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 absolute value of the number A 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)

Mathematics knowledge points of senior high school entrance examination

1, the concept of inverse proportional function

Generally speaking, a function (k is a constant and k0) is called an inverse proportional function. The analytical expression of inverse proportional function can also be written as. The value range of the independent variable X is all real numbers of x0, and the value range of the function is also all non-zero real numbers.

2. Inverse proportional function image

The image of the inverse proportional function is a hyperbola, which has two branches, which are located in the first and third quadrants, or the second and fourth quadrants respectively, and they are symmetrical about the origin. Because of the independent variable x0 and the function y0 in the inverse proportional function, its image does not intersect with the X axis and the Y axis, that is, the two branches of the hyperbola are infinitely close to the coordinate axis, but they will never reach the coordinate axis.

3. The properties of inverse proportional function

Symbol k of inverse proportional function k >: 0k <; The value range of 0 image yO xyO x attribute ①x is x0,

The value range of y is y0;

2 when k >; 0, the two branches of the function image are

In the first and third quadrants. In each quadrant, y

It decreases with the increase of x.

① The value range of x is x0,

The value range of y is y0;

② when k

In the second and fourth quadrants. In each quadrant, y

It increases with the increase of x.

4. Determination of inverse proportion of resolution function.

The method of determining sum is, er, the undetermined coefficient method. Because there is only one undetermined coefficient in the inverse proportional function, only a pair of corresponding values or the coordinates of a point on the image can be used to find the value of k, thus determining its analytical formula.

5. Geometric meaning of inverse proportional function

Let it be any point in the inverse proportional function image, the intersection point P is the vertical line between the axis and the axis, and the vertical foot is A, then

(1) △ the area of OPA.

(2) The area of rectangular OAPB. This is the geometric meaning of the coefficient. No matter how P moves, the area of △OPA and the area of rectangular OAPB remain unchanged.

Rectangular PCEF area =, parallelogram PDEA area =

Mathematical knowledge points in quadratic function examination

The analytic expression of quadratic function has three forms:

(1) general formula:

(2) Vertex:

(3) When the parabola intersects the X axis, that is, the corresponding quadratic good equation has real roots and exists, the quadratic function can be transformed into two equations according to the factorization of quadratic trinomial. If there is no intersection, you can't express it like this.

Note: The position of parabola is determined by.

(1) Determines the opening direction of the parabola.

① The opening is upward.

② The opening is downward.

(2) Determine the intersection position of parabola and Y axis.

① The intersection of the image and the Y axis is above the X axis.

② The image passes through the origin.

③ The intersection of the image and the Y axis is below the X axis.

(3) Determine the position of the parabola axis of symmetry (axis of symmetry:)

① The symmetry axis of the same symbol is on the left side of the Y axis.

② The axis of symmetry is the Y axis.

③ The symmetry axes of different symbols are on the right side of the Y axis.

(4) Vertex coordinates.

(5) Determine the intersection of parabola and X axis.

①△& gt； Parabola and x axis have two different intersections.

②△=0 Common point of parabola and X axis (tangent).

③△& lt； Parabola and x axis have nothing in common.

(6) Use A to judge whether the quadratic function has the minimum value.

① when a >; 0, the parabola has the lowest point and the function has the minimum value.

② when a

(7) the symbol of judgment:

Expression, please substitute the value, and the corresponding y value is positive and negative;

Symmetry axis has many uses, and three formulas satisfy it;

Judging from both sides of the axis, the left side is the same as the right side, and the difference is 0;

1 Judging from both sides, the left-right difference is 0;

-1 judging from both sides, the left side is different and the right side is the same.

(8) Function image translation: left-right translation becomes X, left+right-; Shift the constant term up and down, up+down-; Knowing the translation results first, reverse translation is the trick; I don't know the way of translation, but find it through the vertex.

(9) Symmetry: the analytical formula about the X axis symmetry is, the analytical formula about the Y axis symmetry is, the analytical formula about the origin axis symmetry is, and the analytical formula after the vertex is folded is (A is opposite, the fixed-point coordinates are unchanged).

(10) Conclusion: ① Quadratic function (there is only one intersection with X axis, and the vertex of quadratic function is δ = 0 on X axis;

(2) The vertex of the quadratic function is on the Y axis, and the image of the quadratic function is symmetrical about the Y axis;

③ Quadratic function (after the origin, then.

Analytical formula of (1 1) quadratic function;

① General formula: (,used to understand three points.

(2) Vertex: used to know the vertex coordinates or maximum or symmetry axis.

(3) Intersection point:, where is the abscissa of the two intersections of the quadratic function and the X axis. This formula can also be used if the symmetry axis and the intercept on the X axis are known.

