Chapter 1: Entering the world of mathematics.

Chapter II Rational Numbers

1. number axis: three elements of number axis: origin, positive direction and unit length; There is a one-to-one correspondence between points on the number axis and real numbers.

2. the reciprocal of real number a is-a; If A and B are opposites, then a+b=0, and vice versa; Geometric meaning: on the number axis, two points representing the opposite number are located on both sides of the origin, and the distance to the origin is equal.

3. Reciprocal: If the product of two numbers is equal to 1, then these two numbers are reciprocal.

4. Absolute value: Algebraic meaning: the absolute value of a positive number is itself, the absolute value of a negative number is its opposite number, and the absolute value of 0 is 0; Geometric meaning: the absolute value of a number is the distance from the point representing this number to the origin on the number axis.

5. Scientific symbol:, in which. 6. Real number comparison: compare the size according to the law; Use the number axis to compare sizes.

7. In the range of real numbers, addition, subtraction, multiplication, division and power operations can be performed, but the root operation may not work, for example, negative numbers can't even be opened. The operation basis of real numbers is rational number operation, and all the operation properties and laws of rational numbers are applicable to real number operation. It is the key to master the real number operation to correctly determine the symbol of the operation result and flexibly use the algorithm.

Chapter III Addition and subtraction of algebraic expressions

First, the related concepts of algebraic expression

1, monomial: the product of numbers and letters. Such algebraic expressions are called monomials. A single number or letter is also a monomial.

2. Coefficient of single item: numerical factor in single item.

3. The number of monomials: the exponential sum of all the letters in the monomials.

4. Polynomial: The sum of several monomials is called polynomial.

5. Term and degree of polynomial: the monomial in polynomial is called the term of polynomial, and the degree of the highest term in polynomial is called the degree of polynomial. Please note that the degree of a polynomial is not the exponential sum of all the letters that make up the polynomial! ! !

Algebraic expression: monomials and polynomials are collectively called algebraic expressions. (Algebraic expressions with letters in denominator are not algebraic expressions)

Second, the operation of algebraic expressions

(1) The basic steps of algebraic expression addition and subtraction: removing brackets and merging similar items.

(2) Multiplication of algebraic expressions

1, the law of power with the same base: multiply the power of the base, the base is unchanged, and the exponents are added. Mathematical symbol: _ _ _ _ (where m and n are positive integers)

2. Power Law: Power, constant base, exponential multiplication. Mathematical symbols: _ _ _ _ _ _ (where m and n are positive integers)

3. Product power law: the power of the product, first multiply the factors in the product respectively, and then multiply the obtained power. (that is, equal to the product of the power of each factor in the product. Mathematical symbol: _ _ _ _ _ _ (where n is a positive integer)

4. The law of power division with the same base: power division with the same base, exponential subtraction with the same base. Mathematical symbol: _ _ _ _ (where m and n are positive integers)

5. The law of multiplying the monomial by the monomial: when the monomial is multiplied by the monomial, their coefficients are multiplied by the power of the same letter respectively, while the other letters, together with their exponents, remain unchanged as a factor of the product.

6. The rule of polynomial multiplying by monomial: Polynomial multiplying by monomial is to multiply each term of polynomial by monomial according to the distribution law, and then add the obtained products.

7. Polynomial Multiplication Polynomial Rule: When a polynomial is multiplied by a polynomial, first multiply each term of one polynomial by each term of another polynomial, and then add the products.

8. Rule of square difference formula: Two numbers are multiplied by the difference of these two numbers, which is equal to the square difference of these two numbers. Mathematical symbol: _ _ _ _ (where A and B can be both numbers and algebraic expressions) Description: The square difference formula is obtained by multiplying a polynomial, which is the product of the sum of two numbers and the difference of the same two numbers.

9. Complete square formula rule: the square of the sum (or difference) of two numbers is equal to the sum of the squares of these two numbers plus (or minus) twice the product of these two numbers.

