Connecting junior and senior high school teaching with classification thought
1. Configuration
● One of the three basic ideas;
● Can be directly tested with paper and pen;
● Required content for large-scale exams.
2. Analysis of the thinking process of classified thinking to solve problems.
When using the idea of classification to solve problems, its thinking process can usually be divided into: first, it is necessary to clarify whether it is necessary to discuss by classification; Second, determine the object of classification; Third, determine the classification standard; Fourth, discuss it step by step; Fifth, comprehensively summarize the conclusions.
First, it is clear whether it is necessary to discuss in different categories.
To solve problems with the idea of classification, we first need to make clear the reasons for classification discussion, that is, which problems often need to be solved with the idea of classification. When faced with a mathematical problem, it is difficult for most students to judge whether the problem needs to be solved by classification, that is, it is impossible to quickly identify the quantitative relationship or positional relationship related to classification in the problem according to the conditions and conclusions of the problem. Therefore, from the given problem situation, it is the basis of solving problems to correctly and quickly identify the quantitative relationship or positional relationship related to classification in the topic. Generally speaking, when we study the following situations, we can consider using classified thinking methods to solve problems.
● It involves the concept of classification definition.
Some concepts are defined by classification, such as rational number, real number, absolute value, isosceles triangle, square root, rational number formula and so on. When we apply these concepts, we must consider using the method of classified discussion.
Example 1: The circumference of an isosceles triangle is 16 and the length of one side is 6. Find the length of the other two sides.
When defining some concepts, the range of objects to be considered is limited, such as the concept of fraction and the quadratic equation of one variable. When we need to break through these limitations in solving problems, we must consider the method of classified discussion.
Example 2: Solve the equation about X (A- 1) X-2AX+A = 0.
● Directly apply the theorems, properties, formulas and rules of classification research.
The requirements of Mathematics Curriculum Standard directly apply the theorems, properties, formulas and rules of classification research;
Size comparison rules of rational numbers; Rules of addition, multiplication, division and power of rational numbers; The relationship between the symbol of rational number multiplication and the symbol of factor: the rules of parenthesis and brackets removal; Multiply (or divide) the same non-zero number on both sides of the equation, and the solution of the equation remains unchanged; Both sides of inequality multiply (or divide) the same positive (negative) number, and the direction of inequality remains unchanged; The root formula of quadratic equation in one variable; The discriminant formula of the root of a quadratic equation with one variable: the positional relationship between a straight line and a circle (the comparison of the number of intersection points, the radius and the distance from the center of the circle to the straight line); The positional relationship between two circles ((compare the number of intersection points, the sum of the radii of two circles, and the number of center distances); Properties of linear functions; Properties of inverse proportional function; Properties of quadratic function, etc.
When we apply the discriminant of the root of a quadratic equation in one variable, we can consider the method of classified discussion on the positional relationship between a straight line and a circle (the comparison of the number of intersections, the radius and the distance from the center of the circle to the straight line) and the positional relationship between two circles (the comparison of the number of intersections, the sum of the radii of two circles and the distance from the center of the circle).
If we need to break through the conditions of theorems, properties, formulas and laws when applying other theorems, properties, formulas and laws limited by the conditions of application scope, we can consider using the method of classified discussion.
Example 3: If all points on the image of the function y = kx+3 (- 1 ≤ x ≤ 1, and k≠0) are above the x axis, then the value range of k is.
● Perform some limited operations.
When solving problems, we should consider using classified discussion when we encounter operations such as division, even power and absolute value sign.
In the process of calculation and reasoning, quantity is uncertain.
In the process of calculation and reasoning, we often encounter the same known condition with different values (within the range of values), and different values lead to different results. In this case, we can consider using the classification method to solve the problem.
In the process of junior high school mathematics teaching, it is not only in line with the standards of the new curriculum, but also an excellent starting point for mathematics quality education to gradually and moderately infiltrate mathematical thinking methods, cultivate students' thinking ability and form good mathematical thinking habits. The classification thought in mathematics is not only an important mathematical thought, but also an important mathematical logic method. The idea of classification is not only very important in the inquiry of mathematical knowledge and concept learning, but also plays an irreplaceable role in solving mathematical problems.
The idea of classification in mathematics is to divide mathematical objects into several different categories according to their similarities and differences in essential attributes, and then study the mathematical idea of solving problems. It is not only an important mathematical thought, but also an important mathematical logic method.
The so-called mathematical classification discussion method is a mathematical method that divides mathematical objects into several categories and discusses them separately to solve problems. It is logical, comprehensive and exploratory to discuss mathematical problems related to thinking in categories, which can train people's thinking order and generality. Different from ordinary mathematics knowledge, the idea of classification can make students master the application through several classes of teaching. Instead, according to the age characteristics of students, students' cognitive level in each learning stage gradually permeates and spirals, constantly enriching their own connotations, so as to achieve the purpose of solving problems by using mathematical classification discussion methods.
In teaching, students can form an active application of classification ideas through analogy, observation, analysis, synthesis, discussion and generalization in the process of learning mathematics from the following aspects.
