The most comprehensive summary of junior high school mathematics knowledge points 1. Number and algebra a, number and formula: 1, rational number rational number: ① integer → positive integer /0/ negative integer ② score → positive fraction/negative fraction.
Number axis: ① Draw a horizontal straight line, take a point on the straight line to represent 0 (origin), select a certain length as the unit length, and specify the right direction on the straight line as the positive direction to get the number axis. ② Any rational number can be represented by a point on the number axis. (3) If two numbers differ only in sign, then we call one of them the inverse of the other number, and we also call these two numbers the inverse of each other. On the number axis, two points representing the opposite number are located on both sides of the origin, and the distance from the origin is equal. The number represented by two points on the number axis is always larger on the right than on the left. Positive numbers are greater than 0, negative numbers are less than 0, and positive numbers are greater than negative numbers.
Absolute value: ① On the number axis, the distance between the point corresponding to a number and the origin is called the absolute value of the number. (2) The absolute value of a positive number is itself, the absolute value of a negative number is its reciprocal, and the absolute value of 0 is 0. Comparing the sizes of two negative numbers, the absolute value is larger but smaller.
Operation of rational numbers: addition: ① Add the same sign, take the same sign, and add the absolute values. ② When the absolute values are equal, the sum of different symbols is 0; When the absolute values are not equal, take the sign of the number with the larger absolute value and subtract the smaller absolute value from the larger absolute value. (3) A number and 0 add up unchanged.
Subtraction: Subtracting a number equals adding the reciprocal of this number.
Multiplication: ① Multiplication of two numbers, positive sign of the same sign, negative sign of different sign, absolute value. ② Multiply any number by 0 to get 0. ③ Two rational numbers whose product is 1 are reciprocal.
Division: ① Dividing by a number equals multiplying the reciprocal of a number. ②0 is not divisible.
Power: the operation of finding the product of n identical factors A is called power, the result of power is called power, A is called base, and N is called degree.
Mixing order: multiply first, then multiply and divide, and finally add and subtract. If there are brackets, calculate first.
2. Real irrational numbers: Infinitely circulating decimals are called irrational numbers.
Square root: ① If the square of a positive number X is equal to A, then this positive number X is called the arithmetic square root of A. If the square of a number X is equal to A, then this number X is called the square root of A. (3) A positive number has two square roots /0 square root is 0/ negative number without square root. (4) Find the square root of a number, which is called the square root, where a is called the square root.
Cubic root: ① If the cube of a number X is equal to A, then this number X is called the cube root of A. ② The cube root of a positive number is positive, the cube root of 0 is 0, and the cube root of a negative number is negative. The operation of finding the cube root of a number is called square root, where a is called square root.
Real numbers: ① Real numbers are divided into rational numbers and irrational numbers. ② In the real number range, the meanings of reciprocal, reciprocal and absolute value are exactly the same as those of reciprocal, reciprocal and absolute value in the rational number range. ③ Every real number can be represented by a point on the number axis.
3. Algebraic expressions
Algebraic expression: A single number or letter is also an algebraic expression.
Merge similar items: ① Items with the same letters and the same letter index are called similar items. (2) Merging similar items into one item is called merging similar items. (3) When merging similar items, we add up the coefficients of similar items, and the indexes of letters and letters remain unchanged.
4. Algebraic expressions and fractions.
Algebraic expression: ① The algebraic expression of the product of numbers and letters is called monomial, the sum of several monomials is called polynomial, and monomials and polynomials are collectively called algebraic expressions. ② In a single item, the index sum of all letters is called the number of times of the item. ③ In a polynomial, the degree of the term with the highest degree is called the degree of this polynomial.
Algebraic expression operation: when adding and subtracting, if you encounter brackets, remove them first, and then merge similar items.
Power operation: AM+AN=A(M+N)
(AM)N=AMN
(A/B)N=AN/BN division.
Multiplication of algebraic expressions: ① Multiply the monomial with the monomial, respectively multiply their coefficients and the power of the same letter, and the remaining letters, together with their exponents, remain unchanged as the factors of the product. (2) Multiplying polynomial by monomial means multiplying each term of polynomial by monomial according to the distribution law, and then adding the products. (3) Polynomial multiplied by polynomial. Multiply each term of one polynomial by each term of another polynomial, and then add the products.
There are two formulas: square difference formula/complete square formula.
Algebraic division: ① monomial division, which divides the coefficient and the power of the same base as the factor of quotient respectively; For the letter only contained in the division formula, it is used as the factor of quotient together with its index. (2) Polynomial divided by single item, first divide each item of this polynomial by single item, and then add the obtained quotients.
Factorization: transforming a polynomial into the product of several algebraic expressions. This change is called factorization of this polynomial.
Methods: Common factor method, formula method, grouping decomposition method and cross multiplication were used.
