I. Algebra section:
(1) real numbers (rational numbers, irrational numbers) positive and negative numbers
rational number
number axis
Corresponding thing
absolute value
Addition and subtraction of rational numbers addition of rational numbers
Subtraction of rational numbers; Mixed operation of rational number addition and subtraction.
Rational number multiplication
Division of rational numbers
Power of rational number
Scientific symbol
Factors and significant numbers
Rational number mixed operation
Use of calculator
square root
cube root
Real number and number axis
Write a prescription with a calculator
Mathematical activities
(2) Algebra (Algebra, Fraction, Quadratic Radical)-algebra.
Two algebraic expressions.
1 algebraic expression Acceleration and deceleration algebraic expression
Multiplication of 2 algebraic expressions
Division of 3 algebraic expressions
Three factory decomposition
1 common factor method
2 using the formula method
3-component solution
quadrant
1 Fractions and Basic Properties of Fractions
Multiplication and division of two fractions
Addition and subtraction of three fractions
Quintic radical
Six One-dimensional Linear Inequalities and One-dimensional Linear Inequalities
The second method
Holistic thinking method
Two alternative methods.
A combination of three numbers and shapes
Four types of discussion thoughts
Five-party thought
Six-factor decomposition method
Seven, the method of adding items.
Eight-parameter method
Nine matching method
Ten undetermined coefficient method
(3) Equation (group) and inequality (group) (quadratic equation in one yuan, quadratic equation in two (three) yuan, quadratic equation in one yuan, quadratic equation in two yuan, linear inequality in one yuan, linear inequality in one yuan) The first one is about linear equation in one yuan.
Solutions of one-dimensional linear equations
Binary linear equations and ternary linear equations
Binary linear equations and their solutions
Three-dimensional linear equations and their solutions
One-dimensional linear inequality and one-dimensional linear inequality system
One-dimensional linear inequality and its solution
One-dimensional linear inequality system and its solution
Direct Kaiping method
Factorization method
Formula method
Relationship between root and coefficient
Fractional equation and its solution (1)
Fractional Equation and Its Solution (2)
Binary Quadratic Equation and Its Solution (1)
Binary Quadratic Equation and Its Solution (2)
Innovative application problems
Query application problem
(4) Quadratic function of function (rectangular coordinate system, linear function, proportional function, inverse proportional function, quadratic function)
I. Definition and definition of expressions
Generally speaking, there is the following relationship between independent variable x and dependent variable y:
y=ax？ +bx+c(a, b, c are constants, a≠0)
Y is called the quadratic function of X.
The right side of a quadratic function expression is usually a quadratic trinomial.
Two. Three Expressions of Quadratic Function
General formula: y=ax? +bx+c(a, b, c are constants, a≠0)
Vertex: y=a(x-h)? +k[ vertex P(h, k) of parabola]
Intersection point: y = a(X-X 1)(X-x2)[ only applicable to parabolas with intersection points a (x 1, 0) and b (x2, 0) with the x axis]
Note: Among these three forms of mutual transformation, there are the following relations:
h=-b/2a k=(4ac-b？ )/4a x 1，x2=(-b √b？ -4ac)/2a
Three. Image of quadratic function
Do quadratic function y=x in plane rectangular coordinate system? Images,
It can be seen that the image of quadratic function is a parabola.
Four. Properties of parabola
1. Parabola is an axisymmetric figure. The axis of symmetry is a straight line
x = -b/2a .
The only intersection of the symmetry axis and the parabola is the vertex p of the parabola.
Especially when b=0, the symmetry axis of the parabola is the Y axis (that is, the straight line x=0).
2. The parabola has a vertex p, and the coordinates are
P [ -b/2a，(4ac-b？ )/4a ].
-b/2a=0, p is on the y axis; When δδ= b? When -4ac=0, p is on the x axis.
3. Quadratic coefficient A determines the opening direction and size of parabola.
When a > 0, the parabola opens upward; When a < 0, the parabola opens downward.
