Junior high school mathematics knowledge point concourse

I. Numbers and Algebra A: Numbers and Formulas:

1: rational number

Rational Numbers: ① Integer → Positive Integer /0/ Negative Integer ② Fraction → 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. & ltbr & gt

(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. & ltbr & gt

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. & ltbr & gt

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. & ltbr & gt

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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. & ltbr & gt

The absolute value of a positive number is himself/the absolute value of a negative number is his opposite number/the absolute value of 0 is 0. Comparing the sizes of two negative numbers, the absolute value is larger but smaller. & ltbr & gt

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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. & ltbr & gt

Subtraction: Subtracting a number equals adding the reciprocal of this number. & ltbr & gt

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. & ltbr & gt

Division: ① Dividing by a number equals multiplying the reciprocal of a number. ②0 is not divisible. & ltbr & gt

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. & ltbr & gt

Mixing order: multiply first, then multiply and divide, and finally add and subtract. If there are brackets, calculate first. & ltbr & gt

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2: Real number

Irrational number: Infinitely circulating decimals are called irrational numbers.

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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. & ltbr & gt

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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 /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. & ltbr & gt

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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. & ltbr & gt

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3. Algebraic

Algebraic expression: A single number or letter is also an algebraic expression. & ltbr & gt

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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. & ltbr & gt

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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. & ltbr & gt

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Algebraic expression operation: when adding and subtracting, if you encounter brackets, remove them first, and then merge similar items. & ltbr & gt

Operation of power: morning. AN=A(M+N) (AM)N=AMN (AB)N=AN .BN division. & ltbr & gt

A0= 1，A-P = 1/AP & lt； br & gt

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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. & ltbr & gt

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There are two formulas: square difference formula/complete square formula.

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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. & ltbr & gt

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Factorization: transforming a polynomial into the product of several algebraic expressions, which is called factorization of the polynomial.

Methods: Common factor method/formula method/grouping decomposition method/cross multiplication;

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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. & ltbr & gt

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Fractional operation: 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. & ltbr & gt

Division: dividing by a fraction is equal to multiplying the reciprocal of this fraction. & ltbr & gt

Addition and subtraction: ① Add and subtract fractions with the same denominator, and add and subtract molecules with the same denominator. ② Fractions with different denominators shall be divided into fractions with the same denominator first, and then added and subtracted. & ltbr & gt

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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. & ltbr & gt

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B: equations and inequalities

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1: equations and equations

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. & ltbr & gt

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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. & ltbr & gt

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Binary linear equation: An equation that contains two unknowns and whose terms are 1 is called binary linear equation. & ltbr & gt

Binary linear equations: The equations composed of two binary linear equations are called binary linear equations. & ltbr & gt

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A set of unknown values suitable for binary linear equation is called the solution of this binary linear equation. & ltbr & gt

The common * * * solution of each equation in a binary linear system of equations is called the solution of this binary linear system of equations. & ltbr & gt

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Methods of solving binary linear equations: substitution elimination method/addition and subtraction elimination method. & ltbr & gt

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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. & ltbr & gt

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Second, space and graphics.

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A: Understanding of graphics:

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1: point, line and 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. & ltbr & gt

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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. & ltbr & gt

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Cutting a geometric figure: cutting a figure with a plane, and the cutting surface is called a section. & ltbr & gt

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Three views: main view, left view and top view. & ltbr & gt

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Polygon: It is a closed figure composed of some line segments that are not on the same straight line. & ltbr & gt

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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. & ltbr & gt

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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. & ltbr & gt

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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. & ltbr & gt

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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. & ltbr & gt

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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. & ltbr & gt

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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. & ltbr & gt

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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. & ltbr & gt

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3. Intersecting lines and parallel lines

Angle: ① If the sum of two angles is a right angle, then the sum of the two angles is complementary; If the sum of two angles is a right angle, then these two angles are called complementary angles. ② The complementary angle/complementary angle of the same angle or equal angle is equal. ③ The vertex angles are equal. ④ congruent angle/internal dislocation angle are equal/internal angles on the same side are complementary, and two straight lines are parallel, and vice versa. & ltbr & gt

