A summary of senior high school students' collective knowledge

The inequality whose concept contains an unknown number and the highest degree is 2 is called unary quadratic inequality, and its general form is AX 2+BX+C > 0 or AX 2+BX+C.

The solution of the unary quadratic inequality 1) When v ("v" means yes, the same below) = b 2-4ac > =0, the quadratic trinomial, ax 2+bx+c has two real roots, then ax 2+bx+c can always be decomposed into a (x-x1). In this way, solving a quadratic inequality can be reduced to solving two linear inequalities. The solution set of unary quadratic inequality is the union of the solution sets of these two unary linear inequalities.

For example. Let's look at a set of examples:

1) All the students in Class Three, Senior One, Li Antang No.1 Middle School.

2) All prime numbers less than 10

3) All countries participating in the 2006 World Cup

4) The set of all solutions of the equation

5) Tall people in China

6) A number very close to 10.

Teacher: Through the above example, we find an intriguing question. Some objects are certain and some are uncertain, so we regard the objects that can be determined as a whole and say that the whole is a collection of all these objects.

1. Definition: Generally, some specified objects are grouped together to form a set. Each object in a collection is called an element of the collection.

Teacher: Which of the above are collections? What is the element?

Students: 1), 2), 3), 4), 5), 6) and so on.

Teacher: It seems that everyone has a different opinion. A collection consists of elements. If you want to determine the set, you must first determine the elements. What are the characteristics of elements?

2. Characteristics of elements in the set

1) Determinism: The elements in the set must be deterministic and cannot be ambiguous.

2) Reciprocity: Any two elements in a set must be different from each other.

3) Disorder: A set has nothing to do with the order of its elements.

Teacher: at this point, let's judge which sets are right.

Health: 1), 2), 3), 4), because 5) and 6) do not satisfy the uncertainty.

Teacher: Good!

Teacher: Set is usually represented by capital letters a, b, c, d, etc. Elements are usually represented by lowercase letters a, b, c, etc.

3. The relationship between elements and sets

1) If it is an element of set A, say that A belongs to set A and write it as: A A.

2) if a is not an element of set a, say that a does not belong to set a, and record it as: a a.

Attention; It just represents the relationship between elements and sets.

Example:

1) A={2，4，6} 2 A 8 A

2) Please consider: A = {1, 2}, B = {{ 1, 2}, {2,3}}, the relationship between sets A and B?

4. Common specific collection of symbols: n, n, z, q, r.

Third, classroom exercises.

1, exercises on page 5 of the textbook.

2. Fill in the blanks with the correct symbols: () r, -2( )Q, () q, 6.5 () n, 0 () n.

3. Can each of the following objects form a set? Explain why.

1) famous mathematician

2) All teachers in Li Antang No.1 Middle School.

3) All points in rectangular coordinate system

4) Real numbers with absolute values less than 8

5) Small rivers in China

Comments:

Sanctity: The word "gathering" refers to collection, which refers to the whole of something, not to individual things.

Certainty: "Designated object" means that the set is completely sure that there are elements belonging to it, and an object is either his element or not, and the two must be one of them.

The teacher immediately explained the above example.

First of all, introduce the changes of learning characteristics of high school mathematics and junior high school mathematics to help students actively adjust their learning psychology.

1, the mathematical language has a sudden change in abstraction.

There are significant differences between senior high school mathematics language and junior high school mathematics language. Junior high school mathematics is mainly expressed in vivid and popular language. Mathematics in senior one involves abstract set of symbolic language, logical operation language, functional language, graphic language and so on. The thinking gradient of junior one students is so great that the concepts of set, mapping and function are difficult to understand, and they feel far away from life and seem to be "mysterious". In teaching, we can integrate theory with practice, reduce the difficulty of thinking, train and train students step by step to transform images and popular written language with symbolic language and graphic language, and improve their language understanding ability.