The larger the |a|, the smaller the opening of the parabola.
4. Both the linear coefficient b and the quadratic coefficient a*** determine the position of the symmetry axis.
When the signs of A and B are the same (that is, AB > 0), the symmetry axis is left on the Y axis;
When the signs of A and B are different (that is, AB < 0), the symmetry axis is on the right side of the Y axis.
5. The constant term c determines the intersection of parabola and Y axis.
The parabola intersects the Y axis at (0, c)
6. Number of intersections between parabola and X axis
δ= b？ When -4ac > 0, the parabola has two intersections with the x-axis.
δ= b？ When -4ac=0, the parabola has 1 intersections with the X axis.
δ= b？ When -4ac < 0, the parabola has no intersection with the x axis.
Verb (abbreviation of verb) quadratic function and unary quadratic equation
Especially quadratic function (hereinafter referred to as function) y=ax? +bx+c，
When y=0, the quadratic function is a univariate quadratic equation about x (hereinafter referred to as equation).
Is that an axe? +bx+c=0
At this point, whether the function image intersects with the X axis means whether the equation has real roots.
The abscissa of the intersection of the function and the X axis is the root of the equation.
linear function
I. Definitions and definitions:
Independent variable x and dependent variable y have the following relationship:
Y=kx+b(k, b is a constant, k≠0)
It is said that y is a linear function of x.
In particular, when b=0, y is a proportional function of x.
Two. Properties of linear functions:
The change value of y is directly proportional to the corresponding change value of x, and the ratio is K.
That is △ y/△ x = K.
Three. Images and properties of linear functions;
1. exercises and graphics: through the following three steps (1) list; (2) tracking points; (3) Connecting lines can make images of linear functions-straight lines. So the image of a function only needs to know two points and connect them into a straight line.
2. Property: any point P(x, y) on the linear function satisfies the equation: y = kx+b.
3. Quadrant where k, b and function images are located.
When k > 0, the straight line must pass through the first and third quadrants, and y increases with the increase of x;
When k < 0, the straight line must pass through the second and fourth quadrants, and y decreases with the increase of x.
When b > 0, the straight line must pass through the first and second quadrants; When b < 0, the straight line must pass through three or four quadrants.
Especially, when b=O, the straight line passing through the origin o (0 0,0) represents the image of the proportional function.
At this time, when k > 0, the straight line only passes through one or three quadrants; When k < 0, the straight line only passes through two or four quadrants.
Four. Determine the expression of linear function:
Known point A(x 1, y1); B(x2, y2), please determine the expressions of linear functions passing through points A and B. ..
(1) Let the expression (also called analytic expression) of a linear function be y = kx+b.
(2) Because any point P(x, y) on the linear function satisfies the equation y = kx+b, two equations can be listed:
Y 1 = KX 1+B 1，Y2 = KX2+B2。
(3) Solve this binary linear equation and get the values of K and B. ..
(4) Finally, the expression of the linear function is obtained.
The application of verb (verb's abbreviation) linear function in life
1. When the time t is constant, the distance s is a linear function of the velocity v .. s=vt.
2. When the pumping speed f of the pool is constant, the water quantity g in the pool is a linear function of the pumping time t, and the original water quantity s in the pool is set. G = S- feet.
inverse proportion function
A function in the form of y = k/x (where k is a constant and k≠0) is called an inverse proportional function.
The range of the independent variable x is all real numbers that are not equal to 0.
The image of the inverse proportional function is a hyperbola.
As shown in the figure, the function images when k is positive and negative (2 and -2) are given above.
trigonometric function
Trigonometric function is a kind of transcendental function in elementary function in mathematics. Their essence is the mapping between the set of arbitrary angles and a set of ratio variables. The usual trigonometric function is defined in the plane rectangular coordinate system, and its domain is the whole real number domain. The other is defined in a right triangle, but it is incomplete. Modern mathematics describes them as the limit of infinite sequence and the solution of differential equation, and extends their definitions to complex system.