The arrangement of mathematics knowledge points in grade three is 2 knowledge points 1. concept

Figures with the same shape are called similar figures. (i.e. graphs with equal corresponding angles and equal corresponding edge ratios)

Interpretation: (1) Two graphs are similar, and one graph can be seen as being enlarged or reduced by the other graph.

(2) Conformity can be regarded as a special similarity, that is, not only the same shape, but also the same size.

(3) To judge whether two graphs are similar is to see whether the shapes of the two graphs are the same, which has nothing to do with other factors.

Knowledge point 2. Proportional line segment

For four line segments A, B, C and D, if the ratio of the lengths of two of them is equal to the ratio of the lengths of the other two line segments, that is (or A: B = C: D), then these four line segments are simply called proportional line segments.

Knowledge point 3. Properties of similar polygons

Properties of similar polygons: the corresponding angles of similar polygons are equal, and the proportions of corresponding edges are equal.

Interpretation: (1) Understand the definition of similar polygons correctly and make clear the "corresponding" relationship.

(2) It is clear that the "correspondence" of similar polygons comes from writing, and the similarity ratio is sequential.

Knowledge point 4. Similar triangles's concept

A triangle with equal corresponding angles and equal ratio of corresponding sides is called similar triangles.

Interpretation: (1) similar triangles is one of the similar polygons;

(2) similar triangles should be understood by combining the properties of similar polygons;

(3) similar triangles should have the same shape, but different sizes;

(4) Similarity is indicated by "√" and pronounced as "similar to";

(5) The ratio of corresponding sides in similar triangles is called similarity ratio.

Knowledge point 5. Similar triangles's judgment method

(1) Definition: Two triangles with equal corresponding angles and proportional corresponding sides are similar;

(2) The triangle formed by cutting other two sides (or extension lines of other two sides) with a straight line parallel to one side of the triangle is similar to the original triangle.

(3) If two angles of a triangle are equal to two angles of another triangle, then the two triangles are similar.

(4) If two sides of a triangle are proportional to two sides of another triangle, and the included angles are equal, then the two triangles are similar.

(5) If three sides of a triangle are proportional to three sides of another triangle, then the two triangles are similar.

(6) The right triangle is divided into two right triangles by the height on the hypotenuse, which is similar to the original triangle.

Knowledge point 6. The nature of similar triangles

(1) The corresponding angles are equal, and the ratio of the corresponding sides is equal;

(2) The ratio corresponding to the height, the ratio corresponding to the center line and the ratio corresponding to the angular bisector are all equal to the similarity ratio;

(3) The ratio of similar triangles perimeter is equal to the similarity ratio; The area ratio is equal to the square of the similarity ratio.

(4) Projective theorem

Sort out the three triangles of mathematics knowledge points in grade three.

Classification: (1) Classification by edge;

(2) according to the angle.

1. Definition (including internal angle and external angle)

2. Relationship between angles of triangle: ⑴ Angle and angle :⑴Sum and inference of internal angles; ② sum of external angles; (3) the sum of the internal angles of the N-polygon; (4) the sum of the external angles of the N-polygon. ⑵ Edge and edge: The sum of two sides of a triangle is greater than the third side, and the difference between the two sides is less than the third side. ⑶ Angle and edge: In the same triangle,

3. The main part of the triangle

Discussion: ① Define ② the nature of triangle center at the intersection of straight lines ③.

① High line ② Middle line ③ Angle bisector ④ Middle vertical line ⑤ Middle line.

⑵ General triangle ⑵ Special triangle: right triangle, isosceles triangle and equilateral triangle.

4. Determination and properties of special triangles (right triangle, isosceles triangle, equilateral triangle and isosceles right triangle)

5. congruent triangles

(1) Determine the consistency of general triangles (SAS, ASA, AAS, SSS).

⑵ Determination of congruence of special triangle: ① General method ② Special method.

6. Area of triangle

⑵ General calculation formula ⑵ Properties: The areas of triangles with equal bases and equal heights are equal.

7. Important auxiliary lines

(1) The midpoint and the midpoint form the midline; (2) Double the center line; (3) Add auxiliary parallel lines

8. Proof method

(1) direct proof method: synthesis method and analysis method.

(2) Indirect proof by reduction to absurdity: ① Counterhypothesis ② Reduction to absurdity ③ Conclusion.

(3) Prove that line segments are equal and angles are equal, often by proving triangle congruence.

(4) Prove the folding relationship of line segments: folding method and folding method.

5. Prove the sum-difference relationship of line segments: continuation method and truncation method.

[6] Prove the area relationship: indicate the area.

The arrangement of mathematics knowledge points in grade three is a linear equation of 4 yuan;

(1) In an equation, there is only one unknown, and the exponent of this unknown is

1, such an equation is called a 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:

Denominator is removed, items are shifted, similar items are merged, and the unknown coefficient is changed to 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.