Fraction: ① Algebraic expression A is divided by algebraic expression B. If the divisor B contains a denominator, then this is a fraction. For any fraction, the denominator is not 0. ② The numerator and denominator of the fraction are multiplied or divided by the same algebraic expression that is not equal to 0, and the value of the fraction remains unchanged.
Operation of fraction:
Multiplication: take the product of molecular multiplication as the numerator of the product, and the product of denominator multiplication as the denominator of the product.
Division: dividing by a fraction is equal to multiplying the reciprocal of this fraction.
Addition and subtraction: ① Addition and subtraction with denominator fraction, denominator unchanged, numerator addition and subtraction. ② Fractions with different denominators shall be divided into fractions with the same denominator first, and then added and subtracted.
Fractional equation: ① The equation with unknown number in denominator is called fractional equation. ② The solution whose denominator is 0 is called the root increase of the original equation.
Equations and inequalities
1, equation and equation
Unary linear equation: ① In an equation, there is only one unknown, and the exponent of the unknown is 1. Such an equation is called a one-dimensional linear equation. ② Adding or subtracting or multiplying or dividing (non-0) an algebraic expression on both sides of the equation at the same time, the result is still an equation.
Steps to solve a linear equation with one variable: remove the denominator, shift the term, merge the similar terms, and change the unknown coefficient into 1.
Binary linear equation: An equation that contains two unknowns and all terms are 1 is called binary linear equation.
Binary linear equations: The equations composed of two binary linear equations are called binary linear equations.
A set of unknown values suitable for binary linear equation is called the solution of this binary linear equation.
The common * * * solution of each equation in a binary linear system of equations is called the solution of this binary linear system of equations.
Methods of solving binary linear equations: substitution elimination method/addition and subtraction elimination method.
Quadratic equation of one variable: an equation with only one unknown and the highest coefficient of the unknown term is 2.
1) The relation of quadratic function of quadratic equation in one variable.
Everyone has studied quadratic function (parabola) and has a deep understanding of it, such as solution and representation in images. In fact, the quadratic equation of one variable can also be expressed by quadratic function. In fact, the quadratic equation of one variable is also a special case of quadratic function, that is, when y is 0, it constitutes the quadratic equation of one variable. Then, if expressed in a plane rectangular coordinate system, the quadratic equation of one variable is the intersection of the X axis in the image and the quadratic function. Which is the solution of the equation.
2) Solution of quadratic equation in one variable.
As we all know, a quadratic function has a vertex (-b/2a, 4ac-b2/4a), which is very important. Remember, as mentioned above, the quadratic equation with one variable is also a part of the quadratic function, so he also has his own solution, and he can find all the solutions of the quadratic equation with one variable.
(1) matching method
Using this formula, the equation is transformed into a complete square formula and solved by direct Kaiping method.
(2) Factor decomposition method
Select the common factor, apply the formula, and cross multiply. The same is true for solving quadratic equations with one variable. Using this, the equation can be solved in the form of several products.
(3) Formula method
This method can also be used as a general method to solve quadratic equations with one variable. The roots of the equation are x 1 = {-b+√ [B2-4ac]}/2a, and x2 = {-b-√ [B2-4ac]}/2a.
3) the step of solving a quadratic equation with one variable:
(1) Matching method steps:
First, the constant term is moved to the right of the equation, then the coefficient of the quadratic term is changed to 1, and the square of half the coefficient of 1 is added, and finally the complete square formula is obtained.
(2) The steps of factorization:
Turn the right side of the equation into 0, and then see if you can extract the common factor, formula (here refers to the formula in factorization) or cross multiplication, and if you can, turn it into the form of product.
(3) Formula method
Simply substitute the coefficient of quadratic equation into a variable, where the coefficient of quadratic term is a, the coefficient of linear term is b, and the coefficient of constant term is c.
4) Vieta theorem
Understand with Vieta theorem, Vieta theorem in a quadratic equation, the sum of two roots =-b/a, the product of two roots = c/a.
It can also be expressed as x 1+x2 =-b/a, and x1x2 = c/a. By using Vieta's theorem, we can find out the coefficients in a quadratic equation with one variable, which is very common in the topic.
5) The case of the root of a linear equation with one variable
Using the discriminant of roots to understand, the discriminant of roots can be written as "Delta" and read as "Tune ta", and Delta = B2-4ac can be divided into three situations:
I am △ > 0, a quadratic equation with one variable has two unequal real roots;
II When △=0, the quadratic equation of one variable has two identical real roots;
Three dang △
2. Inequality and unequal groups
Inequality: ① When the symbol > = 0 is used, it passes through quadrant124; When k > 0 and b < 0, pass through quadrant134; When k > 0 and b > 0, pass through quadrant 123. ④ When k > 0, y value increases with the increase of x value, and when x < 0, y value decreases with the increase of x value.
