★ Emphasis★ The related concepts and properties of real numbers, and the operation of real numbers.

☆ Summary ☆

I. Key concepts

The classification and concept of 1 figure

Digital series table:

Note: "Classification" principle: 1) Proportionality (no weight, no leakage)

2) There are standards

2. Non-negative number: the collective name of positive real number and zero. (Table: x≥0)

Common non-negative numbers are:

Property: If the sum of several non-negative numbers is 0, then every unburdened number is 0.

3. Reciprocity: ① Definition and characterization

② attribute: a.a ≠1/a (a ≠1); B. 1/a，a≠0； c . 0 < a < 1/a > 1； When a > 1,1/a <1; D. the product is 1.

4. Reciprocal: ① Definition and representation

② Properties: A.a when A.a≠0; The position of a and -a on the number axis; The sum of c is 0 and the quotient is-1.

5. Number axis: ① Definition ("three elements")

② Function: a. Visually compare real numbers; B. clearly reflect the absolute value; C. establish a one-to-one correspondence between points and real numbers.

6. Odd number, even number, prime number and composite number (positive integer-natural number)

Definition and expression:

Odd number: 2n- 1

Even number: 2n(n is a natural number)

7. Absolute value: ① Definition (two kinds):

Algebraic definition:

Geometric definition: the geometric meaning of the absolute value top of the number A is the distance from the point corresponding to the real number A on the number axis to the origin.

② A ≥ 0, and the symbol "│ │" is a sign of "non-negative number"; ③ There is only one absolute value of number A; ④ When dealing with any type of topic, as long as "│ │" appears, the key step is to remove the "│ │" symbol.

Second, the operation of real numbers

1. Arithmetic (addition, subtraction, multiplication, division, power and root)

2. Algorithm (five plus [multiplication] commutative law and associative law; [Multiplication versus Addition]

Distribution law)

3. Operation sequence: a. Advanced operation to low-level operation; B. (Operation at the same level) From "Left"

To the "right" (such as 5 ÷ 5); C (when there are brackets) from "small" to "medium" to "large".

Third, the application examples (omitted)

Attachment: Typical examples

1. It is known that the positions of a, b and x on the number axis are as follows. Please verify: │x-a│+│x-b│.

=b-a。

2. a-b=-2 and AB are known.

Chapter II Algebraic Formulas

★ Key points ★ Related concepts, properties and operations 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

Numbers or letters are 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. Monomial and Polynomial

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.

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 a ≥ 0-and "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.

Crossing out the root sign in the denominator is called denominator rationalization.

9. Index

(1)(- power supply, power supply operation)

(1) when a > 0, > 0; ② when a < 0, > 0 (n is even) and < 0 (n is odd)

(2) Zero index: = 1(a≠0)

Negative integer index: = 1/ (a≠0, 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 properties: = (m≠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 essence of power operation: ① =; ② ÷ = ; ③ = ; ④ = ; ⑤

Skills:

5. Multiplication rule: (1) single× single; (2) single × many; 3 more x more.

6. Multiplication formula: (plus or minus)

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

(a b) =

7. Division rules: (1) single-single; (2) Too many orders.

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

9. The nature of arithmetic roots: =; ; (a≥0，b≥0)； (a ≥ 0, b > 0) (positive and negative)

10. radical algorithm: (1) addition rule (merging similar quadratic roots); (2) multiplication and division; (3) The denominator is reasonable: a; b； c。

1 1. Scientific notation: (1 ≤ A < 10, n is an integer =

Third, the application examples (omitted)

Four, comprehensive operands (omitted)

Chapter III Preliminary Statistics

★ main points ★

☆ Summary ☆

I. Key concepts

1. population: all the respondents.

2. Individuals: each respondent in the group.

3. Sample: a part of individuals extracted from the population.

4. Sample size: the number of individuals in the sample.

5. Mode: the data with the highest frequency in a set of data.

6. Median: the number of a set of data in order of size (or the average of two data in the middle).

Second, the calculation method

1. Average sample: (1); (2) If,,,, then (A- constant,,,, is close to a more integer constant A); (3) Weighted average: (4) Average is a characteristic number that describes the trend (concentration position) in data concentration. Sample average is usually used to estimate the overall average. The larger the sample size, the more accurate the estimation.