2. Inequality and unequal groups

Inequality:

Formulas connected by "=" symbols are called inequalities.

② Add or subtract the same algebraic expression on both sides of the inequality, and the direction of the inequality remains unchanged.

③ Both sides of inequality are multiplied or divided by a positive number, and the direction of inequality remains unchanged.

④ Both sides of inequality are multiplied or divided by the same negative number, and the unequal numbers are in opposite directions.

Solution set of inequality;

(1) can make the value of an unknown inequality known as the solution of inequality.

(2) All solutions of an inequality with unknowns constitute the solution set of this inequality.

③ The process of finding the solution set of inequality is called solving inequality.

One-dimensional linear inequality: an inequality with algebraic expressions on both sides and only one unknown number of degree 1 is called one-dimensional linear inequality.

One-dimensional linear inequality system;

(1) Several linear inequalities about the same unknown quantity are combined into a linear inequality group.

② The common part of the solution set of each inequality in a linear inequality group is called the solution set of this linear inequality group.

③ The process of finding the solution set of inequality group is called solving inequality group.

3. Function

Variable: dependent variable, independent variable. When using images to represent the relationship between variables, we usually use points on the horizontal axis as independent variables and points on the vertical axis as dependent variables.

Linear function:

(1) If the relationship between two variables X and Y can be expressed as y = kx+b (where B is constant and K is not equal to 0), then Y is said to be a linear function of X..

When B=0, y is said to be a proportional function of X. ..

Linear function image:

① Take the values of the independent variable X and the corresponding dependent variable Y of a function as the abscissa and ordinate of a point respectively, and trace the corresponding point in the rectangular coordinate system. A graph composed of all these points is called an image of a function.

② The image with the proportional function Y=KX is a straight line passing through the origin.

③ In a linear function, when k < 0 and b < 0, it passes through 234 quadrants; When k < 0, b > 0, pass 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.

Space and graphics

Understanding of graphics:

1, point, line, surface

Points, lines and faces:

① Graphics are composed 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.

Expand and collapse:

(1) In a prism, the intersection of any two adjacent faces is called an edge, and a side is the intersection of two adjacent edges. All sides of the prism have the same length, the upper and lower bottom surfaces of the prism have the same 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, sector:

(1) A graph 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.

corner

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 representation of angles;

The (1) 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.

Angle comparison:

The angle (1) 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.

Parallel:

(1) In the same plane, two disjoint straight lines 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.

Vertical:

Two straight lines are perpendicular to each other if they intersect at right angles.

(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.

2. 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.

The concept and nature of five key algebraic expressions in the arrangement of mathematics knowledge points in grade three, and the operation of algebraic expressions

☆ Summary ☆

I. Key concepts

Classification:

1. Algebras and Rational Expressions

Formulas that associate numbers or letters representing numbers with operational symbols are called algebraic expressions. independent

The number or letter of is also an algebraic expression.

Algebraic expressions and fractions are collectively called rational forms.

2. Algebraic expressions and fractions

Algebraic expressions involving addition, subtraction, multiplication, division and multiplication are called rational expressions.

Rational expressions without division or division but without letters are called algebraic expressions.

Rational number formula has division, and there are letters in division, which is called fraction.

3. Monomial and Polynomial

Algebraic expressions without addition and subtraction are called monomials. (The product of numbers and letters includes a single number or letter)

The sum of several monomials is called polynomial.

Note: ① According to whether there are letters in the division formula, algebraic expressions and fractions are distinguished; According to whether there are addition and subtraction operations in algebraic expressions, monomial and polynomial can be distinguished. ② When classifying algebraic expressions, the given algebraic expressions are taken as the object, not the deformed algebraic expressions. When we divide the category of algebra, we start from the representation. For example,

=x, =│x│ and so on.

4. Coefficients and indices

Difference and connection: ① from the position; (2) In the sense of representation.

5. Similar projects and their combinations

Conditions: ① The letters are the same; ② The indexes of the same letters are the same.

Basis of merger: law of multiplication and distribution

6. Radical form

The algebraic expression of square root is called radical.

Algebraic expressions that involve square root operations on letters are called irrational expressions.

Note: ① Judging from the appearance; ② Difference: It is a radical, but it is not an irrational number (it is an irrational number).

7. Arithmetic square root

(1) The positive square root of a positive number (the difference between 0 and the square root);

⑵ Arithmetic square root and absolute value

① Contact: all are non-negative, =│a│.

② Difference: │a│, where A is all real numbers; Where a is a non-negative number.

8. Similar quadratic root, simplest quadratic root, denominator of rational number.

After being transformed into the simplest quadratic root, the quadratic roots with the same number of roots are called similar quadratic roots.

The following conditions are satisfied: ① the factor of the root sign is an integer and the factor is an algebraic expression; (2) The number of roots does not include exhausted factors or factors.