2. Transition of thinking mode to rational level.

The thinking method of high school mathematics is very different from that of junior high school. In junior high school, many teachers have established a unified thinking mode for students to solve various problems, such as how to solve fractional equations in several steps, and what to look at first and then what to look at in factorization, so they have determined common thinking routines. Therefore, junior high school students are used to this mechanical and easy-to-operate fixed way in mathematics learning. However, the thinking form of high school mathematics has changed greatly, and the abstraction of mathematical language puts forward higher requirements for thinking ability. This sudden change in ability requirements makes many freshmen feel uncomfortable, which leads to a decline in their grades, which is another reason why senior one students have difficulty in mathematics learning. Pay attention to heuristic teaching and use discussion teaching to cultivate students' ability. Of course, the development of students' ability is gradual, not overnight. As long as freshmen can get rid of the mindset of junior high school, they can quickly transition from empirical abstract thinking to theoretical abstract thinking, and finally need to form dialectical thinking.

3. The total amount of knowledge content has increased dramatically.

Compared with junior high school mathematics, the knowledge content of senior high school mathematics has increased sharply, and the amount of knowledge and information received per unit time has increased a lot compared with junior high school mathematics, and the class hours for assisting practice and digestion have decreased accordingly. This also makes many freshmen who are passive and psychologically dependent feel uncomfortable. This requires us to conduct psychological counseling, put forward learning requirements and timely check and supervise in class: First, do a good job in pre-class preparation and after-class review every day, and try to remember key knowledge; Second, we should distinguish the old and new knowledge in time every week and after each unit, understand their internal relations, and make the new knowledge assimilate into the original knowledge structure smoothly; Third, correct the mistakes in time after each unit test, otherwise, when the amount of knowledge is too large, its memory effect will not be very good, which will affect students' learning confidence. Fourth, it is necessary to summarize and classify more and establish a knowledge structure network of disciplines.

Therefore, it is necessary to teach students to sort out the knowledge structure, form a plate structure, and implement "full container", such as tabulation, so that the knowledge structure can be seen at a glance; Experience several learning methods: from special to general analogy, from one case to one lesson, from one lesson to many lessons, from many lessons to unity; The special case method from general to special makes several kinds of problems isomorphic to the same knowledge method for divergent thinking.

Second, learn to distinguish between normal learning psychological state and bad learning state.

1. Cultivate a positive learning attitude and realize the difference between "I want to learn" and "I want to learn".

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Junior high school students are obviously dependent on learning and want me to learn. There are many reasons, such as: 1) In order to improve scores, teachers list various types of questions in junior high school mathematics teaching, and students' mathematics learning depends on teachers to provide them with "models" to apply; 2) Parents are eager for their children to succeed, and often "participate in learning" and conduct after-school counseling and inspection. After entering the third year of senior high school, students in Grade One are faced with the change of teachers' teaching methods, and the "model" they once relied on is gone, so their parents' counseling ability can't keep up. After entering high school, many students do not make study plans, do not preview before class, are busy taking notes in class, and can't hear the "doorway". His study lags behind because of his dependence, and he has a strong dependence. He follows the teacher's inertia and has no initiative to learn. I pay attention to cultivating students' active learning attitude in teaching, requiring students to preview before class, review after class, summarize units and correct mistakes in time. Take students with excellent study habits as examples and let them learn from them.

2. Correctly distinguish between normal mental state and abnormal mental state. After the senior high school entrance examination, some thoughts began to relax, especially in the first and second days of junior high school. They only worked hard for a month or two before the middle school entrance examination, and even mistakenly thought that high school students and high school students didn't need to work so hard at all. As long as you work hard for a month or two before the middle school entrance examination, you will still be admitted to an ideal university. The difficulty of high school mathematics is far less difficult than that of junior high school mathematics, and it takes three years of hard work. In addition, the content of college entrance examination comes from textbooks and is higher than textbooks, which has strong selectivity. It is very difficult to complete a lot of knowledge until the third year of high school. In my teaching, I advocate students to make a study plan for senior three: to lay a solid foundation for senior one, to focus on senior two, and to make achievements for senior three. It is conducive to the formation of a good psychological development environment for schools, with different emphasis in three years to cultivate students' self-psychological adjustment ability.

