Mathematics knowledge points of senior high school entrance examination
1. axis
The concept of (1) number axis: A straight line with origin, positive direction and unit length is called number axis.
Three elements of the number axis: origin, unit length and positive direction.
(2) Points on the number axis: All rational numbers can be represented by points on the number axis, but not all points on the number axis represent rational numbers. (Generally, the right direction is the positive direction, and the points on the number axis correspond to any real number, including irrational numbers. )
(3) Compare the size with the number axis: Generally speaking, when the number axis is to the right, the number on the right is always greater than the number on the left.
Key knowledge:
The first lesson of junior high school mathematics, know positive and negative numbers! From the new junior high school ~
2. Inverse number
(1) The concept of antipodal: Only two numbers with different symbols are called antipodal.
(2) The meaning of opposites: Grasp that opposites appear in pairs and cannot exist alone. From the number axis, except 0, they are two mutually opposite numbers, both on both sides of the origin, and the distance from the origin is equal.
(3) Simplification of multiple symbols: No matter the number of "+",the odd number of "﹣" is negative and the even number of "﹣" is positive.
(4) Summary of conventional methods: The way to find the reciprocal of a number is to add "﹣" in front of this number. For example, the reciprocal of A is ﹣a, and the reciprocal of m+n is ﹣(m+n). At this time, m+n is a whole. When you put a minus sign before an integer, use parentheses.
3. Absolute value
1. Concept: The distance between a number and the origin on the number axis is called the absolute value of this number.
(1) The absolute values of two opposite numbers are equal;
② There are two numbers whose absolute values are equal to positive numbers, one number whose absolute values are equal to 0, and no number whose absolute values are equal to negative numbers.
③ The absolute values of rational numbers are all non-negative.
2. if the letter a is used to represent rational numbers, then the number a.
The absolute value should be determined by the value of the letter a itself:
(1) When a is a positive rational number, the absolute value of a is itself a;
(2) When A is a negative rational number, the absolute value of A is its inverse-A;
③ When a is zero, the absolute value of a is zero.
That is | a | = {a(a >；; 0)0(a=0)﹣a(a<； 0)
Key knowledge:
The second lesson of junior high school mathematics, the related knowledge of rational numbers! From the new junior high school ~
4. Comparison of rational numbers
Comparison of 1. rational numbers
The number axis can be used to compare the sizes of rational numbers, and their order is from left to right, that is, from big to small (the number on the right of two rational numbers represented on the number axis is always greater than the number on the left); You can also use the nature of numbers to compare the sizes of two numbers with different symbols and 0, and use absolute values to compare the sizes of two negative numbers.
2. The rational number size comparison law:
① Positive numbers are all greater than 0;
② Negative numbers are all less than 0;
③ Positive numbers are greater than all negative numbers;
(4) Two negative numbers, the greater the absolute value, the smaller it is.
Regularization of rational numbers and three comparison methods;
(1) Rule comparison: all positive numbers are greater than 0, all negative numbers are less than 0, and all positive numbers are greater than all negative numbers. Two negative numbers are bigger, but the absolute value is bigger.
(2) Number axis comparison: the number represented by the right point on the number axis is greater than that represented by the left point.
(3) Compare the differences:
If a-b > 0, then a & gtb；;
If a-b
If a-b = 0, then a-b = 0 b.
5. Rational number subtraction
Rational number subtraction rule
Subtracting a number is equal to adding the reciprocal of this number. Namely: a-b = a+(-b)
Method guide:
(1) In the subtraction operation, first find the symbol of subtraction;
② When rational numbers are converted into addition, two symbols should be changed at the same time: one is the operation symbol (minus sign changes to plus sign); The second is the symbol of the nature of meiosis (the reciprocal of meiosis);
Note: In rational number subtraction, the positions of the minuend and the minuend cannot be interchanged at will; Because subtraction has no commutative law.
The law of subtraction cannot be compared with the law of addition. 0 plus any number remains unchanged, and 0 minus any number should be calculated according to the law.
6. Rational number multiplication
(1) rational number multiplication rule: two numbers are multiplied, the same sign is positive, the different sign is negative, and the absolute value is multiplied.
(2) Any number multiplied by zero will get 0.
(3) Multiplication law of multiple rational numbers:
① Multiply several numbers that are not equal to 0, and the sign of the product is determined by the number of negative factors. When there are odd negative factors, the product is negative; When there are even negative factors, the product is positive.