2. Sample variance: (1); (2) If,,,,, then (a) is a relative "integer" constant, which is close to the average value of,,,; If … is less than "whole", then; ⑶ Sample variance is a characteristic number that describes the degree of data dispersion (fluctuation). When the sample size is large, the sample variance is very close to population variance, which is usually used to estimate population variance.

3. Sample standard deviation:

Third, the application examples (omitted)

Chapter IV Linetype

★ Emphasis★ Concepts, judgments and properties of intersecting lines and parallel lines, triangles and quadrangles.

☆ Summary ☆

I. Straight lines, intersecting lines and parallel lines

1. Differences and connections between line segments, rays and straight lines

This paper analyzes the graph, representation, boundary, number of endpoints and basic properties.

2. The midpoint of the line segment and its representation

3. Basic properties of straight lines and line segments (using "basic properties of line segments" to demonstrate that "the sum of two sides of a triangle is greater than the third side")

4. The distance between two points (three distances: point-point; Dotted line; Line-line)

5. Angle (flat angle, rounded corner, right angle, acute angle, obtuse angle)

6. Complementary angle, complementary angle and their expressions

7. The bisector of an angle and its representation

8. Vertical line and its basic properties (use it to prove that "the hypotenuse of a right triangle is greater than the right")

9. Vertex angle and its properties

10. Parallel lines and their judgments and properties (reciprocal) (differences and connections between them)

1 1. Common theorems: ① parallel to two straight lines and parallel to one straight line (transitivity); ② Two straight lines parallel to and perpendicular to a straight line.

12. Definition, proposition and composition of proposition

13. Axioms and theorems

14. Inverse proposition

Second, the triangle

Classification: (1) Classification by edge;

(2) according to the angle.

1. Definition (including internal angle and external angle)

2. The relationship between angles of triangle: (1) the sum and inference of angles and angles: (1) inner 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 the intersection of ②×× lines-the property of the× center of triangle.

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

⑴ General calculation formula ⑴ 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-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.

Third, quadrilateral.

Classification table:

1. General Properties (Angle)

⑴ Sum of internal angles: 360.

(2) Parallelogram connecting the midpoint of each side in turn.

Inference 1: Connect the midpoints of the sides of the quadrilateral in turn with equal diagonal lines to get a diamond.

Inference 2: Connect the midpoints of the sides of the quadrilateral in turn with diagonal lines perpendicular to each other to get a rectangle.

⑶ Sum of external angles: 360.

2. Special quadrilateral

(1) General methods to study them:

(2) parallelogram, rectangle, diamond and square; Definition, properties and judgment of trapezoid and isosceles trapezoid

⑶ Determination steps: quadrilateral → parallelogram → rectangle → square.

┗→ Diamond-=

(4) diagonal tie rod:

3. Symmetric graphics

(1) axis symmetry (definition and properties); (2) Central symmetry (definition and nature)

4. Related Theorems: ① Parallel bisection theorem and its inference 1, 2.

② The midline theorem of triangle and trapezoid.

③ The distance between parallel lines is equal everywhere. (For example, find triangles with equal areas in the figure below)

5. Important auxiliary lines: ① Diagonal lines of quadrangles are often connected; ② Trapezoids are often transformed into triangles by translating a waist, translating a diagonal, making a height, connecting the midpoint between the vertex and the waist and extending the intersection with the bottom.

6. Drawing: Divide the line segments randomly.

Four, application examples (omitted)

The fifth chapter equation (group)

★ Emphasis★ Solution of one-dimensional linear equation, one-dimensional quadratic equation and two-dimensional linear equations; Related application problems of the equation (especially travel and engineering problems)

☆ Summary ☆

I. Basic concepts

1. equation, its solution (root), its solution, its solution (group)

2. Classification:

Second, the basis of solving equation-the nature of equation

1.a=b←→a+c=b+c

2.a=b←→ac=bc (c≠0)

Third, the solution

1. Solution of linear equation with one variable: remove denominator → remove brackets → move terms → merge similar terms →

The coefficient becomes 1→ solution.

