1, will not establish the relationship between knowledge and topics, and will not analyze and prove with theorems when encountering proof problems;
2. I can't write in geometric language, and the steps are incoherent.
3, the auxiliary line will not be added, and the law will not be summarized.
Students are not interested in boring mathematics knowledge.
Second, the measures taken are: 1. Guide students to establish the connection between topics, find the similarities and differences between topics, and guide students to solve more problems.
2. Let students remember "common geometric expressions" and often organize students to read and speak in class to improve their oral expression ability. Draw pictures from basic sentences, give basic sentences, let students draw pictures, combine sentences with pictures, and train students to remember sentences.
3, combined with graphics, familiar with graphics, summed up the law of adding auxiliary lines.
4. When learning geometric reasoning and proving knowledge, the key is to cultivate students' interest in learning.
Interest is the best motivation for learning. When learning a certain knowledge, I always set up some practical activities unexpectedly, so that students can feel the joy of success personally, so that they will quickly enter the role of learning, increase the motivation of learning, and really let students learn actively.
Teachers should constantly update teaching methods and master math skills.
Mathematics teaching under the new curriculum standard can not meet the requirements only by traditional chalk and blackboard. There are many pictures and images that need multimedia display, and many processes of knowledge generation and development need computer demonstration. With the help of multimedia-assisted teaching, these phenomena can be activated, which is particularly intuitive and vivid, and students can feel the knowledge of mathematics from it without the teacher's multi-language. .
6. Pay attention to individual differences and promote everyone's development.
Mathematics education should promote the development of every student, that is, to lay a solid foundation for all students, and also to pay attention to the development of students' personalities and specialties. Due to the influence of various factors, students have differences in mathematics knowledge, skills, abilities and interests. Teachers should recognize this difference in teaching, teach students in accordance with their aptitude and guide them according to the situation. We should proceed from the reality of students and take into account students with learning difficulties and spare capacity, and adopt various ways and methods.
Third, how to let students learn geometric proof?
Under the implementation of the new curriculum standards, teachers should pay attention to process teaching, with students as the main body and teachers as the supplement. Geometric proof is the key and difficult point in teaching. It is the key for students to master the method of geometric proof. Let's talk about the proof of geometry and the proof method of geometry:
Learning geometry is inseparable from proof. The so-called proof is the process of reasoning and drawing conclusions according to known conditions and definitions, axioms and theorems. Proof is an important means to learn geometry, so we must learn it well. I'll talk about some views on the geometric proof (1) in the second volume of Grade 8 for your reference.
(1) Learn the basic knowledge of geometry.
This is the premise of learning geometric proof well. The basic knowledge of geometry such as definition, axiom and theorem is the theoretical basis of geometric proof, so we must master it carefully, and deeply understand the meaning of each concept and the topics and conclusions of theorems and axioms when studying. Only in this way can we use them correctly for relevant proof.
(2) You must practice several basic skills.
To learn geometry proof well, we must practice several basic skills.
1, learn to read and draw correctly.
The so-called figure recognition does not mean observing, analyzing and understanding geometric figures, so as to recognize not only simple figures representing various concepts, but also that part of complex figures representing a certain concept. The so-called drawing means that you can draw various figures representing concepts independently and correctly, pay attention to the corresponding relationship between "topic" and "picture", and make the drawn figures conform to the meaning of the topic. In the process of doing the problem, if the pictures given in the problem do not meet the known conditions, students are advised to draw accurate pictures according to the meaning of the problem.
2. Learn to use geometric language correctly.
Geometric language is a special language of geometry, including written language, symbolic language and graphic language. Learning geometric language well is very important for learning geometric proof. The key to learning geometric language is to link graphics with words and symbols, and master the skills of mutual transformation between words, symbols and graphics. For example, the written language "two straight lines AB and CD are perpendicular to each other" is translated into the symbolic language AB⊥CD, and its graphic language is shown in figure 1.
(3) Master the basic structure of the certificate.
The basic structure of proof is:
∵ ... (known/confirmed)
∴……(……,……
)
The "⊙" is followed by the "reason" of reasoning.