(2) Multiply several numbers, with a factor of 0 and a product of 0.
(4) Method guide
(1) using the multiplication rule, first determine the symbol, and then multiply by the absolute value.
② Multiplying multiple factors, first look at the sign of the product of factor 0, and the operation is accurate and simple.
7. Mixed operation of rational numbers
1. rational number mixed operation order: first calculate the power, then calculate the multiplication and division, and finally calculate the addition and subtraction;
Operations at the same level should be calculated from left to right; If there are brackets, do the operation in brackets first.
2. In the mixed operation of rational numbers, pay attention to the application of various operation rules to simplify the operation process.
Four operation skills of rational number mixed operation;
(1) transformation method: first transform division into multiplication; Second, multiplication is transformed into multiplication; Thirdly, in the mixed operation of multiplication and division, decimals are usually converted into fractions for reduction calculation.
(2) Rounding method: In the mixed operation of addition and subtraction, two numbers with zero, two numbers with denominator, two numbers with integer, and two numbers with integer product are usually combined to solve.
(3) Split method: first split the band score into the sum of an integer and a true score, and then calculate it.
(4) Clever use of arithmetic: Clever use of addition arithmetic or multiplication arithmetic in calculation can often make the calculation easier.
8. Scientific notation-representing a larger number
1. Scientific notation: Write numbers greater than 10 in the form of a× 10n, where a is a number with only one integer and n is a positive integer. This notation is called scientific notation.
(scientific notation: a× 10n, where1≤ a.
2. Overview of conventional methods
The requirement of a in (1) scientific notation and the expression law of the exponent n of 10 are the key, because the exponent of 10 is less than the original integer digits1; According to this law, the exponent n of 10 can be obtained by counting the integer digits of the original number first.
(2) notation requires that numbers greater than 10 can be expressed by scientific notation, and negative numbers with absolute values greater than 10 can also be expressed by this method, except that they are preceded by a negative sign.
Key knowledge:
Junior high school mathematics lesson 8: scientific counting method, junior high school freshmen are coming ~
9. Algebraic evaluation
(1) algebraic value: replace the letters in the algebraic expression with numerical values, and the calculated result is called algebraic value.
(2) Evaluation of algebraic expression: the value of algebraic expression can be directly substituted for calculation. If a given algebraic expression can be simplified, it should be simplified before evaluation.
The following is a brief summary of three types of problems:
① The known conditions are not simplified, but the given algebraic expression is simplified;
② given conditions are simplified, given algebraic expressions are not simplified;
③ Both known conditions and given algebraic expressions should be simplified.
10. General type: diversity of graphics
First of all, it is necessary to find out which parts of the graph have changed and according to what law. After finding the changing rules of each part through analysis, it can be solved directly by the rules. To explore the law, we should carefully observe and think, and make good use of association to solve such problems.
Properties of 1 1. equation
Properties of 1. equation
Adding the same number (or formula) on both sides of the property 1 equation still leads to the equation;
Property 2 Multiply both sides of the equation by the same number or divide by a non-zero number, and the result is still an equation.
2. Use the properties of the equation to solve the equation.
Using the properties of the equation, the equation is transformed into the form of x = a.
Pay attention to two aspects when applying:
① How to deform;
According to which one, the correct deformation can be guaranteed by reasonable evidence step by step.
Summary of knowledge points in the second chapter of the new junior one: addition and subtraction of algebraic expressions, children's collection!
12. Solutions of one-dimensional linear equations
Definition: The value of an unknown quantity that makes the left and right sides of a linear equation equal is called the solution of a linear equation.
Substituting the solution of the equation into the original equation, the left and right sides of the equation are equal.
13. Solving linear equations with one variable
1. General steps for solving linear equations with one variable
Removing the denominator, removing brackets, moving terms, merging similar terms, and converting the coefficient into 1 are just the general steps to solve the linear equation with one variable. According to the characteristics of the equation, all the steps are to gradually transform the equation into the form of x = a.
2. When solving a linear equation with one variable, first observe the form and characteristics of the equation. If there is a denominator, generally go to the denominator first;
If there are both denominators and brackets, and the denominator can be eliminated after the items outside the brackets are multiplied by the items inside the brackets, the brackets should be removed first.
3. When solving an equation similar to "ax+bx=c", merge the left side of the equation into one term according to the method of merging similar terms, that is, (A+B) x = C.
The equation is gradually transformed into the simplest form of ax=b, which embodies the idea of reduction.