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4: Triangle

Triangle: ① A figure composed of three line segments that are not on the same straight line is called a triangle. ② The sum of any two sides of a triangle is greater than the third side. The difference between any two sides of a triangle is less than the third side. ③ The sum of the three internal angles of a triangle is equal to 180 degrees. ④ Triangle is divided into acute triangle/right triangle/obtuse triangle. ⑤ The two acute angles of a right triangle are complementary. ⑥ The bisector of the inner angle of a triangle intersects its opposite side, and the line segment between the vertex and the intersection of this angle is called the bisector of the triangle. ⑦ In a triangle, the line segment connecting the vertex and the midpoint of its opposite side is called the center line of the triangle. Today, the three bisectors of a triangle intersect at one point, and the three median lines intersect at one point. Pet-name ruby straight line from a vertex of a triangle to its opposite side, the line segment between the vertex and the vertical foot is called the height of the triangle. Attending the triangle three height straight lines intersect at one point. & ltbr & gt

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Congruence of figures: congruent figures have the same shape and size. Two graphs that can overlap are called congruent graphs. & ltbr & gt

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Congruent triangles: ① The corresponding edges/angles of congruent triangles are equal. ② Conditions: SSS/AAS/ASA/SAS/HL. & ltbr & gt

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Pythagorean Theorem: The sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse, and vice versa. & ltbr & gt

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5: Quadrilateral

Properties of parallelogram: ① Two groups of parallelograms with parallel opposite sides are called parallelograms. (2) The line segment connected by two nonadjacent vertices of a parallelogram is called its diagonal. ③ The opposite sides/diagonals of parallelogram are equal. (4) The diagonal of the parallelogram is equally divided. & ltbr & gt

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The judgment condition of parallelogram: a quadrilateral with two diagonal lines bisecting each other/a group of quadrilaterals with parallel and equal opposite sides/two groups of quadrilaterals with equal opposite sides/definition. & ltbr & gt

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Diamond: ① A group of parallelograms with equal adjacent sides is a diamond. (2) The four sides of the collar are equal, and the two diagonal lines are equally divided vertically, and each diagonal line is equally divided into a set of diagonal lines. ③ Judgment conditions: define a parallelogram with vertical diagonal and a quadrilateral with equal sides. & ltbr & gt

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Rectangular and square: ① A parallelogram with a right angle is called a rectangle. ② The diagonals of a rectangle are equal, and all four corners are right angles. ③ Parallelograms with equal diagonals are rectangles. ④ A square has all the properties of parallelogram, rectangle and diamond. ⑤ A set of rectangles with equal adjacent sides is a square. & ltbr & gt

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Trapezoid: ① A group of quadrangles with parallel opposite sides and another group of quadrangles with non-parallel opposite sides are called trapeziums. ② Two trapezoid with equal waist are called isosceles trapezoid. A trapezoid with a vertical waist bottom is called a right-angled trapezoid. ④ The two internal angles on the same base of the isosceles trapezoid are equal, and the diagonal is equal, and vice versa. & ltbr & gt

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Polygon: ① The sum of the internal angles of the N-polygon is equal to (N-2) 180 degrees. (2) The angle formed by the extension line between one side and the other side of the inner corner of a polygon is called the outer corner of this polygon. Take an outer angle of the polygon at each vertex, and their sum is called the sum of the inner angles of the polygon (both equal to 360 degrees).

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Dense laying of plane graphics: triangles, quadrangles and regular hexagons can be laid densely. & ltbr & gt

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Centrally symmetric figure: ① On the plane, a figure rotates around a point 180 degrees. If the figures before and after rotation overlap, then this figure is called a central symmetric figure, and this point is called its symmetric center. ② The line segments connected by each pair of corresponding points on the central symmetric figure are equally divided by the symmetric center. & ltbr & gt

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B: graphics and transformation:

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1: Axisymmetry of graphs

Axisymmetric: If a figure is folded along a straight line, and the parts on both sides of the straight line can overlap each other, then the figure is called an axisymmetric figure, and this straight line is called an axis of symmetry. & ltbr & gt