Mathematical symbol: _ _ _ _ _ _

(b) division of algebraic expressions

1, the law of monomial divided by monomial: monomial divided by monomial, divided by their coefficients and the power of the same letter, as a factor of quotient. For letters only contained in the division formula, it is considered as a factor of quotient together with its index.

2. Polynomial divided by single term rule: Polynomial divided by single term, that is, each term of polynomial is divided by single term, and then the obtained quotients are added.

The fourth chapter is the preliminary understanding of graphics.

1. Point, line and surface: Learn more about point, line and surface through rich examples (for example, a city is represented by a point on a traffic map, and the picture on the screen is composed of points). 2. Angle ① Through abundant examples, we can further understand the angle. ② I will compare angles, estimate the size of an angle, calculate the sum and difference of angles, identify minutes and seconds, and make a simple conversion. ③ Understand the angular bisector and its properties.

Intersecting line and parallel line

I. Basic concepts

1. Straight line: (1) The straight line extends infinitely to _ _ _ _ _ _ _ _ _, and the straight line has no end point. (2) There is only one _ _ _ _ _ _ after two o'clock.

2. Ray: A point on a straight line and the part next to it are called _ _ _ _ _ _ _ _, this point is called the endpoint of the ray, and the ray has only one endpoint.

2. Line segment: (1) The part between two points on a straight line is called _ _ _ _ _ _ _ _ _, and _ _ _ _ _ _ has two endpoints. (2) Between two points, _ _ _ _ _ is the shortest.

(3) The point where a line segment is divided into two equal line segments is called _ _ _ _ _ _ _ _ of the line segment.

4. Vertical line; When two straight lines intersect, one of the four angles is _ _ _ _ _ _, which means that the two straight lines are perpendicular to each other; One of the lines is called the perpendicular of the other line, and their intersection is called _ _ _ _ _ _ _.

5. The nature of the vertical line: (1) passes through a point, and only _ _ lines are perpendicular to the known lines; (2) _ _ is the shortest of all the line segments connecting the outer point and the point on the line.

6. Distance between two points: the length of the line segment connecting _ _ _ _ _ _ _.

7. Distance from point to straight line: the length of the vertical line from a point outside the straight line to _ _ _ _ _ _ _ _.

8. Distance between two parallel lines: the distance from one of the two parallel lines to the other.

9. Angle: A figure consisting of two _ _ _ _ _ _ points with a male * * end is called an angle. This common endpoint is called the vertex of the angle, and these two _ _ _ are called the edges of the angle.

10, angle bisector: starting from the vertex of an angle, the ray that divides this angle into two _ _ _ _ _ _ _ _ angles is called the angle bisector.

/ Continue to rotate back to the _ _ _ _ _ _ _ position. This angle is called fillet.

12. Measurement of angle: 1 fillet = _ _ angle = _ _ right angle = 360, 1 = _ _',1'= _ _ "

13. Classification of angles smaller than right angles: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

14. Complementary angle and complementary angle: If the sum of two angles is _, then these two angles are called complementary angles; If the sum of two angles is _, they are called complementary angles.

15. Properties of correlation angle: (1) vertex angle _ _ _ _ _ (2) congruent angle with the same angle or equal angle _ _ _ _ _; (3) the complementary angle of the same angle or equal angle _ _ _ _ _.

Second, the intersection line and parallel line

1. Parallel lines: In the same plane, two straight lines of _ _ _ _ _ are called parallel lines.

2. In the same plane, there are only two positional relationships between two straight lines: _ _ _ _ _ _ _ _. When intersecting, the vertex angles are equal.

3. Determination of parallel lines: (1) isomorphic angle _ _, two straight lines are parallel. (2) The internal dislocation angles are equal, and the two straight lines are _ _ _ _ _.