When the coefficient of ax=b is changed to 1, the calculation should be accurate. Once it is clear whether the two sides of the equation are divided by a or b, especially when a is a fraction; Second, we must accurately judge symbols. The same sign X of A and B is positive, and the different sign X of A and B is negative.
14. Application of one-dimensional linear equation
1. Types of applied problems for solving linear equations of one variable
(1) Explore the problem of regularity;
(2) Quantity;
(3) Sales problem (profit = selling price-purchase price, profit rate = profit purchase price ×100%);
(4) engineering problems (① workload = per capita efficiency × number of people × time; (2) If a job is completed in several stages, the sum of workload in each stage = total workload);
(5) Travel problem (distance = speed × time);
(6) the problem of equivalent transformation;
(7) Sum, difference, multiplication and division;
(8) Distribution problem;
(9) Competition score;
(10) Current navigation problem (downstream speed = still water speed+current speed; Water velocity = still water velocity-water velocity).
2. The basic idea of using equations to solve practical problems
First, find out the unknown quantity and all known quantities in the problem through examination, set the required unknown quantity as X directly or indirectly, and then use the formula containing X to express the related quantity, find out the equation between them, and solve it to get the answer, that is, set, column, solution and answer.
List five steps of solving application problems by linear equations of one variable.
(1) Examination: Carefully examine the questions, determine the known quantity and the unknown quantity, and find out the equivalent relationship between them.
(2) Assumptions: Assumptions about the unknown (X). According to the actual situation, it can be directly unknown (ask whatever you want) or indirectly unknown.
(3) Column: list the equations according to the equivalence relation.
(4) Solution: Solve the equation to obtain the value of the unknown quantity.
(5) Answer: Check whether the unknown value is correct and write a complete answer.
15. Characters on opposite sides of the cube
(1) The general method to solve this kind of problem is to fold the paper according to the diagram, or directly imagine it on the basis of understanding the unfolded diagram.
(2) It is the key to solve this kind of problem to distinguish the geometric expansion diagram from the real object, and to establish the concept of space by combining the transformation between the three-dimensional figure and the plane figure.
(3) There are 1 1 cases in the cubic expansion diagram. After analyzing various situations in the plane expansion diagram, carefully judge which two surfaces are relative.
16. Lines, rays and line segments
(1) Representation methods of lines, rays and line segments
① Straight line: represented by a lowercase letter, such as straight line L, or represented by two uppercase letters, such as straight line AB.
② Ray: a part of a straight line, represented by lowercase letters, such as ray L; It is represented by two capital letters, with the endpoint in front, such as ray OA. Note: When it is represented by two letters, the endpoint letter comes first.
③ Line segment: A line segment is a part of a straight line and is represented by lowercase letters, such as line segment A; It is represented by two letters representing the endpoint, such as line segment AB (or line segment BA).
(2) The positional relationship between a point and a straight line:
(1) point through a straight line, said the point in a straight line;
(2) The point does not pass through the straight line, which means that the point is outside the straight line.
The distance between two points.
(1) Distance between two points: The length of the line segment connecting two points is called the distance between two points.
(2) There is a certain distance between any two points on the plane, which refers to the length of the line segment connecting these two points. When learning this concept, pay attention to the last two words "length", that is, it is a quantity with size, which is different from a line segment, which is a figure. The length of a line segment is the distance between two points. It can be said that it is a line segment, not a distance.
18. The concept of angle
Definition of (1) angle: A graph with two common endpoints is called an angle, where the common endpoint is the vertex of the angle and the two rays are two sides of the angle.
(2) Representation of angle: The angle can be represented by one capital letter or three capital letters. The letter of the vertex should be written in the middle. Only when the vertex has only one angle can the angle be recorded with a letter of the vertex, otherwise it is not clear which angle this letter represents. The angle can also be expressed by a Greek letter (for example, ∠ α, ∠β, ∞).
(3) Straight corners and rounded corners: A corner can also be regarded as a graph formed by the rotation of light around its endpoint. When the start edge and the end edge are in a straight line, a straight angle is formed, and when the start edge and the end edge rotate and overlap, a rounded corner is formed.
(4) Measurement of angle: degrees, minutes and seconds are commonly used units of angle measurement. 1 degree =60 minutes, that is, 1 degree =60', 1 minute = 60 seconds, that is, 1'= 60 ".
19. Definition of angular bisector
Starting from the vertex of an angle, the ray that divides the angle into two equal angles is called the bisector of the angle.