2. Solution of linear equations: ① Basic idea: "elimination method" ② Method: ① Replacement method.

② addition and subtraction

Fourth, a quadratic equation

1. Definition and general form:

2. Solution: (1) direct leveling method (pay attention to characteristics)

(2) Matching method (pay attention to the step-inferring the root formula)

(3) Formula method:

(4) factorization method (feature: left =0)

3. The discriminant of the root:

4. The relationship between root and coefficient top:

Inverse theorem: If, then the quadratic equation with one root is:.

5. Common equation:

5. Equations that can be transformed into quadratic equations

1. Fractional equation

(1) definition

(2) Basic ideas:

⑶ Basic solution: ① Denominator removal ② Substitution method (such as).

(4) Root test and method

2. Unreasonable equation

(1) definition

(2) Basic ideas:

(3) Basic solution: ① Multiplication method (pay attention to skills! ! (2) substitution method (example), (4) root test and method.

3. Simple binary quadratic equation

A binary quadratic equation consisting of a binary linear equation and a binary quadratic equation can be solved by method of substitution.

Six, column equation (group) to solve application problems

summary

Solving practical problems by using equations (groups) is an important aspect of integrating mathematics with practice in middle schools. The specific steps are as follows:

(1) review the questions. Understand the meaning of the question. Find out what is a known quantity, what is an unknown quantity, and what is the equivalent relationship between problems and problems.

⑵ Set an element (unknown). ① Direct unknowns ② Indirect unknowns (often both). Generally speaking, the more unknowns, the easier it is to list the equations, but the more difficult it is to solve them.

⑶ Use algebraic expressions containing unknowns to express related quantities.

(4) Find the equation (some are given by the topic, some are related to this topic) and make the equation. Generally speaking, the number of unknowns is the same as the number of equations.

5] Solving equations and testing.

[6] answer.

To sum up, the essence of solving application problems by column equations (groups) is to first transform practical problems into mathematical problems (setting elements and column equations), and then the solutions of practical problems (column equations and writing answers) are caused by the solutions of mathematical problems. In this process, the column equation plays a role of connecting the past with the future. Therefore, the column equation is the key to solve the application problem.

Two commonly used equality relations

1. Travel problem (uniform motion)

Basic relationship: s=vt

(1) Meeting problem (at the same time):

+ = ;

(2) Follow-up questions (start at the same time):

If A starts in t hours, B starts, and then catches up with A at B, then

(3) sailing in the water:

2. batching problem: solute = solution × concentration

Solution = solute+solvent

3. Growth rate:

4. Engineering problems: Basic relationship: workload = working efficiency × working time (workload is often considered as "1").

5. Geometric problems: Pythagorean theorem, area and volume formulas of geometric bodies, similar shapes and related proportional properties.

Third, pay attention to the relationship between language and analytical formula.

Such as more, less, increase, increase to (to), at the same time, expand to (to), ...

Another example is a three-digit number, where A has 100 digits, B has 10 digits and C has one digit. Then this three-digit number is: 100a+ 10b+c, not abc.

Fourth, pay attention to writing equal relations from the language narrative.

For example, if X is greater than Y by 3, then x-y=3 or x=y+3 or X-3 = Y, and if the difference between X and Y is 3, then x-y=3. Pay attention to unit conversion

Such as the conversion of "hours" and "minutes"; Consistency of s, v and t units, etc.

