For example, the absolute values of 10 and-10 are both 10, that is
Obviously.
The absolute value of 1
The absolute value of a number is 7. Find this number.
2. Absolute value of rational number:
(1) The absolute value of a positive number is itself.
(2) The absolute value of a negative number is its reciprocal.
(3) The absolute value of 0 is 0.
that is
In other words, the absolute value of any rational number is nonnegative.
When finding the absolute value of a number expressed by letters, we must first judge whether the number is positive, zero or negative, and then find the absolute value according to the definition.
3, the geometric meaning of absolute value:
The absolute value of the number A is the distance between the point representing the number A on the number axis and the origin.
With the help of the number axis, students can see two negative numbers, the larger absolute value is smaller, resulting in
4. Comparison of rational numbers
(1) Positive number is greater than 0, 0 is greater than negative number, and positive number is greater than negative number;
(2) Two negative numbers, the larger one has the smaller absolute value.
Example 3 Compare the following logarithms:
(1) -(- 1) and -(+2)
(2) and
(3)-(0.3) and
Example 4 Judge whether the following conclusions are correct and explain the reasons:
(1) If, then A = B.
(2) If yes, a>b
Example 5 uses ">" to indicate the following figures. Connection:
Example 6 The positions of rational numbers A, B and C on the number axis are simplified as shown in the figure.
Exercise: Textbook 17, 18.
Summary: the meaning of absolute value
Thinking:
1, if, find a, b.
2. Fill in the blanks:
(1) If, then a 0.
(2) If a 0.
(3) If a 0.
(4) If yes, it is 0.
Homework: textbook 19, pages 4 and 5.
Question 2: What is the teaching focus of absolute value of Zhejiang Education Edition? Geometric meaning, algebraic meaning, and the relationship between equality and inequality are based on sign transformation.
Question 3: When the number is extended from rational number to real number, does the meaning of rational number about inverse number and absolute value also apply to real number? The meaning of rational numbers about reciprocal and absolute value also applies to real numbers.
Question 4: What is rational reasoning? What are the important forms? What is wise reasoning? What are the important forms?
For a long time, junior high school mathematics teaching has emphasized the rigor of reasoning, overemphasized the importance of logical reasoning and neglected vivid and reasonable reasoning, which made people mistakenly think that mathematics is a purely deductive science. In fact, every important discovery in the history of mathematics development, besides deductive reasoning, perceptual reasoning also plays an important role. Therefore, in classroom teaching, teachers should cultivate students' reasonable reasoning ability according to the content of teaching materials. It can not only improve the quality of classroom teaching, but also help to cultivate students' innovative consciousness and improve their innovative ability.
Keywords: junior high school mathematics teaching reasoning ability training
I used to be confused: I thought that the new textbook ignored the accurate definition of concepts and the reasoning and demonstration of theorems, and did not analyze and discuss them. I only asked students to remember concepts and theorems and emphasized that they should be able to use them. This is called knowing what it is but not knowing why, which is obviously not conducive to the long-term development of students. For example, there is no proof process in the textbook of "triangle interior angle sum theorem", but students are required to explain it with paper-cut splicing experiments. For another example, the symmetry figures, lines, the center line on the bottom and the coincidence of high lines (three lines in one) in the textbook have not been proved, and students are required to confirm their existence with origami. This is the taboo of logical reasoning, which is not conducive to the cultivation of students' logical reasoning ability and loses the rigor of mathematics. The misunderstanding is eliminated by carefully reading the mathematics curriculum standard, which stipulates that "students can develop their reasonable reasoning ability and preliminary deductive reasoning ability through mathematics learning, experience observation, experiment, guess and proof in compulsory education."
Mathematician Paulia said: "Mathematics can be regarded as the science of proof, but this is only one aspect. The mathematical theory has been completed and expressed in the final form. It seems that only proof can be purely provable. Strict mathematical reasoning is based on deductive reasoning, and the process of drawing mathematical conclusions and proving them is discovered by perceptual reasoning. " Deducing another unknown judgment from one or several known judgments is called reasoning. Reasonable reasoning is based on the existing knowledge and experience, in a certain situation and process, to deduce the conclusion of possibility. Reasonable reasoning is a kind of reasonable reasoning, which mainly includes observation, comparison, incomplete induction, analogy, conjecture, estimation, association, consciousness, epiphany and inspiration. The result of reasonable reasoning is accidental, but it is not completely fictional. It is an exploratory judgment based on certain knowledge and methods. Therefore, how to teach students reasonable reasoning in normal classroom teaching is a topic worthy of discussion.