(3) The inner angle on the same side is _ _ _ _ _ _ _ _ _ _, and the two straight lines are parallel. (4) Two straight lines parallel (or perpendicular) to the same straight line _ _ _ _ _ _ _.

4. Nature of parallel lines: (1) After passing a point outside the straight line, there are only _ _ _ straight lines parallel to this straight line.

(2) Two straight lines are parallel, and the same angle is _ _ _ _ _. (3) Two straight lines are parallel, and the internal dislocation angle is _ _ _ _ _ _.

(4) Two straight lines are parallel with equal internal angles. (5) A straight line is perpendicular (or parallel) to one of the two parallel lines, and this straight line is also perpendicular (or parallel) to _.

(6) The distance between parallel lines is _ _ _ _ _. (7) A straight line passing through the midpoint of one side of a triangle and parallel to the other side must be equally divided into _ _ _ _ _ _ _.

Third, parallel lines are divided into segments in proportion.

1. Parallel lines bisect line segments theorem: If a group of parallel lines have equal line segments on a straight line, then the line segments on other straight lines are also _ _ _ _.

2. Inference of the theorem that parallel lines bisect the line segment: (1) A straight line passing through the midpoint of one waist and the bottom of the trapezoid will bisect the other waist. (2) A straight line passing through the midpoint of one side of a triangle and parallel to the other side must be equally divided into _ _ _ _ _ _ _.

3. Proportional theorem of parallel lines: three parallel lines cut two straight lines, and the corresponding line segment is _ _ _ _ _ _ _.

4. Inference of the proportion theorem of parallel line segments: A straight line on one side of a triangle intersects with the other two sides (or extension lines on both sides), and the corresponding line segments are proportional. 5. Theorem: If a straight line cuts two sides of a triangle (or the extension of two sides), then the straight line is on the third side of the triangle.

Chapter V Data Collection and Expression

Learn how to collect data, organize data, analyze data and finally draw corresponding conclusions; In addition, we should also master knowledge points such as frequency and frequency.

Clearly investigate the problem-the use of data;

Determine the survey object-data collection scope;

Select the survey method-the method used to collect data;

Conduct surveys-collect data;

Record results-data collation;

Draw a conclusion-data analysis;

Abstract: Frequency indicates the number of times each object appears.

Frequency indicates the ratio (or percentage) of each object's appearance to the total number of times.

Frequency and frequency can reflect the frequency of each object.

Learn to express data intuitively with statistics, and find the relationship between data from statistical charts. Learn to draw statistical charts by computer.

Chapter VI One-variable Linear Equation

1. Transform the equation appropriately to solve the linear equation of one variable: the basic idea of solving the equation is transformation, that is, to transform the equation, and two points should be paid attention to when transforming. At one time, the two sides of the equation could not be multiplied (or divided) by algebraic expressions containing unknowns, otherwise the obtained equation might be different from the original equation; Second, when removing the denominator, don't omit the multiplication of items without the denominator. One-dimensional linear equations are the basic contents of learning binary linear equations, one-dimensional quadratic equations, one-dimensional linear inequalities and function problems.

2. Correctly understand the definition of the equation solution and skillfully solve the test questions by using the properties of the equation: the solution of the equation should be understood as being appropriate to substitute the original equation, and its method is to substitute the solution of the equation into the original equation to transform the problem.

3. Understand all kinds of solutions of the equation ax=b under different conditions, and simply apply it: (1)a≠0, the equation has a unique solution X =；;

(2) When a = 0 and b=0, the equation has countless solutions; (3) When a = 0 and b≠0, the equation has no solution.

4. Correctly establish a linear equation of one variable to solve the application problem: the key to establishing an equation to solve the application problem is to find the equivalent relationship in the problem. According to the analysis of examination questions in recent years, when solving application problems, we should pay more attention to social hotspots, closely link with reality, collect and process more information, and pay attention to whether the results are in line with practical significance.

5. Several common problems: sum-difference multiplication, isoelectric deformation, labor distribution, proportional distribution, numerical problems and engineering problems.