(1) ∠ AOB is the sum of ∠AOC and ∠BOC, marked as ∠AOB=∠AOC+∠BOC. ∠AOC is the difference between∠ ∠AOB and∠ ∠BOC, marked as ∞.
② If the ray OC is the bisector of ∠AOB, ∠AOB=3∠BOC or ∠BOC= 13∠AOB.
20. Operation of degrees, minutes and seconds
(1) Add and subtract degrees, minutes and seconds.
When adding and subtracting degrees, minutes and seconds, add and subtract degrees and degrees, minutes and minutes, plus points and seconds, every 60 decimal places. When doing subtraction, we need to borrow 1 to 60.
(2) Multiplication and division of degrees, minutes and seconds
① Multiplication: Multiply degrees, minutes and seconds respectively, and the result is rounded to the nearest 60.
② Division: Remove degrees, minutes and seconds respectively, and transfer the remainder of each time to the next unit for further removal.
2 1. Judging geometry from three views
(1) Imagine geometric shapes from three views. Imagine the front, top and left shapes of geometry from the front view, top view and left view respectively, and then consider the overall shape comprehensively.
(2) It is difficult to imagine the shape of the geometric figure from the three views of the object, which can be analyzed by the following methods:
① According to the front view, top view and left view, imagine the shapes of the front, top and left sides of the geometric body, as well as the length, width and height of the geometric body;
② Imagine the contours of visible and invisible parts of geometric figures from solid lines and dotted lines;
(3) Memorizing three views of simple geometry will help to imagine complex geometry;
(4) Using the reciprocal process of drawing geometry with three views and drawing geometry with three views, practice repeatedly and summarize the methods constantly.
Summary of key and difficult points in mathematics for senior high school entrance examination
Construct a complete knowledge framework
1. Building a complete knowledge framework is the basis for us to solve problems. In order to learn mathematics well, we must attach importance to basic concepts and deepen our understanding of knowledge points. Then we will use knowledge points to solve problems, learn multi-dimensional reflection and thinking when encountering problems, and finally form our own ideas and methods. However, many junior high school students do not attach importance to the concept of books, have a half-knowledge of some concepts, have an incomplete understanding of knowledge points and an incomplete knowledge system, which will lead to erratic grades.
2. Correctly understand and master some basic concepts, laws, formulas and theorems of mathematics, and master their internal relations. Because mathematics is a subject with strong knowledge coherence and logic, mastering every concept, rule, formula and theorem you have learned correctly can lay a good foundation for future study. If you encounter difficulties in learning something or solving a problem, it is probably because you have not mastered some basic knowledge related to it. Therefore, you should always check for leaks and fill gaps, find problems and solve them in time, and try to find a problem and solve it in time. Only with a solid foundation can we easily solve problems and improve our grades.
Analysis on the difficulties and difficulties of junior high school mathematics entrance examination knowledge
1, and functions (linear function, inverse proportional function and quadratic function) account for about 15% of the total score.
Quadratic function, in particular, is the key and difficult point of the senior high school entrance examination. It will appear in filling in the blanks, selecting topics and solving problems, with many knowledge points and changeable questions.
Moreover, the last two questions in the test paper usually have a solution, and the general quadratic function and the images and properties of quadratic function, as well as the triangle and quadrilateral synthesis questions are difficult to apply. There is a certain difficulty.
If you don't master this link well, it will directly affect the algebra foundation and have a great influence on the results of the senior high school entrance examination.
2. Simplification of algebraic expressions, fractions and quadratic roots.
Algebraic expression operation, factorization, quadratic root, scientific counting and fractional simplification are the key points of junior high school learning, which run through the whole junior high school mathematics knowledge and are the basis of our mathematical operation, among which factorization and understanding the relationship between factorization and algebraic expression multiplication and fractional operation are difficult.
The senior high school entrance examination generally appears in the form of multiple-choice questions and fill-in-the-blank questions, but it is the basis for solving complete answers. The proficiency of computing ability is directly related to the correct rate of answering questions. If you don't master it well, the correct answer rate will not be very high, and then you can't learn the following equations, inequalities and functions well.
3. The application questions in the senior high school entrance examination account for about 30% of the total score.
Including equation (group) application, linear inequality (group) application, function application, triangle solution application and probability statistics application.
Generally, there will be two or three answers (about 30 points) and two or three multiple-choice questions and fill-in-the-blank questions (10-15 points), accounting for about 30% of the total score of the senior high school entrance examination.