Nowadays, the field of education is advancing in an all-round way, aiming at cultivating students' innovative ability. However, for a long time, mathematics teaching in middle schools has attached great importance to the rigor of reasoning, overemphasized the importance of logical reasoning and neglected vivid and reasonable reasoning, making people mistakenly think that mathematics is a pure deductive science. In fact, every important discovery in the history of mathematics development, besides deductive reasoning, rational reasoning also plays an important role, and rational reasoning and deductive reasoning complement each other. Before proving a theorem, you should guess and discover the content of a proposition. Before making a complete proof, you should constantly test, improve and modify the conjecture put forward, and you should guess the idea of proof. You should synthesize the observed results before making an analogy. You must try again and again. In this series of processes, what you need to make full use of is not argumentation reasoning, but reasonable reasoning. The essence of rational reasoning is "discovery-conjecture". Newton has long said: "Without bold guesses, there will be no great discoveries." As early as 1953, Paulia, a famous mathematics educator, shouted: "Let's teach guessing!" "Guess before you prove it-this is the method of most discoveries. Rational reasoning in solving problems is characterized by not thinking according to logical procedures, but it is actually a leap-forward form of expression in which students organically combine their own experience with logical reasoning methods. Therefore, in mathematics learning, we should not only emphasize the rigor of thinking and the correctness of results, but also attach importance to the intuitive exploration and discovery of thinking, that is, we should attach importance to the cultivation of mathematical reasoning ability.
First, cultivate the rational reasoning ability in "Number and Algebra"
In the teaching of "number and algebra", there must be certain "rules" in calculation-formulas, rules, reasoning rules and so on. Therefore, there is reasoning in calculation, and the quantitative relationship in the real world often has its own laws. Algebraic operation not only needs to know how to operate, but also needs to understand arithmetic, and can tell the concepts involved in the basis of each step in the operation. & gt
Question 5: What are the factors that affect the learning of mathematical concepts? 1. Students' existing experience.
Students' ability to acquire concepts develops with age, intelligence and experience. The research shows that in terms of the influence of intelligence and experience on concept learning, experience plays a more important role, and rich experience background is the premise of understanding the essence of concepts, otherwise it will easily lead to memorizing the literal definition of concepts and failing to understand the connotation of concepts. The "experience" here is not only obtained from school study, but also the experience gained by students from daily life plays a very important role. In fact, many scientific concepts mastered by students are formed and developed from everyday concepts. Therefore, teachers should pay attention to guiding students to accumulate experience beneficial to concept learning from daily life, and at the same time pay attention to using students' daily experience to serve concept teaching.
As far as mathematical concept learning is concerned, the influence of "experience" on new concept learning is more manifested in the expansion of concept system. Some students can find concepts related to new concepts from past experience and establish new concepts on the basis of comparing their similarities and differences, while some students will be disturbed by this experience and have a wrong understanding of concepts. For example, students have been exposed to square operation since primary school. In their experience, the square operation is only associated with "positive"; In addition, they are familiar with linear equations, that is, one equation corresponds to one solution. When learning the concepts of "square root" and "arithmetic square root", it is very difficult to learn the concept of "square root" because the square root of positive numbers involves positive numbers and negative numbers, but in fact these two numbers are the two roots of the equation x2=a, which is very different from their experience. At the same time, we have to learn the concept of "arithmetic square root", so that sometimes we have to take two values and sometimes we can only take a positive number, which leads to confusion in understanding.