Chapter VII Binary Linear Equations

1. Binary linear equation (group) and its solution application: Note: the solution of equation (group) is applicable to equation. Any binary linear equation has countless solutions, and sometimes the integer solution is examined, and the algebraic value is often calculated skillfully by using the concept of equations.

2. Solving binary linear equations: The basic idea of solving equations is elimination method, and the commonly used methods are substitution elimination method and addition and subtraction elimination method. Transformation ideas and overall ideas are also the focus of this chapter.

The substitution elimination method will be used to solve the binary linear equations with unknown coefficient 1. Method of substitution will be used to solve binary linear equations with unknown coefficients other than 1. Addition and subtraction can be used to find the solution of binary linear equations with equal or opposite unknown coefficients. Learn to use equation deformation, and then solve binary linear equations by adding, subtracting and eliminating. Flexible use of substitution elimination method and addition and subtraction elimination method to solve problems.

3. Application of binary linear equations: The key to listing binary linear equations is to correctly analyze the equivalence relation in the topic. The content of the topic is often close to the reality of life and related to hot issues of social relations. Please pay attention to collection, observation and analysis.

Chapter VIII One-dimensional Linear Inequalities

1. Judging whether inequality is established: The key is to analyze and judge the change of inequality, which is based on the nature of inequality. When both sides of inequality are multiplied (or divided) by the same negative number, it is particularly important to change the direction of inequality. On the other hand, if the direction of inequality changes, it means that both sides of inequality are multiplied (or divided) by a negative number. Therefore, we should carefully observe the form and direction of inequality when judging whether inequality is established or looking for the range of some letters from inequality deformation.

2. Solving linear inequality of one variable (group): The steps of solving linear inequality of one variable are almost the same as those of solving linear equation of one variable. It should be noted that there are positive and negative numbers of multiplication (or division) on both sides of inequality, and its properties can be used flexibly according to different situations. One-dimensional linear inequalities (groups) are often associated with fractions, roots, one-dimensional quadratic equations, functions and other knowledge to solve comprehensive problems.

3. Find the special solution of inequality (group): Inequality (group) often has countless solutions, but their special solutions are all limited to certain ranges, such as integer solutions and non-negative integer solutions. To find these special solutions, we must first determine the solution set of inequalities (groups), and then find the corresponding answers. Pay attention to the application of the idea of combining numbers with shapes.

4. Column inequality (group) solving application problems: pay attention to the analysis of inequality relations in the problem, and the hot spot of examination is inequality (group) application problems closely related to real life.

Chapter 9 Polygons

1. Polygon: Generally speaking, a polygon is a closed figure surrounded by some end-to-end line segments. We usually divide polygons into triangles, quadrilaterals and pentagons according to the number of sides.

2.N-polygon: A closed figure surrounded by N line segments end to end is called an N-polygon (n is an integer greater than or equal to 3).

3. Polygon segmentation: Starting from a certain vertex of a polygon and connecting this vertex with other vertices respectively, the polygon can be divided into several triangles.

4. There are (n- 3) diagonal lines from a vertex of an n- polygon, and the N-polygon is divided into (n-2) triangles. N polygon * * * has n vertices, n edges and n(n-3)÷2 diagonals.

5. Circle: The figure formed by the rotation of a line segment around one end is called a circle.

6. The line segment between two points on a circle is called an arc, and the figure consisting of an arc and two radii passing through the end points of the arc is called a fan.

7. A circle can be divided into several sectors.

8. Two points on the circle (the line segment connecting the two points is not the diameter) divide the circle into two parts, one part is larger than the semicircle and the other part is smaller than the semicircle, so the two points on the circle are divided into two arcs, and each arc corresponds to a sector.

Understand the concept of triangle (inner angle, outer angle, midline, height and angular bisector), and draw the angular bisector, midline and height of any triangle. Understand the stability of triangles. The sum of two sides of a triangle is greater than the third side. ② Explore and master the nature of the triangle midline.