Second, space and graphics.
First, the understanding of graphics
1, point, line, surface
Points, lines and surfaces: ① A figure consists of points, lines and surfaces. (2) Lines intersecting face to face and points where lines intersect. (3) Points become lines, lines become surfaces, and surfaces become bodies.
Unfolding and folding: ① In a prism, the intersection of any two adjacent faces is called an edge, and the side edge is the intersection of two adjacent edges. All sides of the prism are equal in length, the upper and lower bottom surfaces of the prism are the same in shape, and the side surfaces are cuboids. (2) N prism is a prism with N faces on its bottom.
Cutting a geometric figure: cutting a figure with a plane, and the cutting surface is called a section.
Views: main view, left view and top view.
Polygon: It is a closed figure composed of some line segments that are not on the same straight line.
Arc and sector: ① A figure consisting of an arc and two radii passing through the end of the arc is called a sector. ② The circle can be divided into several sectors.
2. Angle
Line: ① A line segment has two endpoints. (2) The line segment extends infinitely in one direction to form a ray. A ray has only one endpoint. ③ A straight line is formed by the infinite extension of both ends of a line segment. A straight line has no end. Only one straight line passes through two points.
Comparison length: ① Of all the connecting lines between two points, the line segment is the shortest. ② The length of the line segment between two points is called the distance between these two points.
Measurement and expression of angle: ① An angle consists of two rays with a common endpoint, and the common endpoint of the two rays is the vertex of the angle. ② One degree of 1/60 is one minute, and one minute of1/60 is one second.
Comparison of angles: ① An angle can also be regarded as a light rotating around its endpoint. (2) The ray rotates around its endpoint. When the ending edge and the starting edge are on a straight line, the angle formed is called a right angle. The starting edge continues to rotate, and when it coincides with the starting edge again, the angle formed is called fillet. (3) The ray from the vertex of an angle divides the angle into two equal angles, and this ray is called the bisector of the angle.
Parallelism: ① Two straight lines that do not intersect in the same plane are called parallel lines. ② One and only one straight line is parallel to this straight line after passing through a point outside the straight line. If both lines are parallel to the third line, then the two lines are parallel to each other.
Perpendicular: ① If two lines intersect at right angles, they are perpendicular to each other. (2) The intersection of two mutually perpendicular straight lines is called vertical foot. ③ On the plane, there is one and only one straight line perpendicular to the known straight line at one point.
Perpendicular bisector: A straight line perpendicular to and bisecting a line segment is called perpendicular bisector.
The perpendicular bisector in perpendicular bisector must be a line segment, not a ray or a straight line, which is related to the infinite extension of rays and straight lines. Look at the back, the middle vertical line is a straight line, so when drawing the middle vertical line, the line segment should pass through two points and then two points (about drawing, we will talk about it later).
Perpendicular bisector theorem;
Property theorem: the distance between the point on the middle vertical line and the two ends of the line segment is equal;
Decision Theorem: The point with the same distance from the endpoint of line segment 2 is on the middle vertical line of this line segment.
Angular bisector: The ray bisecting an angle is called the angular bisector of the angle.
There are several points to note in the definition, that is, the bisector of an angle is a ray, not a line segment or a straight line. Many times there will be a straight line in the topic, which is the symmetry axis of the bisector, which also involves the problem of trajectory. The bisector of an angle is a point with equal distance to both sides of the angle.
Property theorem: the distance between the point on the bisector of an angle and both sides of the angle is equal.
Decision theorem: the points with equal distance to both sides of the angle are on the bisector of the angle.
Square: A set of rectangles with equal adjacent sides is a square.
Properties: Square has all the properties of parallelogram, rhombus and rectangle.
Judgment: 1, rhombus 2 with equal diagonals and rectangle with equal adjacent sides.
Ten problem-solving skills of senior high school entrance examination mathematics 1, matching method
The so-called formula is to change some items of an analytical formula into the sum of positive integer powers of one or more polynomials by using the method of constant deformation. The method of solving mathematical problems with formulas is called matching method. Among them, the most common method is to make it completely flat. Matching method is an important method of constant deformation in mathematics. It is widely used in factorization, simplifying roots, solving equations, proving equality and inequality, finding extreme values of functions and analytical expressions.
2, factorization method
Factorization is to transform a polynomial into the product of several algebraic expressions. Factorization is the basis of identity deformation. As a powerful mathematical tool and method, it plays an important role in solving algebra, geometry and trigonometry problems. There are many methods of factorization, such as extracting common factors, formulas, grouping decomposition, cross multiplication and so on. Middle school textbooks also introduce the use of decomposition and addition, root decomposition, exchange elements, undetermined coefficients and so on.
3. Alternative methods
Method of substitution is a very important and widely used method to solve problems in mathematics. We usually refer to unknowns or variables as elements. The so-called method of substitution is to replace a part of the original formula with new variables in a complicated mathematical formula, thus simplifying it and making the problem easy to solve.