In order to prevent the negative influence of experience on new concept learning, we should first work hard on the teaching of basic concepts, put basic concepts in the central position, make them become links of related knowledge, and highlight the internal relations between concepts. Some concepts in mathematics have overall effects, such as "* *", "function", "equation" and "distance". These concepts should give students the opportunity to contact again and again, and on this basis, other concepts can be deduced. In Ausubel's words, it is often the most effective to learn the most general concepts first and then gradually differentiate into more specific concepts. For example, the arrangement of algebra textbooks in senior high school is from correspondence to mapping to power function, exponential function and logarithmic function according to the principle of gradual differentiation. Of course, not all the content can be arranged in this way. For example, "number system" cannot be arranged in the order of "complex number, real number, rational number, irrational number, integer number, fraction number and natural number" because this order is contrary to people's daily experience in understanding the concept of "number". For such content, we should pay attention to giving an appropriate number of examples in teaching, so that students have the opportunity to sum up the same characteristics from each specific example, and then abstract the essential characteristics (in fact, we should pay attention to using the teaching strategy of "concept formation" to learn from the shallow to the deep, from the easy to the difficult, and from the known to the unknown. At the same time, we should pay attention to guiding students to explore the relationship between old and new concepts in time and find out their similarities and differences, so that students can have sufficient practical opportunities to establish this sense of connection and difference. Here we emphasize the importance of letting students practice concepts repeatedly. In our opinion, this practice cannot be equated with mechanical repetition, because the mathematical concept is far from the students' reality. If they don't have the opportunity to practice concepts repeatedly, it will be difficult to build up the feeling that they need to understand. For example, in the concepts of "rational number" and "irrational number", students can distinguish, understand and master some numbers that are not cyclic decimals, but finite decimals or cyclic decimals in the process of calculating the square roots of numbers 2, 3 and 5. Of course, this kind of repeated training should be adapted to students' cognitive level, and high-standard understanding should be put forward to students in time. With the growth of students' age and the deepening of mathematics learning, practical training can be gradually carried out under the guidance of abstract concepts, so that the understanding and application of concepts can promote each other, thus speeding up understanding and improving training efficiency.
Second, perceptual materials or perceptual experience.
Concept formation mainly depends on the abstract generalization of perceptual materials, while concept assimilation mainly depends on the abstract generalization of perceptual experience. Therefore, perceptual materials ... >>
Question 6: How do math teachers play a leading role? "New Mathematics Curriculum Standard" points out: "Students are the masters of learning, and teachers are the organizers and guides of learning activities." In other words, students should have the awareness and habit of autonomous learning, and teachers are their helpers to cultivate their awareness and habit of autonomous learning.
Classroom teaching activities are the most basic ways and means of school education. According to the requirements of the above new curriculum standards, in classroom teaching, teachers must update their concepts, put their positions in a correct position, and guide students to study happily and in good faith. As a math teacher, how to correct the relationship between teaching and learning in math teaching practice, that is, how to correct the position relationship between teacher-led and student-centered, how to give full play to the leading role of teachers and guide students to learn math knowledge independently, so as to achieve the teaching purpose of mathematics, is a subject worthy of in-depth discussion. The following is my exploration in mathematics teaching practice.
First, stimulate students' interest in learning and make them like learning.
Good interest is a force to promote people's knowledge. People always pay special attention to what they are interested in, try to understand it and study it, so as to gain rich knowledge and skills about it quickly; On the other hand, if you feel meaningless and boring, even if you do it reluctantly, it will be difficult to get good results, and the effort itself is often not lasting. This is especially true in mathematics learning. Confucius said, "Those who know are not as good as those who know; Good people are not as good as happy people. " He emphasized interest. Interest is the cognitive tendency of students to actively explore something. In order to cultivate and stimulate students' interest in learning mathematics, I stimulate students' interest in learning through various channels in teaching. To this end, I pay attention to let students collect teaching-related materials through various channels in the teaching preview. For example, in geometry teaching, using the background, pictures and materials related to the teaching content, using modern information technology to make various geometric models and teaching demonstration pictures, at the same time, let students make physical models themselves, attract students' attention through the intuitive image of objects, stimulate students' curiosity and thirst for knowledge, cultivate students' subjective participation consciousness, and stimulate students' thinking interest, thus having the motivation for further study. For example, when studying the contents of the chapter "Quadrilateral", I used the graphics introduced in the textbook, related materials in "Reading" and examples in life (irregular tiles on the floor) to make pictures and demonstrations. Through common graphic examples in life, let students realize that mathematics knowledge comes not only from books, but also from life. Studying them has practical significance, which can make students have a correct learning attitude and curiosity about mathematics and stimulate their interest in learning.