At present, there will be more and more investigations on the practical application of mathematics in the senior high school entrance examination, and the relationship between mathematics and life is getting closer and closer. Application questions require students to have strong understanding and discrimination ability, read the necessary mathematical information from the questions, and seek strategies and methods to solve the problems from the mathematical point of view. Equation thought, function thought and the combination of numbers and shapes are also very important mathematical thoughts in middle schools and tools to solve many problems.
4. Triangles (congruence, similarity, bisector, vertical line, high line and right triangle) and quadrilaterals (parallelogram, rectangle, diamond and square) account for about 25% of the total score in the senior high school entrance examination.
Triangle is a piece of knowledge with the most content in junior high school geometry, and it is also a necessary basis for learning plane geometry well. It runs through the geometry knowledge from the second grade to the third grade, and geometric proof and calculation of line segment length and angle are difficult for many students.
Only by learning the proof of triangles and quadrangles and even the circle behind them can we easily understand and master them. On the contrary, all the geometric proofs behind will be at a loss and have no clear ideas.
Among them, the learning of solving triangles in the second volume of the third grade is based on right triangle. There will be a big question about the ship hitting the rocks, the height of the building and the shadow in the senior high school entrance examination. Therefore, it is also a key point in junior high school mathematics learning.
There are many special quadrilateral properties and judgment theorems in learning quadrilateral in senior two, which are easy to be confused. A deep understanding of these properties and judgments and a clear understanding of their relationship are the basis for solving the problems of proof and calculation. The types of quadrangles are changeable and difficult to calculate and prove. It often appears in the last question (the last question) of multiple-choice questions, fill-in-the-blank questions and solution questions in the senior high school entrance examination, which requires students' comprehensive application ability of knowledge.
5. Round, accounting for about 10% of the total score of the senior high school entrance examination.
Including the basic properties of a circle, the positional relationship between points, straight lines and circles, the central angle and the peripheral angle, the nature and judgment of tangents, and the arc length and area of a sector, the knowledge in this chapter was learned in Grade Three.
Among them, the nature and judgment of tangent line, the understanding and application of basic properties in circle, the positional relationship between straight line and circle, and the calculation of length and angle of some line segments in circle are the key and difficult points.
Mathematics learning methods in grade three
First, the study plan
In order to make the purpose of learning more clear, it is necessary to arrange the study time reasonably, calmly and steadily, which is the internal motivation to promote students' active learning and overcome difficulties. But the plan must be practical, with both long-term plans and short-term arrangements. In the process of implementation, we must be strict with ourselves and hone our will to learn.
Second, reflection on wrong questions
We should not generally blame ourselves for being "careless" in solving problems, but should study the wrong questions, whether because of inattention, we pay attention to one thing and can't see another; Or review the questions carelessly and misunderstand the meaning of the questions; Or remember the wrong concepts, formulas and theorems; Or you are in a hurry, jumping steps at will, causing operational mistakes and so on.
As long as the root cause is found, the same mistake can be prevented from happening again; As long as you do all the right questions, you can get excellent results.
Third, review is very important.
Mathematics learning is often to consolidate knowledge, deepen understanding and learn to use it by doing homework, thus forming skills and developing intelligence and mathematical ability. Students should pay attention to the following four points when doing homework to improve learning efficiency. First, review before doing your homework. You need to review before you do your homework, and then do it on the basis of having a basic understanding and mastery of the textbooks you have learned. Otherwise, you will get twice the result with half the effort, take time and get the desired result.
Fourth, establish a knowledge network.
To learn how to build a knowledge network, mathematical concept is the starting point of building a knowledge network, and it is also the focus of mathematics in the senior high school entrance examination. Therefore, we should master the concepts, classifications, definitions, properties and judgments of numbers, formulas, inequalities, equations, functions, trigonometric ratios, parallel lines, triangles, quadrangles and circles in algebra, and apply these concepts to solve some problems.
Five, actively carry out extracurricular learning
Extracurricular learning is the supplement and continuation of in-class learning, including reading extracurricular books and newspapers, participating in academic competitions and lectures, and visiting senior students or teachers to exchange learning experiences. It can not only enrich students' cultural and scientific knowledge, deepen and consolidate what they have learned in class, but also satisfy and develop students' hobbies, cultivate students' ability to study and work independently, and stimulate students' thirst for knowledge and enthusiasm for learning.
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