4. Discriminant method and Vieta theorem.
The root discrimination of unary quadratic equation ax2+bx+c=0(a, B, C belongs to R, a≠0) and△ = B2-4ac is not only used to judge the properties of roots, but also widely used in algebraic deformation, solving equations (groups), solving inequalities, studying functions and even geometric and trigonometric operations as a problem-solving method.
Vieta's theorem not only knows one root of a quadratic equation, but also finds another root. Knowing the sum and product of two numbers, we can find the symmetric function of the root, calculate the sign of the root of quadratic equation, solve the symmetric equation and solve some problems about quadratic curve. , has a very wide range of applications.
5, undetermined coefficient method
When solving mathematical problems, it is first judged that the obtained results have a certain form, which contains some undetermined coefficients, then the equations about undetermined coefficients are listed according to the problem setting conditions, and finally the values of these undetermined coefficients or some relationship between these undetermined coefficients are found. This method is called undetermined coefficient method to solve mathematical problems. It is one of the commonly used methods in middle school mathematics.
6. Construction method
When solving problems, we often use this method to construct auxiliary elements by analyzing conditions and conclusions, which can be a figure, an equation (group), an equation, a function, an equivalent proposition and so on. And establish a bridge connecting conditions and conclusions, so that the problem can be solved. This mathematical method of solving problems is called construction method. Using construction method to solve problems can make algebra, trigonometry, geometry and other mathematical knowledge permeate each other, which is beneficial to solving problems.
7. reduce to absurdity
Reduction to absurdity is an indirect proof method. First, a hypothesis contrary to the conclusion of the proposition is put forward, and then from this hypothesis, through correct reasoning, contradictions are led out, thus denying the opposite hypothesis and affirming the correctness of the original proposition. The reduction to absurdity can be divided into reduction to absurdity (with only one opposite conclusion) and exhaustive reduction to absurdity (with more than one opposite conclusion). The steps to prove a proposition by reduction to absurdity can be roughly divided into: (1) reverse design; (2) return to absurdity; (3) conclusion.
Anti-design is the basis of reduction to absurdity. In order to make correct anti-design, we need to master some commonly used negative expressions, such as: yes/no; Existence/non-existence; Parallel/non-parallel; Vertical/not vertical; Equal to/unequal to; Large (small) inch/small (small) inch; Both/not all; At least one/none; At least n/ at most (n-1); At most one/at least two; Only/at least two.
Reduction to absurdity is the key to reduction to absurdity. There is no fixed model in the process of derivation of contradiction, but it must be based on reverse design, otherwise the derivation will become passive water without roots. Reasoning must be rigorous. There are the following types of contradictions: contradictions with known conditions; Contradicting with known axioms, definitions, theorems and formulas; There are dual contradictions; Contradictions
8. Find the area method
The area formula in plane geometry and the property theorems related to area calculation derived from the area formula can be used not only to calculate the area, but also to prove that plane geometry problems sometimes get twice the result with half the effort. The method of proving or calculating plane geometric problems by using area relation is called area method, which is commonly used in geometry.
The difficulty in proving plane geometry problems by induction or analysis lies in adding auxiliary lines. The characteristic of area method is to connect the known quantity with the unknown quantity by area formula, and achieve the verification result through operation. Therefore, using the area method to solve geometric problems, the relationship between geometric elements becomes the relationship between quantities, and only calculation is needed. Sometimes there may be no auxiliary lines, even if auxiliary lines are needed, it is easy to consider.
9, geometric transformation method
In the study of mathematical problems, the transformation method is often used to transform complex problems into simple problems and solve them. The so-called transformation is a one-to-one mapping between any element of a set and the elements of the same set. The transformation involved in middle school mathematics is mainly elementary transformation. There are some exercises that seem difficult or even impossible to start with. We can use geometric transformation to simplify the complex and turn the difficult into the easy. On the other hand, the transformed point of view can also penetrate into middle school mathematics teaching. It is helpful to understand the essence of graphics by combining the research of graphics under isostatic conditions with the research of motion.
Geometric transformation includes: (1) translation; (2) rotation; (3) symmetry.
10, objective problem solving method
Multiple-choice questions are questions that give conditions and conclusions and require finding the correct answer according to a certain relationship. Multiple-choice questions are ingenious in conception and flexible in form, which can comprehensively examine students' basic knowledge and skills, thus increasing the capacity and knowledge coverage of test papers.
Fill-in-the-blank question is one of the important questions in standardized examination. Like multiple-choice questions, it has the advantages of clear test objectives, wide knowledge coverage, accurate and fast marking, and is conducive to examining students' analytical judgment and calculation ability. The difference is that the fill-in-the-blank question does not give an answer, which can prevent students from guessing the answer.