⒑ Key points: 1. Basic concepts of quadrilateral:

(1) quadrilateral: In a plane, four line segments are connected end to end. If any two line segments are not on the same straight line, the figure formed is called quadrilateral.

(2) Name of each part: Edge: Vertex of the line segment that constitutes each side of the quadrilateral: Internal angle of the common point of two adjacent sides: angle formed by two adjacent sides when viewed from the inside of the quadrilateral, which is called angle for short. Diagonal: A line segment connecting two nonadjacent vertices of a quadrilateral. External angle: sum of one side of quadrilateral

Chapter 10 Axisymmetric

Axisymmetric and axisymmetric figures are different concepts: "Axisymmetric" refers to the shape and position relationship between two figures; "Axisymmetric figure" refers to the shape of a figure.

Definition: An equilateral triangle is an isosceles triangle.

The nature of isosceles triangle;

The two base angles of an isosceles triangle are equal. (abbreviated as "equilateral angle")

The bisector of the vertex of an isosceles triangle, the median line on the bottom, and the coincidence of the heights on the bottom (abbreviated as "three lines in one")

The bisectors of the two base angles of an isosceles triangle are equal. (The midline of the two waists is equal and the height of the two waists is equal)

The distance from the base of an isosceles triangle to the waist is equal.

The angle between the waist height and the bottom of an isosceles triangle is equal to half of the top angle.

Determination of isosceles triangle: A triangle with two equal angles is an isosceles triangle.

Some properties of triangles:

1. The sum of any two sides of the triangle must be greater than the third side, which also proves that the difference between any two sides of the triangle must be less than the third side.

2. The sum of the internal angles of the triangle is equal to 180 degrees.

3. The bisector of the vertex, the midline of the bottom and the height of the bottom of the isosceles triangle coincide, that is, the three lines are one.

The symmetry of graphics is a new hot topic in the senior high school entrance examination. The score is generally 3-4 points, and the questions are mainly filled in the blanks, multiple-choice questions and lottery, with occasional answers.

Contents of investigation: ① Discrimination of axisymmetric and axisymmetric graphic properties. ② Pay attention to mirror symmetry and solve practical problems. Breakthrough methods: ① Master the basic symmetry and drawing methods of graphics. (2) Combined with specific problems, try boldly and operate by hands to explore and discover its internal laws. (3) Pay attention to the research on the transformation of graphics in grids and coordinates, and master its common problem-solving methods skillfully. ④ Pay attention to the innovation of graphics and transformation, find out its essence, and master basic problem-solving methods, such as hands-on operation, folding and rotation.

Chapter 11 Experience Uncertainty

1, inevitable event: the event that must occur in each experiment has a probability of 100%.

2. Impossible event: an event that will never happen in each experiment, with a probability of 0.

(Necessary events and impossible events are collectively referred to as definite events)

3. Uncertain events (random events): events that cannot be determined in the experiment occur.

The probability of is a number between 0 and 1.

4. "Unlikely" does not mean "impossible", and a small possibility does not mean that it will not happen.

5. Opportunity: The success rate of an uncertain event or a random event is the stable state after many experiments. We call this success rate the possibility of random events, that is, opportunities.

6. Equality and inequality of opportunities: When the probability of success or failure of uncertain events is half, that is, 0.50, we say that the opportunities of uncertain events are equal, otherwise the opportunities are unequal.

7. Estimation of the occurrence probability of uncertain phenomena.

(1) experimental method: estimate by a large number of repeated experiments.

(2) Analysis method: It is determined from all possible situations of experimental results.

8. Stability of the frequency of uncertain events in a large number of repeated experiments.

7. The experiment must be carried out under the same conditions. The more experiments, the more accurate the chance estimation.

8. Experiment is a method to estimate opportunities.