abstract
I. Key concepts
1. Table of numerical classification and conceptual number system;
Note: the principle of "classification":
(1) ratio (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: Nature: If the sum of several non-negative numbers is 0, then each non-burden number is 0.
3. Countdown:
① Definition and representation
② properties: a. (a ≠1);
B. medium, a ≠ 0;
C.0<a< in 1, >1; A> when 1, <1;
The product of D.a and is 1.
4. The opposite number:
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(except 0).
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 (2):
Algebraic definition: the absolute value of positive numbers and 0 is itself, and the absolute value of negative numbers is its inverse.
The absolute values of two opposite numbers are equal.
The absolute value of a is represented by |a |, which is read as "the absolute value of a".
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; Distribution law of multiplication to addition)
3. Operation sequence:
A. high-level operation to low-level operation;
B. (Operation at the same level) From "left" to "right" (for example, 5 ÷ 5);
C (when there are brackets) from "small" to "medium" to "large".
Third, the application examples (omitted)
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.
abstract
I. Classification of important concepts:
1. Algebras and Rational Expressions
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. Monomial and Polynomial
Algebraic expressions without addition and subtraction are called monomials.
(The product of numbers and letters-including a single number or letter) The sum of several monomials is called a 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. When we divide the category of algebra, we start from the representation. For example =x, =│x│ and so on.
4. The difference and connection between coefficient and index:
(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 exponential combination of the same letters is based on the same principle: the 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: radical, but not irrational (irrational).
7. Arithmetic square root
(1) the positive square root of a positive number;
⑵ Arithmetic square root and absolute value
① Contacts: all are non-negative, =│a│.
② Difference: │a│, where A is all real numbers; Where a is a non-negative number.
8. The same kind of secondary roots, after rationalizing the simplest secondary roots and denominator into the simplest secondary roots, the secondary roots with the same number of roots are called the same kind of secondary roots.
meet a condition
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 and power supply operation ① a > 0，an > 0； ②a & lt； 0, an & gt0(n is an even number), an < 0(n is an odd number)
(2) Zero index: = 1(a≠0)
(3) Negative integral index: (a≠0)
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
5. Multiplication rule: (1) single× single; (2) single × many; 3 more x more.
6. Multiplication formula: (plus or minus) (A+B) (A-B) = (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。
Third, the application examples (omitted)
Four, comprehensive operand (omitted) ★ key ★
abstract
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. Sample average:
⑴ ; (2) If,,,, then (A- constant,,,, is close to a more integer constant A);
(3) Weighted average:
(4) The average is the characteristic number that describes the trend (concentration position) in the data set. 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. Standard deviation of samples
Three. Application examples (omitted) ★ Key points ★
1 The concept, judgment and properties of intersection line and parallel line, triangle and quadrilateral.
abstract
I. Straight lines, intersecting lines and parallel lines
1. This paper analyzes the differences and relations among line segments, rays and straight lines from the aspects of graphs, representations, boundaries, the 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 2. Triangle classification: (1) divided by edge; (2) according to the angle.
Second, the triangle
1. Definition (including internal angle and external angle)
2. Angle relation of triangle:
(1) Angle and angle:
① Sum 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 the intersection of ②×× lines-the nature of triangle ③ High line ② Midline ③ Angular bisector ④ Midline ⑤ General triangle ② Special triangle: right triangle, isosceles triangle and equilateral triangle.
Note: the center of gravity of the triangle-the intersection of the center lines.
4. Determination and properties of special triangles (right triangle, isosceles triangle, equilateral triangle and isosceles right triangle)
5. congruent triangles ① congruence judgment of general triangles (SAS, ASA, AAS, SSS) ② congruence judgment of special triangles: ① general methods ② special methods.
6. The area of triangle (1) General calculation formula (2) Properties: The areas of triangles with equal bases and equal heights are equal.
7. The midpoint of the important auxiliary line (1) cooperates with the midpoint to 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) inverse hypothesis; (2) reduction to absurdity; (3) conclusion; (3) Equivalence between line segments and angles is often proved by triangle congruence.
(4) Prove the doubling relation of line segments: doubling method and half method; (5) Prove the sum and difference relation of line segments: delay method and truncation method; (6) Prove the area relation: express the area.
Third, quadrilateral.
Classification table: 1. The parallelogram (2) with the general property (angle) (1) and an internal angle of: 360 degrees connects the midpoints 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) The general methods to study them are: (2) parallelogram, rectangle, diamond and square; Definition, properties and judgment of trapezoid and isosceles trapezoid (3) Judgment steps: quadrilateral → parallelogram → rectangle → square ┗→ diamond —↑ (4) Diagonal system:
3. Symmetric graph (1) axis symmetry (definition and properties); (2) Central symmetry (definition and nature)
4. Relevant theorems: ① the bisection theorem of parallel lines and its inference 1, 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.