3. Cultivate good study methods and habits, and understand the difference between "rote learning" and "living learning". Teachers usually explain the ins and outs of knowledge in class, analyze the connotation of concepts, analyze key and difficult points, and highlight thinking methods. But some students can't grasp the key points and difficulties in class and can't understand the way of thinking. They just do their homework, confuse the questions, have a little knowledge of concepts, laws, formulas and theorems, imitate mechanically and memorize them by rote. As a result, they get twice the result with half the effort and have little effect. At the beginning of the school year, I asked students who scored well in the college entrance examination to introduce their learning experience in high school to freshmen, so that freshmen can be prepared to change their learning methods and habits. At the same time, study and discuss various difficult problems in class, so that freshmen can experience and strengthen good learning methods.

4. Pay attention to the basic cultivation of sound personality and change the learning misunderstanding of "knowing at a glance", "knowing at a glance" and "being wrong at a glance". Compared with junior high school mathematics, senior high school mathematics is a leap in depth, breadth and ability requirements. This requires you to master basic knowledge and skills to prepare for further study. Such as quadratic function, parametric variable, application of trigonometric formula, space and plane, practical application, etc. , are not mentioned in junior high school textbooks, which need to be made up by senior high school, otherwise it will inevitably fail to keep up with the requirements of senior high school. Some students who "feel good about themselves" often despise basic training and don't calculate and write carefully, but they are very interested in difficult problems, emphasizing "quantity" over "quality" and falling into the sea of questions. They either make mistakes in calculation or give up halfway in formal homework or exams. In teaching, we should attach importance to basic teaching, help students understand the difference in depth and breadth between high school mathematics and junior high school mathematics knowledge, and use the teaching mode of "asking", "thinking", "doing" and "evaluating" to encourage thinking, so that students can form a sound personality in their study.

Third, optimize learning strategies, strengthen achievement motivation and learn scientifically.

High school students should not only learn, but also "learn", pay attention to scientific learning methods, improve learning efficiency, and change passive learning into active learning in order to improve their academic performance.

1, cultivate good study habits. Good study habits include making plans, self-study before class, paying attention to class, reviewing in time, working independently, solving problems, systematically summarizing and studying after class.

(1) Make a plan and define the learning purpose. A reasonable study plan is the internal motivation to promote our active study and overcome difficulties. The plan should be supervised by the teacher first, and then completed by yourself. There are both long-term plans and short-term arrangements. In the process of implementation, we must be strict with ourselves and temper our learning will.

(2) Pre-class preparation is the basis for achieving better learning results. Preview before class can not only cultivate self-study ability, but also improve interest in learning new lessons and master the initiative in learning. Preview can't go through the motions, we should pay attention to quality, try to understand the teaching materials before class, pay attention to the teacher's ideas in class, grasp the key points, break through the difficulties and solve the problems in class as much as possible.

(3) Classroom is the key link to understand and master basic knowledge, skills and methods. "Learning is not enough" allows you to concentrate on the key points and difficulties in class and record what the teacher added, instead of copying everything and remembering everything.

(4) Reviewing in time is an important part of improving learning efficiency. By reading textbooks repeatedly and consulting relevant materials in multiple ways, we can strengthen our understanding and memory of the basic conceptual knowledge system, link the new knowledge we have learned with the old knowledge, analyze and compare the results, and arrange the review results in the notebook at the same time, so that the new knowledge we have learned can be changed from "understanding" to "knowing".

(5) Independent homework is a process of thinking independently, analyzing and solving problems flexibly, further deepening the understanding of new knowledge and mastering new skills. This process is also a test of our will and perseverance. Through application, we can change knowledge from "knowing" to "being familiar with".

(6) Problem-solving refers to the process of understanding the knowledge incorrectly exposed in the process of independently completing homework, or missing the answer because of thinking obstruction, so as to clarify the thinking and supplement the answer. Perseverance is needed to solve problems. Did the wrong homework again. Think over what you don't understand wrong. If you can't really solve the problem, you should consult your teachers and classmates, review and strengthen the mistakes that are easy to make, do appropriate repetitive exercises, digest what you ask your teachers and classmates into your own knowledge, and persist in changing knowledge from "familiar" to "alive" for a long time.

(7) Systematic summarization is an important link to master knowledge and develop cognitive ability comprehensively, systematically and profoundly through positive thinking. To sum up, we should, on the basis of systematic review, take teaching materials as the basis, refer to notes and materials, and reveal the internal relationship between knowledge through analysis, synthesis, analogy and generalization, so as to achieve the purpose of mastering what we have learned. Frequent multi-level summary can change knowledge from "living" to "understanding".