Scraping the root sign from the denominator is called denominator rationalization.

9. Index

(1) (power supply, power supply operation)

①0, ②a0, 0(n is even), 0(n is odd)

⑵ Zero index: = 1(a0)

Negative integer index: = 1/0, where p is a positive integer)

Second, the law of operation and the law of nature

1. The law of addition, subtraction, multiplication, division, power and root of fractions.

2. The nature of the score

(1) Basic attribute: =0)

(2) Symbolic law:

⑶ Complex fraction: ① Definition; ② Simplified methods (two kinds)

3. Algebraic expression algorithm (bracket deletion and bracket addition)

4. The operational nature of power: ① = ② = ③ = ④ = ⑤.

Skills:

5. Multiplication rule: (1) single (2) single (3) multiple.

6. Multiplication formula: (plus or minus)

(a+b)(a-b)= 1

(ab)= 1

7. Division rule: (1) single (2) multiple single.

8. Factorization: (1) definition; ⑵ Methods: A. Common factor method; B. formula method; C. cross multiplication; D. group decomposition method; E. root formula method.

9. Properties of arithmetic roots: =0, b0, b0) (positive and negative use)

10. radical algorithm: (1) addition rule (merging similar quadratic roots); (2) multiplication and division; (3) Denominators are physics and chemistry: A.B.C ..

1 1. Scientific notation: a 10, where n is an integer =

Third, the application examples (omitted)

Four, comprehensive operands (omitted)

Sort out six binary linear equations of mathematics knowledge points in grade three.

1. Definition: An integral equation with two unknowns whose degree is 1 is called a binary linear equation.

2. The solution of binary linear equations

(1) substitution method

The equations composed of a quadratic equation and a linear equation are usually solved by method of substitution, which is the basic method of elimination and simplification.

(2) Factor decomposition method

In binary quadratic equation, when at least one equation can be decomposed, it can be solved by factorization, eliminating elements and reducing order.

(3) Matching method

A formula, or a part of a formula, is transformed into a completely flat road or the sum of several completely flat roads through constant deformation.

(4) Vieta's law.

Through the inverse theorem of Vieta's theorem, we can use the sum-product relation of two numbers to construct a quadratic equation with one variable.

(5) Constant elimination method

When the first term is missing in both equations, it can be solved by eliminating the constant term.

Solve a quadratic equation with one variable

The basic idea of solving quadratic equations with one variable is to simplify them into two quadratic equations with one variable.

1, direct Kaiping method:

The equation with the shape of (x-m) 2 = n (n ≥ 0) is solved by direct Kaiping method, and its solution is x = m.

Direct Kaiping method is the inverse operation of square. The result of its operation is usually represented by a root sign.

2. Matching method

The method of obtaining the root of a quadratic equation by matching it completely flat. The solution of this unary quadratic equation is called collocation method, and the formula is based on the complete square formula.

(1) transformation: transform the quadratic equation of one variable into the form of ax 2+bx+c = 0 (that is, the general form of the quadratic equation of one variable).

(2) Coefficient 1: convert the quadratic term into 1.

(3) Move the term: move the constant term to the right of the equal sign.

(4) Formula: The left and right sides of the equal sign are added with the square of half of the first coefficient.

(5) Deformation: Write the algebraic expression on the left of the equal sign as a complete square.

(6) Square root: simultaneous square root.

(7) Solution: The root of the original equation can be obtained by sorting.

3. Formula method

Formula method: Convert the quadratic equation of one variable into a general form, and then calculate the value of the discriminant △ = B2-4ac. When B2-4ac ≥ 0, substitute the values of the coefficients A, B and C into the formula X = (B2-4ac ≥ 0) to get the root of the equation.

algebraic expression

1, algebraic formula and rational formula

Formulas that associate numbers or letters representing numbers with operational symbols are called algebraic expressions. A single number or letter is also algebraic.

Algebraic expressions and fractions are collectively called rational forms.

2. Algebraic expressions and fractions

Algebraic expressions involving addition, subtraction, multiplication, division and multiplication are called rational expressions.

Rational expressions without division or division but without letters are called algebraic expressions.

Rational number formula has division, and there are letters in division, which is called fraction.

3, monomials and polynomials

Algebraic expressions without addition and subtraction are called monomials. (product of numbers and letters-including single numbers or letters)

The sum of several monomials is called polynomial.

Description:

(1) Distinguish algebraic expressions from fractional expressions according to whether there are letters in the division method; According to whether there are addition and subtraction operations in algebraic expressions, monomial and polynomial can be distinguished.

② When classifying algebraic expressions, the given algebraic expressions are taken as the object, not the deformed algebraic expressions.

4, similar projects and their merger

Conditions: ① The letters are the same; ② The indexes of the same letters are the same.

Basis of merger: law of multiplication and distribution.