1: When teaching the trapezoid area formula, we can review the derivation method of the triangle area formula first, so that students can further understand the basic idea of deriving the triangle area formula: transforming the triangle into the learned plane figure. (as shown in figure 1)
Then guide students to analogize and associate, and try to deduce the trapezoidal area formula in the same way. Through observation, comparison and measurement, students can transform the trapezoid into parallelogram, triangle and rectangle, and it is easy to get the area formula of trapezoid.
2. According to the connection, realize the conversion.
Example 2: Find the shaded area in the figure below. (See Figure 3)
The shaded part in the figure is an irregular figure and seems to have no solution. However, if A is translated 2 meters to the right to get Figure 4, it is easy to find the area. Isn't the small rectangle 4 meters long and 2 meters wide in the picture the original shaded part?
Second, simplify complex problems.
Sometimes we encounter complicated and difficult questions, and we can use some skills to help us answer them easily according to some rules.
1. Reasonable division and transformation.
Example 3: Find the shaded area in Figure 5.
There is no ready-made formula to calculate the area of combined graphics, so the original graphics must be divided and transformed. This problem can be divided into three figures. (Figure 6) This has become a very simple problem. Of course, to complete this transformation, you need to have certain observation and analysis skills.
2. Seek difference and simplification, and realize transformation.
If we calculate this problem in the order of operation, it is not only cumbersome, but also prone to errors. It is better to start a new stove and turn it into a fractional form:
= 1. 1
Third, the concretization of abstract problems.
Number-shape transformation is a typical mathematical idea, which can directly and visually translate abstract quantitative relations and facilitate deeper understanding.
1. For example, realize transformation.
Example 5: A number is reduced by 50% and then increased by 50%. What percentage of the original number is the result?
Here you can concretize a number, for example, let a number be 100 to explore. 100-50%)+50%) = 75, and it is easy to get the answer: the result is 75% of the original.
2. Graphic display, realizing conversion
There are 30 students in Grade 6, and everyone has subscribed to at least one magazine. The whole class * * * subscribes to 25 copies of China Youth Daily and 20 copies of Story Club. How many people have subscribed to these two magazines?
Use Figure 7 to help you think. In the picture, the big circle on the left indicates the number of subscribers to China Youth Daily, the small circle on the right indicates the number of subscribers to Story Club, and the shaded part in the middle indicates the number of subscribers to two magazines.
As can be seen from Figure 7, the number of subscribers of the two magazines is 25+20-30= 15 (person) respectively.
Fourth, some systemic problems.
Some mathematical problems give the equivalent relationship between two or more unknowns. If these unknowns are needed, we can choose a basic unknown quantity as the standard and systematize the quantitative relationship of the problem through equivalent substitution.
Example 7: In the grain and oil store, the * * * value of 2kg of rice and 3kg of flour is 1 Yuan 80, and that of 3kg of rice and flour is 1 Yuan 70. 1kg What is the value of rice flour?
Because: 3 kg of rice +2 kg of flour = 17 jiao ...
2 kg of rice +3 kg of flour = 18 jiao ... ②
So: 5 kg of rice +5 kg of flour =35 jiao,
So 2 kg of rice +2 kg of flour =35? 14 angle ... ③
Substituting Formula ③ into Formula ① gives: 1 kg rice = 17- 14=3 jiao.
Substituting Formula ③ into Formula ② gives: 1 kg flour = 18- 14=4 jiao.
In a word, the transformation methods influence and interweave with each other in the actual problem solving process. We should attach importance to teaching students the thinking method of transformation, so that students can master a variety of transformation methods, master problem-solving strategies and improve their problem-solving ability.
(Editor Li Xiang)
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