(8) Extracurricular learning includes reading extracurricular books and newspapers, participating in academic competitions and lectures, and visiting senior students or teachers to exchange learning experiences. Extracurricular learning is a supplement and continuation of in-class learning. It can not only enrich students' cultural and scientific knowledge, deepen and consolidate what they have learned in class, but also satisfy and develop our hobbies, cultivate the ability of independent study and work, and stimulate curiosity and enthusiasm for learning.

2, step by step, positive attribution, to prevent impatience.

Because of their young age and limited experience, many senior one students are prone to impatience. Some students are insatiable, eager for quick success, and want to "sprint" in a few days. Learning is a long-term accumulation process of consolidating old knowledge and discovering new knowledge, which can never be achieved overnight. A very important reason why many excellent students can get good grades is that their basic skills are solid, and their reading, writing and computing abilities have reached the level of automation or semi-automation. Let senior one students learn positive attribution and establish self-confidence, such as: achieving some achievements, achieving success in time and strengthening learning ability; When encountering setbacks, we should adjust our learning methods and strategies in time, make greater efforts to change the setbacks and strive for the success of the college entrance examination step by step.

3. Pay attention to the characteristics of the subject and find the best learning method.

Mathematics is responsible for cultivating computing ability, logical thinking ability, spatial imagination ability, and the ability to analyze and solve problems by using what you have learned. Among them, the cultivation of computing ability must pay attention to "living", not just reading books without doing problems, not burying oneself in doing problems without summing up and accumulating, and optimizing computing strategies in teaching; Logical thinking ability is highly abstract, logical and widely applicable, and requires high ability. Using classification and networking strategies, several concepts are distinguished: the relationship between three-stage reasoning, four propositions and necessary and sufficient conditions; The expansion of plane knowledge by spatial imagination should not only be able to enter, but also be able to jump out, and experience the interaction between graphics, symbols and words in combination with solid geometry; The ability to analyze and solve problems by using the learned knowledge is to pay attention to the transformation training of applied problems, the classification of mathematical models and the understanding of mathematical languages. This is the truth in the learning process of "from thin to thick" and "from thick to thin" advocated by Mr. Hua The methods vary from person to person, but the four links of learning (preview, class, homework and review) and one step (induction and summary) are indispensable.

In a word, mathematics teaching in senior one should be based on textbooks, face all students, focus on key issues, practice common questions repeatedly, rationally use unit review and hierarchical teaching, teach students in accordance with their aptitude, and improve efficiency and self-confidence. Starting from the reality of cultivating innovative talents, we should guide top students at different levels at ordinary times, pay attention to the understanding of mathematical ideas in teaching, and improve the innovative consciousness and ability of top students. At the same time, taking into account the guidance of learning methods, the key point is to digest and solve the problems that have been done wrong, and strive not to make mistakes again. Mathematics learning in senior one is the tempering of students' life and the basic embodiment of teachers' teaching achievements. As long as we proceed from reality and make appropriate goals, long plans and short arrangements, students will enhance their confidence in overcoming difficulties, and math learning will naturally get good results-a hard return and a "win-win" for teachers and students.

2x^2-7x+6<； 0

Use cross multiplication.

2 -3

1 -2

Get (2x-3) (x-2) < 0

Then, it is discussed in two situations:

I. 2x-3

Get x; 2。 wrong

Second, 2x-3 & gt；; 0，x-2 & lt； 0

Get x> 1.5 and x

The solution set of the final inequality is: 1.5

In addition, the collocation method can also be used to solve quadratic inequalities:

2x^2-7x+6

=2(x^2-3.5x)+6

=2(x^2-3.5x+3.0625-3.0625)+6

=2(x^2-3.5x+3.0625)-6. 125+6

=2(x- 1.75)^2-0. 125<； 0

2(x- 1.75)^2<； 0. 125

(x- 1.75)^2<； 0.0625

Both sides are squares, understand?

x- 1.75 & lt； 0.25 and x- 1.75 >: -0.25

X<2 and x> 1.5

The solution set of inequality is 1.5.