Seven, application examples (omitted)

Chapter VI One-dimensional Linear Inequalities (Groups)

★ Essentials ★ Properties and Solutions of One-variable Linear Inequality

☆ Summary ☆

1. Definition: A > B, A < B, a≥b, a≤b, A ≠ B.

2. Unary linear inequality: ax > b, ax < b, ax≥b, ax≤b, ax≠b(a≠0).

3. One-dimensional linear inequalities;

4. the essence of inequality: (1) a > b ←→ a+c > b+c

⑵a & gt； b←→AC & gt； BC (c>0)

⑶a & gt； b←→AC & lt； BC (c<0)

(4) (transitivity) a>b, b & gtc→a & gt；; c

⑸a & gt； b，c & gtd→a+c & gt； b+d。

5. Solution of one-dimensional linear inequality, solution of one-dimensional linear inequality

6. Solution of one-dimensional linear inequality group, solution of one-dimensional linear inequality group (representing the solution set on the number axis)

7. Application examples (omitted)

Chapter VII Similarity

★ Focus★ similar triangles's judgment and nature

☆ Summary ☆

First of all, there are two sets of theorems in this chapter.

The first group (proportion of related properties):

Concepts involved: ① The fourth proportional item ② The front and back items of the third proportional item, the inner item and the outer item ④ The golden section, etc.

The second episode:

Note: ① the meaning of the word "correspondence" in the theorem;

② Parallel → Similar (proportional line segment) → Parallel.

Second, the nature of similar triangles

1. The corresponding line segment ...; 2. Corresponding circumference ...; 3. Corresponding areas ...

Third, correlation mapping.

(1) as the fourth proportion; (2) As a proportional term.

Four, card (solution) problem method, auxiliary line

1. Change "equal product" to "proportion" and find "similarity" in "proportion".

2. If you can't find similarity, find the middle proportion. Methods: Express the ratio of the left and right sides of the equation. ⑴

⑵

⑶

3. Adding auxiliary parallel lines is an important way to obtain proportional line segments and similar triangles.

4. The common method to deal with the ratio problem is to look at K; For the equal ratio problem, the common solution is to set the "common ratio" to K.

5. For complex geometric figures, the method of "extracting" some needed figures (or basic figures) is adopted.

Application examples of verbs (abbreviation of verb) (omitted)

Chapter 8 Functions and Their Images

★ Emphasis★ Positive and negative proportional functions, images and properties of linear and quadratic functions.

☆ Summary ☆

First, the plane rectangular coordinate system

1. Coordinate characteristics of points in each quadrant

2. Coordinate characteristics of each point on the coordinate axis

3. About the characteristics of coordinate axis and symmetry point.

4. The corresponding relationship between points on the coordinate plane and ordered real number pairs.

Second, function

1. Representation method: (1) analysis method; (2) List method; (3) Image method.

2. The principle of determining the range of independent variables: (1) makes algebraic expressions meaningful; (2) The practical problems in manufacturing are as follows

Meaning.

3. Draw a function image: (1) list; (2) tracking points; (3) connection.

Third, several special functions.

(Definition → Image → Attribute)

1. proportional function

⑴ definition: y=kx(k≠0) or y/x = k.

⑵ image: straight line (through the origin)

⑶ nature: ① k > 0，…②k & lt； 0,…

2. Linear function

⑴ definition: y=kx+b(k≠0)

⑵ Image: The straight line passes through the intersection of point (0, b)- and Y axis and the intersection of point (-b/k, 0, b)- and X axis.

⑶ nature: ① k > 0，…②k & lt； 0,…

(4) Four situations of images:

3. Quadratic function

(1) Definition:

In particular, they are all quadratic functions.

⑵ Image: parabola (tracing points: first determine the vertex, symmetry axis and opening direction, and then trace points symmetrically). If the configuration method is changed to, the vertex is (h, k); The symmetry axis is a straight line x = h；; A>0, the opening is upward; A<0, opening down.

⑶ Nature: a>0, on the left and right side of the symmetry axis; A<0, on the left … and right … of the symmetry axis.

4. Inverse proportional function

⑴ Definition: or xy=k(k≠0).

⑵ Image: hyperbola (two branches)-drawn by tracing points.

⑶ nature: ① k > 0, the image is at …, y follows x …; ②k & lt； 0, the image is at …, y follows x …; ③ Two curves are infinitely close to the coordinate axis but can never reach the coordinate axis.