Fourth, application examples (omitted) ★ focus ★ the solution of linear equations, quadratic equations and binary linear equations; Related application problems of the equation (especially travel and engineering problems)
abstract
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 → shift terms → merge similar terms → coefficient becomes 1→ solution.
2. Solution of linear equations: ① Basic idea: "elimination method" ② Method: ① Substitution method ② Addition and subtraction method.
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:
1. Transform the equation into the general formula ax2-bx+c=0.
2. Determine the discriminant and calculate B2-4ac；; ;
3. If B2-4ac >; 0, substituted into the formula;
If B2-4ac
If b2-4ac=0, the equation has a unique solution in the real number domain.
(4) factorization method (feature: left =0)
3. The discriminant of the root:
4. The relationship between the root and the coefficient top: Inverse Theorem: If, then the quadratic equation of one variable has a root of:.
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.
(4) Root test and method
3. A simple binary quadratic equation composed of a binary linear equation and a binary quadratic equation can be solved by substitution method.
Six, column equation (group)
Solving application problems 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 unknown
(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.
Ordinary equality relationship
1. Travel problem (uniform motion) Basic relationship: S = vt (1) Meeting problem (simultaneous departure):+=;
(2) Catch up with the problem (set out at the same time): If A sets out after t hours, B sets out, and then catches up with A at B, then (3) sail in the water:
2. batching problem: solute = solution × concentration solution = solute+solvent
3. The question of 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.
Pay attention to the relationship between language and analytical formula.
For example, "more", "less", "increase", "increase to (to)", "at the same time" and "expand to) ... Another example is a three-digit number, where A has hundreds, B has tens and C has tens, so this three-digit number is: 6544.
Pay attention to writing equal relations from language narration.
. 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 "hours" and "minutes"; Consistency of s, v and t units, etc.
VII. Application Examples (omitted) ★ Emphasis ★ Properties and Solutions of One-variable Linear Inequality
abstract
1. Definition: a >；; b、a & ltb、a≥b、a≤b、a≠b .
2. One-dimensional linear inequalities: 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+ c⑵a & gt； b←→AC & gt； (c>0) BC (3) a > b←→AC & lt； (c & lt0) (4) BC (transitivity) a>b, b & gtc→a & gt；; 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) ★ Focus ★ similar triangles's judgment and nature
abstract
First of all, there are two sets of theorems in this chapter.
The first set (related properties of proportion): involves concepts: ① the fourth proportion item ② the front and back items of the proportion item ③, the internal and external items ④ the golden section, etc.
The second set: attention: ① the meaning of the word "correspondence" in the theorem; ② Parallel → Similar (proportional line segment) → Parallel.
Similar triangles attribute
1. The corresponding line segment ...;
2. Corresponding circumference ...;
3. Corresponding areas ...
Correlation mapping
(1) as the fourth proportion;
(2) As a proportional term.
Laws and auxiliary lines of proving (solving) problems
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.
Verb (abbreviation of verb) application examples (omitted) ★ Emphasis ★ Positive and inverse proportional functions, images and properties of linear and quadratic functions.
abstract
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. Principles for determining the range of independent variables:
(1) makes algebraic expressions meaningful;
(2) Make practical problems meaningful.
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,…
3. Quadratic function
⑴ Definition: Special, 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. Definition of inverse proportional function (1): 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.
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.
Examples of verb (abbreviation of verb) application (omitted) ★ Focus ★ Solving right triangle
abstract
First, trigonometric functions
1. definition: in Rt△ABC, ∠C = Rt∞, then sinA =；; cosA =； tgA =； ctgA=。
2. Trigonometric function values of special angles: 30 degrees, 45 degrees and 60 degrees.
Sin root number 1/2 root number 2/2 root number 3/2 root number 1 root number 3 increases in turn.
Cos root number 3/2 root number 2/2 root number 1/2 root number 3 to root number 1 root number books decrease in turn.
Tangen number 3/3 root number 9/3 root number 27/3 root number is the third power of 1, the third power of 2, and the third power of 3.
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 the two parties; 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. Spacing and height:
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, the application example (omitted) points
① The 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.
abstract
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
Angle relative to the circle:
Definition of (1) 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 calculation center angle: half of the inner angle: ((Related elements can be found by solving Rt△OAM, and so on. )
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 trajectory.
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.
XI. Application Examples (omitted)