As we know, there is a one-to-one correspondence between real numbers and points on the number axis. Among two different points on the number axis, the real number represented by the right point is greater than the real number represented by the left point. For example, in Figure 6- 1, point A represents real number A, point B represents real number B, and point A is to the right of point B, so A > B. 。

Let's take a look at Figure 6- 1. A > B means that the difference between a and b is a number greater than 0, that is, a positive number. Generally speaking:

If a > b, then a-b is a positive number; The counter-proposition is also correct.

Similarly, if a < b, a-b is negative; If a=b, then A-B is equal to 0. Their inverse proposition is correct.

That is to say:

Therefore, to compare the sizes of two real numbers, we only need to check their differences.

Example 1 Compare the sizes of (A+3) (A-5) and (A+2) (A-4).

Solution: (A+3) (A-5)-(A+2) (A-4)

=(a2-2a- 15)-(a2-2a-8)

=-7x4+x2+ 1。

Think about it: In Example 2, if there is no condition of x≠0, what is the size relationship between the two expressions?

practise

1. Compare the sizes of (x+5) (x+7) and (x+6) 2.

By comparing the sizes of real numbers, we can deduce the following properties of inequalities.

Theorem 1 If A > B, then B < A；; If b < a, then a > B.

Prove: ∫a > b,

∴a-b>0.

The reciprocal of a positive number is negative, so

-(a-b) 2A is an isotropic inequality; If the left side of one inequality is greater than (or less than) the right side and the left side of another inequality is less than (or greater than) the right side, these two inequalities are anisotropic inequalities, for example, A2+3 > 2A, and A2 < A+5 is anisotropic inequality.

Theorem 2 if a > b and b > c, then a > c.

Proof: ∫a > b, b > c,

∴a-b>0,b-c>0.

According to the fact that the sum of two positive numbers is still positive, you must

(a-b)+(b-c)>0，

That is, a-c > 0,

∴a>c.

According to the theorem 1, Theorem 2 can also be expressed as:

If c < b and b < a, then c < a.

Theorem 3 if a > b, then a+c > b+C.

Proof: ∫(A+C)-(B+C)

=a-b>0，

∴a+c>b+c.

Theorem 3 shows that adding the same real number to both sides of the inequality leads to the inequality with the same direction as the original inequality.

Think about it: if a < b, is there a+c < b+c?

Using Theorem 3, we can draw the conclusion that:

If a+b > c, then a+b > c-b.

That is to say, any term in inequality can be moved from one side to the other after changing its sign.

Inference If A > B, C > D, then A+C > B+D. 。

Prove: ∫a > b,

∴a+c>b+c. ①

∫c > d，

∴b+c>b+d. ②

Look at A+c > b+d from ① and ②.

Obviously, this inference can be extended to any finite inequality in the same direction. That is to say, two or more inequalities in the same direction are added separately, and the obtained inequality is in the same direction as the original inequality.

Theorem 4 if A > B, C > 0, then AC > BC If a > b, c < 0, then AC < BC.

Proof: AC-BC = (a-b) C.

∫a > b，

∴a-b>0.

Multiply the same symbol according to the positive sign and multiply the negative sign by different symbols.

When c > 0, (a-b) c > 0, that is

ac > bc

When c < 0, (a-b) c < 0, that is

Ac B > 0, C > D > 0, then

ac>bd。

Students can imitate the inference of Theorem 3 to prove the inference of Theorem 4 1.

Obviously, this inference can be extended to any finite inequality with positive numbers on both sides. That is, two or more inequalities with positive numbers on both sides are multiplied separately, and the obtained inequality is in the same direction as the original inequality. From this, we can also get:

Inference 2 if a > b > 0, then an > bn (n ∈ n and n > 1).

We prove it by reducing to absurdity.

These are in contradiction with the known conditions A > B > 0.

Some inequalities can be proved by using the properties and inferences of the above inequalities.

Example 3 a > b, c < d, proving a-c > b-d.

Proof: A-B > 0 is known from A > B, and D-C > 0 is known from C < D. 。

(a-c)-(b-d)

=(a-b)+(d-c)>0，

∴a-c>b-d.

Proof: ∫a > b > 0,

that is

And c < 0,

References:

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Interviewee: ☆ Eros ♂-Trainee Magician II 1-27 13:42