Because of the periodicity of trigonometric function, it does not have the inverse function in the sense of single-valued function.
Trigonometric functions have important applications in complex numbers. Trigonometric function is also a common tool in physics.
It has six basic functions:
Function name sine cosine tangent cotangent secant cotangent
Symbol sin cos tan cot sec csc
Sine function sin(A)=a/h
Cosine function cos(A)=b/h
Tangent function tan(A)=a/b
Cotangent function cot(A)=b/a
In the field of mathematics, a function is a relationship that makes each element in one set correspond to the only element in another (possibly the same) set. The concept of function is the most basic for every branch of mathematics and quantity.
The terms function, mapping, correspondence and transformation usually have the same meaning.
In short, a function is a "rule" that assigns a unique output value to each input. This "rule" can be expressed by a function expression, a mathematical relationship or a simple table listing input values and output values. The most important property of a function is certainty, that is, the same input always corresponds to the same output (note that the opposite is not necessarily true). From this point of view, a function can be regarded as a "machine" or a "black box", which converts a valid input value into a unique output value. Generally, the input value is called the parameter of the function, and the output value is called the value of the function.
The parameters and function values of the most common functions are numbers, and their corresponding relations are expressed by function expressions. The function values can be obtained by directly substituting the parameter values into the function expressions. For example,
F(x) = x2, and the square of x is the function value.
This function can also be simply extended to the case of multi-parameters. For example:
G(x, y) = xy has two parameters x and y, with the product xy as the value. Unlike before, this "rule" is related to two inputs. In fact, these two inputs can be regarded as an ordered pair (x, y), G is a function with this ordered pair (x, y) as a parameter, and the value of this function is xy.
In scientific research, there are often functions that are unknown or can't give expressions. For example, the temperature distribution at different times on the earth, this function takes the place and time of occurrence as parameters, and takes the temperature at a certain place and time as the output.
The concept of function is not limited to the calculation of numbers, or even to calculation. The mathematical concept of function is broader, which includes not only the mapping relationship between numbers. This function links the definition domain (input set) with the mapping domain (possible output set) so that each element of the definition domain uniquely corresponds to an element in the mapping domain. As described below, functions are abstractly defined as explicit mathematical relationships. Because of the generality of function definition, the concept of function is very basic for almost all branches of mathematics.
(5) Probability statistics (sampling survey, data analysis and probability evaluation) 1. Probability theory-the mathematics of studying random phenomena.
Second, probability-the quantitative representation of the probability of random events.
Third, the estimation of frequency and probability
Fourthly, the calculation of equal possibility and probability.
5. Find the probability of events by enumeration.
Six, several typical problems of calculating probability by enumeration method
Seven, clarify some misunderstandings
Eight, the basic requirements and principles of probability teaching in junior high school.
Two. Geometric part
(1) intersection line and parallel line (line segment, angle, verticality, proposition, theorem, axiom) 1. Axiom for judging parallel lines (theorem)
(1) Two straight lines are cut by the third straight line. If the isosceles angles are equal, two straight lines are parallel.
(2) Two straight lines are cut by a third straight line. If the offset angles are equal, two straight lines are parallel (referred to as "offset angles are equal, two straight lines are parallel").
(3) Two straight lines are cut by a third straight line. If they are complementary to the lateral inner angle, then the two straight lines are parallel (referred to as "the lateral inner angle is complementary and the two straight lines are parallel").
2. Axiom of Parallel Lines (Theorem)
If two parallel lines intersect a third line, then
(1) The common angles are equal (meaning "two straight lines are parallel and the common angles are equal").
(2) The internal dislocation angle is equal (meaning "two straight lines are parallel and the internal dislocation angle is equal").
(3) The internal angles on the same side are complementary (meaning "two straight lines are parallel and the internal angles on the same side are complementary").