Second, implement heuristic teaching to guide students how to learn.
Using heuristic teaching method can give full play to students' main role. Subjective participation means that students actively and creatively participate in learning activities, so that students can become the masters of learning and become a new generation with subjective consciousness, thus realizing and promoting their own development. The purpose of subject participation is to cultivate students' subjectivity, and the behavioral characteristics of students' subjectivity are initiative, autonomy and creativity. Students' autonomous learning is a complex, which includes not only the feeling, perception, memory, thinking and intelligence of cognitive psychological system, but also the motivation, attitude, interest, emotion, will and personality of emotional system. Therefore, teachers should pay attention to students' learning motivation, cultivate students' interest in learning, stimulate students' enthusiasm for learning, and guide students to learn to think. In classroom teaching, under the principle of combining typical demonstration with general requirements and combining teaching with guidance, teachers can take various forms of inspiration to help students truly understand and master basic mathematical knowledge and skills, mathematical ideas and methods in the process of independent exploration and cooperative communication, and gain rich experience in mathematical activities.
1. Positive motivation. That is to say, according to the key points and difficulties of teaching, enlightening questions are raised, often in the key points and joints of teaching materials. For example, in the chapter "Factorization", I grasped the teaching of "Square Difference Formula" because it is the first formula introduced in "Factorization" and a widely used formula, which is the basis for learning other contents well. When students master it, they can build up their confidence in learning. In teaching, I first guide students to observe the characteristics of formulas, and then inspire them to think: how can we make formulas conform to the characteristics of formulas? After some thinking and discussion, students come to the conclusion that factorization can be carried out as long as the formula is transformed into the form of () 2-( )2, that is, the key to factorization is whether an algebraic formula can be written in the form of square difference quickly. Add ... >>
Question 7: How to cultivate students' reasoning ability in senior high school mathematics class? Deducing another unknown judgment from one or several known judgments is called reasoning. Reasonable reasoning is based on the existing knowledge and experience, in a certain situation and process, to deduce the conclusion of possibility. Generally speaking, rational reasoning is a kind of rational reasoning, which mainly includes observation, comparison, incomplete induction, analogy, conjecture, estimation, association, consciousness, epiphany and inspiration. Mathematician Paulia said: "Mathematics can be regarded as the science of proof, but this is only one aspect. The mathematical theory has been completed and expressed in the final form. It seems that only proof can be purely provable. Strict mathematical reasoning is based on deductive reasoning, and the process of drawing mathematical conclusions and proving them is discovered by perceptual reasoning. " Mathematicians pointed out the importance of rational reasoning. As a middle school math teacher, how to teach students rational reasoning and cultivate their rational reasoning ability is a topic worthy of discussion.
The result of reasonable reasoning is accidental, but it is not completely fictional. It is an exploratory judgment based on certain knowledge and methods. Nowadays, the field of education is advancing in an all-round way, aiming at cultivating students' innovative ability. However, for a long time, mathematics teaching in middle schools has emphasized the rigor of reasoning, overemphasized the importance of logical reasoning and neglected vivid and reasonable reasoning, making people mistakenly think that mathematics is a purely deductive science. In fact, every important discovery in the history of mathematics development, besides deductive reasoning, rational reasoning also plays an important role, and rational reasoning and deductive reasoning complement each other. Before proving a theorem, you should guess and discover the content of a proposition. Before making a complete proof, you should constantly test, improve and modify the conjecture put forward, and you should guess the idea of proof. You should synthesize the observed results before making an analogy. You must try again and again. In this series of processes, what you need to make full use of is not argumentation reasoning, but reasonable reasoning. The essence of rational reasoning is "discovery-conjecture". Newton has long said: "Without bold guesses, there will be no great discoveries."