First, gradually and year by year, infiltrate the idea of classification and cultivate the sense of classification.
Every student has certain classification knowledge in daily life, such as the classification of people and stationery. We use this knowledge base of students to transfer the classification of life to mathematics, infiltrate the idea of mathematical classification into teaching, tap the opportunities provided by textbooks and seize the opportunities of infiltration. For example, the classification of learning number in grade seven, the meaning of absolute value, the nature of inequality and so on. , is a good opportunity to penetrate the classification thought. Know the number? ㄊ
After representing any number, let the students classify the number A and get three categories: positive number, zero number and negative number. When explaining the meaning of absolute value, guide students to get the following classification: through understanding the absolute values of positive numbers, zero numbers and negative numbers, understand how to learn and understand mathematical concepts through classified discussion. Combined with the teaching of the chapter "Rational Numbers", the idea of mathematical classification is repeatedly strengthened, so that students gradually form the consciousness of classification in mathematics learning. And when discussing classification, we can pay attention to some basic principles, such as the objects of classification are certain and the standards are unified, otherwise the objects are mixed and the standards are different, and there will be errors such as omission and repetition. If rational numbers are divided into positive numbers, negative numbers and integers, it is wrong to make different classification standards. After determining the objects and standards, we should also pay attention to distinguish the levels and not discuss them beyond the levels.
Second, penetrate the classification method to enhance the rigor of thinking.
When the idea of classification is permeated in teaching, students should understand that the so-called classification is to choose appropriate standards, divide the objects into several categories, and then answer the questions of each subcategory. Mastering a reasonable classification method has become the key to solving the problem.
Classification methods generally include the following:
1, classified according to mathematical concepts.
Example 1 Compare the square of a number with its absolute value, can you determine the relationship between them?
Analysis: We know that for numbers in the range of 0 to 1, the square of these numbers is less than or equal to the number itself; For a number greater than 1, its square is greater than the number itself. Because the range of the number given in the title is not clear, we can't determine the size and absolute value of the square of this number, so we need to discuss it in different situations (which can assist the discussion of the number axis).
2. Classification according to graphic features.
In example 2, △ABC, AB = 8°, angle B = 30, AC = 5°, find BC.
Analysis: According to the characteristics of triangle, this topic divides △ABC into acute triangle and obtuse triangle for classification and discussion, thus obtaining two results of BC.
When proving the fillet theorem in Grade Three, because there are three different situations in the position of the center of the circle: on the corner edge, inside the corner and outside the corner, we can guide students to prove that the center of the circle is on the side of the fillet, which is the easiest situation to prove, and then make the diameter of the vertex of the fillet, and then use this situation to solve the two situations in turn. This is a typical theorem in junior high school textbooks. The ideas and methods of classified discussion embodied in the process of theorem proving pave the way for the ideas and methods of classified discussion in the final exam of senior high school entrance examination.
Third, guide students to discuss in categories and improve their ability to solve problems reasonably.
There are many theorems, definitions, formulas, laws and exercises in junior high school textbooks that need to be discussed in categories. In carrying out these contents, we should constantly strengthen the awareness of classified discussion, so that students can realize these problems: only after classified discussion can the conclusion be complete and correct, and if it is not classified discussion, mistakes and omissions will easily occur. In problem-solving teaching, classified discussion is also helpful to help students sum up regular things, thus strengthening the order and meticulousness of students' thinking. Generally speaking, there are two kinds of problems solved by classified discussion ideas and methods:; One is to discuss and solve problems in algebraic expressions, functions or equations in different values according to different values of letters. The second is to discuss and solve problems one by one according to the different positions of points and lines in geometric figures (in recent years, this knowledge point is often examined at the finale of our province).
Example 4. A supermarket offers the following discounts? Scheme {1} One-time shopping does not exceed 100 yuan, and no discount is enjoyed. {2} You can enjoy a discount of 10% for one-time shopping over 65,438+000 yuan, but not more than 300 yuan {3} You can enjoy a discount of 20% for one-time shopping over 300 yuan.
Wang Bo bought it twice for 252 yuan from 80 yuan. If he buys the same goods as the first two times, he should pay ()
A.288 yuan B.332 yuan C.288 yuan or 3 16 yuan D.332 yuan or 3 16 yuan.
Solution: The first shopping obviously does not exceed 100, because 80/0.9=88.88, so the first real shopping value is 80.
Let the first real shopping value be x, then according to the meaning of the question:
1. No more than 300.
X*0.9=252
The solution is X=280.
Then the payment.
(X+80)*0.8=288
More than 2.300
X*0.8=252
X=3 15
Then the payment.
(X+80)*0.8=3 16
As can be seen from the above examples, classified discussion can often make some complex problems simple, and the ideas and steps of solving problems are very clear. On the other hand, in class discussion, students' interest in learning mathematics can be stimulated.
Practice has proved that the mathematical problems of classified discussion ideas are obviously logical and comprehensive, which plays a very key role in cultivating junior middle school students' comprehensive and thorough ability to analyze and solve problems. In junior high school mathematics teaching, we should always infiltrate the idea of classification and guide students to solve problems by means of classified discussion.