Regarding the judgment and nature of parallel lines, we must not confuse their topics and conclusions, but strictly distinguish them, as shown in the following table:
classify
Theme Settings (Reason)
Conclusion (fruit)
Parallel line judgment
Wait, waist angle
Two straight lines are parallel
Equivalence of internal dislocation angle
Complementarity of ipsilateral internal angle
Properties of parallel lines
Two straight lines are parallel
Wait, waist angle
Equivalence of internal dislocation angle
Complementarity of ipsilateral internal angle
Thus, the judgment theorem and the property theorem are two kinds of theorems of causality inversion. The judgment of parallel lines is to determine the positional relationship of lines through angles, and the nature of parallel lines is to determine the quantitative relationship of angles through the positional relationship of lines. For judging theorem, "two straight lines are parallel" is an inference, while for nature, "two straight lines are parallel" is an essential premise, so we can't just say "congruent angle (internal angle)".
Parallel lines also have the following judgments and properties:
The (1) parallel axiom passes through a point outside the straight line, and there is only one straight line parallel to the known straight line.
(2) Transitivity of parallel lines If two straight lines are parallel to the third straight line, then the two straight lines are parallel to each other.
(3) If both straight lines are perpendicular to the third straight line, then the two straight lines are parallel to each other.
(4) The straight line is perpendicular to one of the two parallel lines, so it is also perpendicular to the other (2) triangle (classification, edge, area, median, congruence, similarity, right triangle).
(3) Quadrilateral (trapezoid judging property, parallelogram judging property, other special quadrangles) 1. Teaching objectives
1. Understand the relationship between special quadrangles, and prove their property theorems and judgment theorems; +
2. Applying the obtained conclusions, some problems are solved by calculation and proof;
3. Make students have a further understanding of the necessity of proof through proof.
4. Infiltrate the idea of set through the membership relationship of quadrilateral.
5. Cultivate students' dialectical view by understanding the internal relations of the four quadrangles.
Second, teaching priorities, difficulties and doubts
1. key: apply the obtained conclusions and solve some problems through calculation and proof;
2. Difficulties: the relationship and nature between special quadrangles, and some problems are solved by calculation and proof with the conclusions obtained;
3. Doubts: the * * * nature, characteristics and affiliation of parallelogram, rectangle, diamond and square (list, drawing, sketch and counterexample can be used to explain).
Third, teaching methods.
Induce, talk and practice.
Fourth, teaching methods.
Projection.
Verb (abbreviation of verb) teaching process;
(1) Students complete the following blanks:
The connection and difference of special quadrangles;
edge
corner
Maomaojiao
parallelogram
The opposite sides are parallel and equal.
Diagonal equality
Adjacent angle complementation
Diagonal lines will split each other in two.
rectangle
The opposite sides are parallel and equal.
All four corners are right angles.
Diagonal lines are bisected and equal to each other.
diamond
Opposite sides are parallel and four-sided.
All parties are equal.
Diagonal equality
Diagonal lines vertically bisect each other,
Each diagonal bisects a set of diagonal lines.
square
Opposite sides are parallel and four-sided.
All parties are equal.
All four corners are right angles.
Diagonal lines are bisected and equal to each other.
Each diagonal bisects a set of diagonal lines.
(B) explain the new lesson
1, review the main contents of this chapter.
The content of this chapter: the nature and judgment of rectangle.
The nature and judgment of parallelogram: the nature and judgment of square
The nature and judgment of rhombus
The nature and judgment of isosceles trapezoid
Properties of triangle midline
The parallel lines sandwiched between two parallel lines are equal.
The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.
Exercise 1: (Projection)
(1). In the isosceles trapezoid ABCD, AD‖BC, AB=CD, B = 40, then A = _ _ _ _, C = _ _ _ _, D = _ _.