First, cultivate the rational reasoning ability in "Number and Algebra"
In the teaching of number and algebra, calculation should be based on certain "rules" ―― formulas, rules, inference rules, etc. Therefore, there is reasoning in calculation, and the quantitative relationship in the real world often has its own laws. For algebraic operation, it requires not only knowing how to operate, but also knowing how to reason, and being able to tell the concepts, operation rules and rules involved in each step of the operation. Algebra can't just pay attention to skilled and correct operations. For example, the addition rule of rational numbers is obtained by incomplete inductive reasoning with the help of the east and west movement of points on the number axis. In teaching, we should not only pay attention to the memory and application of rules, but also ignore the thinking of generating rules. For another example, all the algorithms of addition and multiplication are put forward in the form of incomplete inductive reasoning. Paying attention to this reasoning process (though insufficient) can not only explain the rationality of the algorithm, but also strengthen the perceptual knowledge and understanding of the algorithm. For another example, junior high school textbooks introduce the knowledge of mathematical number axis through image analogy and thermometer reasoning. Another example is: find the absolute value |-5|=? |+5|=? |-2|=? |+2|=? |-3/2|=? |+3/2|=? From the above operation, what do you find is the relationship between the absolute values of opposites? And briefly describe it. Through this example, teaching can cultivate students' rational reasoning ability, combine the number axis, let students get in touch with the problem-solving method of combining number and shape, and let students understand the geometric meaning of absolute value; Another example: when learning algebraic multiplication, the textbook uses the area of the graph to get the relevant laws of algebraic multiplication from the whole and local calculation methods. In this intuitive mode of combining numbers and shapes, students can easily understand and express the contents of the law.
In teaching, every knowledge point in the textbook should be prepared for the rationality or inevitability of knowledge before it is put forward, and the reasoning and reasoning process should be fully demonstrated to gradually cultivate students' reasonable reasoning ability.
Second, cultivate the rational reasoning ability in "space and graphics"
In the teaching of "space and graphics", we should not only attach importance to deductive reasoning. We should also attach importance to rational reasoning. The teaching of the new curriculum standard "Space and Graphics" in junior high school mathematics points out: "Reduce the inherent knowledge requirements of space and graphics, strive to follow the students' psychological development and learning rules, pay attention to intuitive perception and operation confirmation, and learn from the facts that students are familiar with ... >>"
Question 8: The relationship between senior high school mathematics elements and * * *
De? Morgan formula
3。 inclusion relation
4。 Capacity incompatibility principle
5。 Number of subsets, appropriate subset-1; Non-empty subset-1-2; Non-empty proper subset, three forms of solving quadratic function
(1) general formula;
(2) Vertex type;
(3) Point type
The solution of inequality is often the following conversion forms.
.
8。 The last and only real root is an inequality equation. The former is a necessary condition, but not a sufficient condition. In particular, the equation has only one root account equal to or and or and.
The quadratic function of 9 is in the closed interval
The maximum value of quadratic function is only at the points at both ends of the time interval between this part and the maximum value in the closed interval, as shown below:
(1) When a >: 0 o'clock;
(2) One yuan
According to the distribution of real roots of quadratic equation, there is at least one real root in the range of the equation.
collect
Equation (1) is a necessary and sufficient condition for the root to be in the range or.
(2) Necessary and sufficient conditions for the root or range of the equation;
(3) Necessary and sufficient conditions within the scope of the root of equation or.
1 1。 Under the condition that the predetermined time interval parameter quadratic inequality is unchanged, according to
(1) subinterval) Necessary and sufficient conditions of quadratic inequality (parameter) (the shape is like a given time interval, but the difference is always the same.
The necessary and sufficient conditions for the interval subinterval quadratic inequality (parameter) given by the parameter in (2) always hold.
(3) Necessary and sufficient conditions are always true or
12 truth table
p
q
Nonp
P or q/>; P and q
real
real
leave
real
real
wrong
wrong
real
wrong
Fake or real?
real
real
wrong
wrong
wrong
real
wrong
wrong
13。 The negative form of * * has a conclusion:
Original conclusion.
The original opposite conclusion.
Contrary to the facts
At least one
be
One by one.
At least two
compare
Bigger than ...
At least one
()BR/& gt； Can't rise
()
At least,
Yes,
Don't take anything
Establish existence,
/& gt； and
The relationship between the four propositions of 14
Inverse proposition of reciprocal of original proposition
If p, then QQ, if p
mutual
As an interaction, there is nothing in each other.