∴ "is followed by the" fruit "of reasoning, while" (...) "is written on the basis of the result of cause, that is, the reason. For example:
∫≈ 1 and ∠2 are antipodal angles (known),
∴∠∠∠∠∠∠∠∠∠∠∠∠∠ 1 =∠∠∠∠∠∠∠∠∠∠∠∠∠8736
Every reasoning should include three parts: cause, effect and cause, and the causal relationship must be reasonable.
(4) Be familiar with three types of reasoning.
1, one cause and one effect, such as the above example, is the reasoning of "one cause and one effect".
2. One cause has many fruits, as shown in the figure (omitted).
∫AD∨BC (known),
∴∠∠ 1 =∠ b (two straight lines are parallel and have the same included angle),
∠2=∠C (two straight lines are parallel and the internal dislocation angles are equal).
This is the reasoning of "one cause has many effects". When proving in detail, one or more effects should be selected as needed.
3, many causes and one fruit, such as:
∫a∨b, b∨c (known),
∴a∥c (two lines parallel to the same line are parallel).
This is the reasoning of "many causes and one fruit", which can only get "fruit" if there are many causes.
(5) Clearly prove the hierarchical relationship.
The proof of geometric propositions usually consists of several inferences, that is, it contains multiple causal relationships. But in the process of actually writing proof, its "cause" is often omitted in the second reasoning (this "cause" is often the "result" of the previous reasoning
)。
Example 1,
It is known that as shown in the figure (omitted), straight lines AB and CD intersect with straight lines EF and GH respectively, and ∠ 1=∠2. Proof: ∠3=∠4
Prove: (1)≈ 1 =∠2 (known), ∠ 1=∠5 (equal to the vertex angle),
∴∠5=∠2 (equivalent substitution),
∴AB∥CD (same angle, two straight lines parallel),
(2)∫AB∨CD (certification)
∴∠ 3 =∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠
The proof of this problem consists of two layers of causality, in which the "cause" of the second reasoning is the "result" of the first reasoning.
Beginners of geometry, in order to better master the reasoning method and ensure that every reasoning is well-founded and orderly, suggestions can be written according to the above format, even if there is no, they should be very clear about the "reasons" of these omissions.
(six), master the commonly used methods of proof.
There are many ways to prove geometry, but there are two commonly used methods:
(1) analysis method. In short, the analysis method is to explore the reasons for its establishment from the results, that is, the "result reason."
(2) Comprehensive method. In short, the comprehensive method is to deduce the result from the condition, that is, "cause leads to effect."
These two methods have their own advantages and disadvantages. Analysis method is easy to find a way to prove the problem, but the writing process is complicated. The comprehensive writing process is simple, but it is not easy to find a way to prove the problem. Therefore, when proving, the two methods are often combined, that is, first find the method to prove the problem by analytical method, and then write the proof process by comprehensive method. We call it "backward deduction smoothing proof". Please look at the following example.
Example 2: As shown in the figure (omitted), AB∨DC, ∠DAB=∠BCD are known. Proof: AD ∨ BC.
First, use the "analysis method" to find a way to prove the problem.
Analysis: If you want to prove AD∨BC, you need to prove ∠1= ∠ 2; To prove ∠ 1=∠2, because ∠DAB=∠BCD (known), it is necessary to prove ∠ 3 = ∠ 4; To prove ∠3=∠4, it is necessary to prove AB∨DC, which is a known condition. At this point, thinking exchange.
Then use the "synthesis method" to write the proof process.
Proof: ∫AB∨DC (known)
∴∠ 3 =∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠∠
Also ∠DAB=∠BCD (known),
That is ∴∠DAB-∠3=∠BCD-∠4.
∠ 1=∠2
∴AD∥BC (internal dislocation angles are equal and two straight lines are parallel).
What I mentioned above is what we must pay attention to when learning geometric proof. In addition, we should also pay attention to prevent mistakes. As long as students are diligent in thinking and good at summing up, they will certainly learn geometric proof well.