Fourth, important problem-solving methods

1. Use the undetermined coefficient method to find the analytical formula (solving the sequence equation [group]). To find the analytic formula of quadratic function, we should reasonably choose the general formula or vertex type, make full use of the characteristics of parabola about the axis of symmetry, and find the coordinates of new points. As shown in the figure below:

2. K and B represent the linear (proportional) function, inverse proportional function and quadratic function in the image; The symbols of a, b and C.

Six, application examples (omitted)

Chapter 9 Solving Right Triangle

★ Key points ★ Solving right triangle

☆ Summary ☆

First, trigonometric functions

1. definition: in Rt△ABC, ∠C = Rt∞, then sinA =；; cosA =； tgA =； ctgA=。

2. The trigonometric function value of special angle:

0 30 45 60 90

sinα

Coase α

tgα /

ctgα /

3. The trigonometric function relation of two complementary angles: sin (90-α) = cos α; …

4. The relationship between trigonometric function value and angle change.

5. Look up the trigonometric function table

Second, solve the right triangle.

1. Definition: known edges and angles (two of which must have one side) → all unknown edges and angles.

2. Basis: ① Relationship between edges:

② Angle relation: A+B = 90.

③ Angular relation: the definition of trigonometric function.

Note: Try to avoid using intermediate data and division.

Third, the handling of practical problems.

1. Pitch and elevation: 2. Azimuth and quadrant angle: 3. Slope:

4. When both right triangles lack the conditions for solving right triangles, they can be solved by column equations.

Four, application examples (omitted)

Chapter 10 Circle

★ Key points ① Important properties of the circle; (2) the positional relationship between straight lines and circles, and between circles; ③ Angle theorem related to circle; ④ Theorem of proportional line segment related to circle.

☆ Summary ☆

First, the basic properties of the circle

Definition of 1. circle (two kinds)

2. Related concepts: chord and diameter; Arc, equal arc, upper arc, lower arc, semicircle; Distance from chord to center; Equal circle, same circle, concentric circle.

3. "Three-point circle" theorem

4. Vertical Diameter Theorem and Its Inference

5. "Equivalence" theorem and its inference

5. Angle related to circle: (1) Definition of central angle (equivalence theorem)

(2) the definition of the angle of circle (the theorem of the angle of circle, the relationship with the angle of center)

⑶ Definition of chord angle (chord angle theorem)

Second, the positional relationship between a straight line and a circle

1. Three positions and their judgments and properties:

2. The nature of the tangent (key point)

3. Judgment theorem of tangent (key point). The determination of the tangent of a circle includes (1)...(2) ...

4. Tangent length theorem

Third, the position relationship between circles.

1. Five positional relationships and their judgments and properties: (emphasis: tangency)

2. The property theorem of the tangent (intersection line) connecting two circles.

3. Common tangent of two circles: (1) Definition (2) Property

Four, proportional line segment related to the circle

1. Intersecting chord theorem

2. Cutting line theorem

Verb (abbreviation for verb) and regular polygon

The inscribed and circumscribed polygons of 1. circle (triangle, quadrilateral)

2. The circumscribed circle, inscribed circle and properties of triangle.

3. The properties of circumscribed quadrangles and inscribed quadrangles of a circle

4. Regular polygon and its calculation

Central angle:

Half of the inner corner: (right)

(Related elements can be found by solving Rt△OAM, etc. )

Six, a set of calculation formulas

1. circumference formula

2. Formula of circular area

3. Sector area formula

4. Arc length formula

5. Calculation method of arch area

6. The side development diagram of cylinder and cone and related calculation.

Seven, the trajectory of the point

Six basic trajectories

Eight, drawings.

1. Draw the circumscribed circle and inscribed circle of the triangle.

2. Divide known arcs equally

3. Find the median ratio of two known line segments.

4. Equal circumference: 4, 8; 6, 3 equal parts

Nine, the basic graphics

Important auxiliary line

1. Production radius

2. Chords are usually regarded as the distance from the center of the chord.

3. See the diameter is usually used as the circumferential angle on the diameter.

4. Don't forget to connect the center of the tangent point

5. Common tangent of two circles (connecting line)

6. Two circles intersect a common chord.