Inverse inverse
No, no.
No>；; Inverse negation of proposition:
Question 9: How do math teachers play a leading role? 1. Stimulate students' interest in learning and make them like learning.
Good interest is a force to promote people's knowledge. People always pay special attention to what they are interested in, try to understand it and study it, so as to gain rich knowledge and skills about it quickly; On the other hand, if you feel meaningless and boring, even if you do it reluctantly, it will be difficult to get good results, and the effort itself is often not lasting. This is especially true in mathematics learning. Confucius said, "Those who know are not as good as those who know; Good people are not as good as happy people. " He emphasized interest. Interest is the cognitive tendency of students to actively explore something. In order to cultivate and stimulate students' interest in learning mathematics, I stimulate students' interest in learning through various channels in teaching. To this end, I pay attention to let students collect teaching-related materials through various channels in the teaching preview. For example, in geometry teaching, using the background, pictures and materials related to the teaching content, using modern information technology to make various geometric models and teaching demonstration pictures, at the same time, let students make physical models themselves, attract students' attention through the intuitive image of objects, stimulate students' curiosity and thirst for knowledge, cultivate students' subjective participation consciousness, and stimulate students' thinking interest, thus having the motivation for further study. For example, when studying the contents of the chapter "Quadrilateral", I used the graphics introduced in the textbook, related materials in "Reading" and examples in life (irregular tiles on the floor) to make pictures and demonstrations. Through common graphic examples in life, let students realize that mathematics knowledge comes not only from books, but also from life. Studying them has practical significance, which can make students have a correct learning attitude and curiosity about mathematics and stimulate their interest in learning.
Second, implement heuristic teaching to guide students how to learn.
Using heuristic teaching method can give full play to students' main role. Subjective participation means that students actively and creatively participate in learning activities, so that students can become the masters of learning and become a new generation with subjective consciousness, thus realizing and promoting their own development. The purpose of subject participation is to cultivate students' subjectivity, and the behavioral characteristics of students' subjectivity are initiative, autonomy and creativity. Students' autonomous learning is a complex, which includes not only the feeling, perception, memory, thinking and intelligence of cognitive psychological system, but also the motivation, attitude, interest, emotion, will and personality of emotional system. Therefore, teachers should pay attention to students' learning motivation, cultivate students' interest in learning, stimulate students' enthusiasm for learning, and guide students to learn to think. In classroom teaching, under the principle of combining typical demonstration with general requirements and combining teaching with guidance, teachers can take various forms of inspiration to help students truly understand and master basic mathematical knowledge and skills, mathematical ideas and methods in the process of independent exploration and cooperative communication, and gain rich experience in mathematical activities.
1. Positive motivation. That is to say, according to the key points and difficulties of teaching, enlightening questions are raised, often in the key points and joints of teaching materials. For example, in the chapter "Factorization", I grasped the teaching of "Square Difference Formula" because it is the first formula introduced in "Factorization" and a widely used formula, which is the basis for learning other contents well. When students master it, they can build up their confidence in learning. In teaching, I first guide students to observe the characteristics of formulas, and then inspire them to think: how can we make formulas conform to the characteristics of formulas? After some thinking and discussion, students come to the conclusion that factorization can be carried out as long as the formula is transformed into the form of () 2-( )2, that is, the key to factorization is whether an algebraic formula can be written in the form of square difference quickly. Then strengthen the training in this field: from the square of numbers, such as: 9=32, ■ =■, 0.0 1(0. 1)2, to simple monomials, such as: m2n2=mn■,16x2xy2 = (4xy) 2. Practice has proved that through such guidance and step-by-step practice, students can easily master what they have learned. With this foundation, it is not difficult to learn other formulas.
2. Situational inspiration. The so-called mathematical problem situation refers to a situation that can make students face various obstacles and difficulties in the learning process, stimulate them to actively find ways and means to solve problems, eliminate such obstacles and difficulties, and then achieve academic and psychological success. The creation of mathematical problem situations can not only stimulate students' interest in learning, but also fully mobilize students' initiative in learning ...
Question 10: How did people come out? They were formed with the joint efforts of mother and father.
Generally, it is the combination of * * and eggs that can produce embryos.