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Axisymmetric figure: ① The distance between the points on the bisector of an angle and both sides of the angle is equal. ② The distance between the point on the vertical line of a line segment and the two endpoints of the line segment is equal. ③ The "three lines in one" of isosceles triangle. & ltbr & gt

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The essence of axial symmetry: the line segments connected by corresponding points are vertically bisected by the axis of symmetry, and the corresponding line segments/corresponding angles are equal. & ltbr & gt

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2. Translation and rotation of graphics

Translation: ① In a plane, a figure moves a certain distance in a certain direction, and the movement of this figure is called translation. ② After translation, the line segments connected by the corresponding points are parallel and equal, the corresponding line segments are parallel and equal, and the corresponding angles are equal. & ltbr & gt

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Rotation: ① In a plane, a figure rotates an angle in a certain direction around a fixed point, and such a figure movement is called rotation. (2) After rotation, every point in the graphic library rotates by the same angle around the rotation center in the same direction, the angle formed by the connecting line of any pair of corresponding points and the rotation center is the rotation angle, and the distances from the corresponding points to the rotation center are equal. & ltbr & gt

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3. Similarity of graphics

Ratio: ①A/B=C/D, then AD=BC, and vice versa. ②A/B=C/D, then A soil B/B=C soil d/d. ③A/B=C/D=. . =M/N，& ltbr & gt

Then A+C+. . . +M/B+D+.。 . N=A/B .& ltbr & gt

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Golden section: Point C divides line AB into AC and BC. If AC/AB=BC/AC, then line segment AB is called golden section by point C, point C is called golden section of line segment AB, and the ratio of AC to AB is called golden section ratio (root number 5- 1/2). & ltbr & gt

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Similarity: ① Two polygons with equal angles and proportional sides are called similar polygons. ② The ratio of corresponding edges of similar polygons is called similarity ratio. & ltbr & gt

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Similar triangles: ① Two triangles with equal triangles and proportional sides are called similar triangles. ② Conditions: AA/SSS/SAS. & ltbr & gt

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The properties of similar polygons are as follows: ① The ratio of similar triangles to height, angle bisector and midline is equal to similarity ratio. ② The perimeter ratio of similar polygons is equal to the similarity ratio, and the area ratio is equal to the square of the similarity ratio. & ltbr & gt

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Zooming in and out of graphs: ① If two graphs are not only similar graphs, but also the straight lines of each group of corresponding points pass through the same point, then such two graphs are called similar graphs, and this point is called similarity center, and the similarity ratio at this time is also called similarity ratio. (2) The ratio of the distance between any pair of corresponding points on the potential diagram and the potential center is equal to the potential ratio. & ltbr & gt

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C: coordinates of the graph

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Plane rectangular coordinate system: On a plane, two mutually perpendicular axes with a common origin form a plane rectangular coordinate system. The horizontal axis is called X axis or horizontal axis, the vertical axis is called Y axis or vertical axis, X axis and Y axis are collectively called coordinate axes, and their common origin O is called the origin of rectangular coordinate system. They are divided into four quadrants. XA and YB are expressed as (a, b). & ltbr & gt

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Definitions and propositions: ① Describe the meaning of names and terms, make clear provisions, that is, give their definitions. (2) Sentences that judge things are called propositions (distinguishing between true and false propositions). ③ Each proposition consists of two parts: conditions and conclusions. (4) To explain that a proposition is false, it is usually necessary to give an ion to make it meet the conditions of the proposition, rather than the conclusion of the proposition. This example is called counterexample. & ltbr & gt

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Axiom: ① The accepted true proposition is called axiom. ② Verifying the correctness of other true propositions by reasoning, and the proved true propositions are called theorems. (3) The same angle is equal, two straight lines are parallel, and vice versa; SAS/ASA/SSS, and vice versa; The internal angles on the same side are complementary, and two straight lines; Parallel, and vice versa; Internal dislocation angles are equal, two straight lines are parallel, and vice versa; The sum of the three internal angles of a triangle is equal to 180 degrees; The diplomacy of a triangle is equal to the sum of two non-adjacent internal angles; The outer angle of the triangle center is greater than any inner angle that is not adjacent to it. A theorem directly derived from an axiom or theorem is called the inference of this axiom or theorem. & ltbr & gt

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Three. Statistics and probability

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1: Statistics

Scientific notation: numbers greater than 10 can be expressed as A* 10N, where 1 is less than or equal to a and less than 10, and n is a positive integer. & ltbr & gt

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Sector statistical chart: ① A circle is used to represent the population, and each sector in the circle represents a different part of the population. The size of the sector reflects the percentage of this part in the population. This kind of statistical chart is called departmental statistical chart. (2) In the sector statistical chart, the percentage of each part in the whole is equal to the ratio of the central angle of the sector corresponding to this part to 360 degrees. & ltbr & gt

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Advantages and disadvantages of various statistical charts: bar chart: the specific figures of each item can be clearly displayed; Broken line statistical chart: can clearly reflect the changes of things; Department statistical chart: it can clearly show the percentage of each part in the total. & ltbr & gt

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Approximate value and significant figures: ① The measurement result is approximate value. (2) When taking the divisor of a number by rounding method, it means that the divisor is rounded to the nearest place. (3) For a divisor, from the first number on the left that is not 0 to the most accurate number, all numbers are called the significant digits of this number. & ltbr & gt

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Average value: for n numbers, X 1, X2. . . XN, we put 1/N(X 1+X2+. . . +XN) is called the arithmetic average of the n numbers, and is recorded as x (a horizontal line above). & ltbr & gt

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Weighted average: the importance of each data in a set of data may be different, so when calculating the average value of this set of data, each data is often given a weight, which is the weighted average. & ltbr & gt

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Median and mode: ①N data are arranged in order of size, and the data in the middle position (or the average of the two data in the middle) is called the median of this group of data. ② The data with the highest frequency in a group of data is called the pattern of this group of data. Advantages and disadvantages: average: all data participate in the operation, which can make full use of the information provided by the data, so it is commonly used in real life, but it is easily affected by extreme values; Median: the calculation is simple, and it is less affected by extreme value, but it can't make full use of all data information; Pattern: when the number of repetitions of each data is roughly equal, the pattern often has no special meaning. & ltbr & gt

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Survey: ① A comprehensive survey of the respondents for a certain purpose is called a general survey, in which all the respondents are called the whole and each object that constitutes the whole is called an individual. (2) Select some individuals from the population for investigation, which is called sampling investigation, and select some individuals from the population as a sample of the population. Sampling survey only investigates a small number of individuals in the population, so it has the advantages of small survey scope and saving time, manpower, material resources and financial resources, but its survey results are often not as accurate as those obtained by census. In order to obtain more accurate survey results, the main samples should be representative and extensive when sampling. & ltbr & gt

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Frequency and frequency: ① frequency is the frequency of each object, and the ratio of the frequency of each object to the total frequency is the frequency. (2) When the collected data take values continuously, we usually group the data properly first, and then draw the histogram of frequency distribution. & ltbr & gt

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Data fluctuation: ① Range refers to the difference between the maximum data and the minimum data in a group of data. ② Variance is the average of the square of the difference between each data and the average. ③ The standard deviation is the arithmetic square root of variance. Generally speaking, the smaller the range, variance or standard deviation of a set of data, the more stable the set of data. & ltbr & gt

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2. Probability

Possibility: ① Some things that we can be sure will happen are called inevitable events; Some things we can be sure will not happen. These things are called impossible events. Inevitable events and impossible events are certain. There are many things that we are not sure will happen. These things are called uncertain events. (3) Generally speaking, the possibility of uncertain events is different. & ltbr & gt

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Probability: ① People usually use 1 (or 100%) to indicate the possibility of inevitable events and 0 to indicate the possibility of impossible events. The fairness of the game means that both sides have the same possibility of winning. (3) The probability of inevitable events is 1, and it is marked as p (inevitable events) =1; The probability of an impossible event is 0, and it is recorded as p (impossible event) = 0; If a is an uncertain event, then 0 < p (a) < 1.