(2) The diagonal length of the diamond is 24 and 10 respectively, so the circumference of the diamond is _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
(3) The diagonal of the rectangular ABCD is 60, the diagonal length is 0 when AB = 2 cm, and the rectangular area is 0;
(4) When the quadrilateral is a figure, the quadrilateral formed by connecting the midpoints of four sides of any quadrilateral in turn is a diamond.
2. The nature and judgment of quadrilateral.
Angle: Angle:
Attribute Edge: Decision Edge:
Diagonal: Diagonal:
1) Let students describe the definitions of parallelogram, rectangle, diamond and square, their special properties, and their connections and differences from three aspects: angle, edge and diagonal.
2) Further explain the internal relations of parallelogram, rectangle, diamond and square through charts.
3. Application of Property Theorem and Judgment Theorem: (Illustration 1)
For example, as shown in figure 1, the middle vertical line EF of diagonal AC of parallelogram ABCD intersects with extension lines on both sides of AB and CD at E and F respectively. Please guess what kind of quadrilateral AECF is? And prove your conclusion.
(3) Consolidation exercises:
Exercise 2 Calculating and Proving Problems:
1), as shown in figure 2, in ABCD, it is known that AB = 4 cm,
Bc = 9 cm, ∠ b = 30, find the area of ABCD.
2) As shown in Figure 3, in the square ABCD
∠ The bisector CF of ∠ACD intersects with AD at point F,
EF⊥AC is at point e,
Please guess what is the relationship between DF and AE?
Prove your conclusion.
② When EF=2cm, find the side length of the square.
Exercise 3 Extension
(3) As shown in Figure 4, it is known that diagonal AC of square ABCD intersects BD O, E is a point on AC, intersection A is AG⊥EB, vertical foot is G, and AG intersects BD at F. Verification: OE=OF.
Variant: For the above proposition, if point E is on the extension line of AC, the extension lines of AG ⊥ EB and EB are at point G, and the extension lines of AG and DB are at point F, other conditions remain unchanged (as shown in Figure 5), does the conclusion "OE=OF" still hold? If yes, please give proof; If not, please explain why.
(4) As shown in Figure 6, in quadrilateral ABCD, ∠ ADC = ∠ ABC = 90, AD = CD, and DP ⊥ AB is at point P. If the area of quadrilateral ABCD is 18, Xiao Ming thought about finding the length of DP:
Cut △ADP along DP to reach △CDF. At this point, PDFB is just a square.
Can you prove that it is a square? ② Can you find the length of DP?
(d) Summary: (1) Special quadrilateral We should understand its particularity and internal relations from the changes of angles, sides and diagonals.
(2) The quadrilateral problem is solved by adding appropriate auxiliary lines to transform it into a triangle problem. +
(5) Homework: Questions 6, 7 and 8 have 59 pages, with 45 to 46 pages attached.
Review and reflection on the third chapter of the ninth grade parallelogram
First, the teaching objectives
1. Understand the relationship between special quadrangles, and prove their property theorems and judgment theorems; +
2. Applying the obtained conclusions, some problems are solved by calculation and proof;
3. Make students have a further understanding of the necessity of proof through proof.
4. Infiltrate the idea of set through the membership relationship of quadrilateral.
5. Cultivate students' dialectical view by understanding the internal relations of the four quadrangles.
Second, teaching priorities, difficulties and doubts
1. key: apply the obtained conclusions and solve some problems through calculation and proof;
2. Difficulties: the relationship and nature between special quadrangles, and some problems are solved by calculation and proof with the conclusions obtained;
3. Doubts: the * * * nature, characteristics and affiliation of parallelogram, rectangle, diamond and square (list, drawing, sketch and counterexample can be used to explain).
Third, teaching methods.
Induce, talk and practice.
Fourth, teaching methods.
Projection.
Verb (abbreviation of verb) teaching process;
(1) Students complete the following blanks:
The connection and difference of special quadrangles;
edge
corner
Maomaojiao
parallelogram
The opposite sides are parallel and equal.
Diagonal equality
Adjacent angle complementation
Diagonal lines will split each other in two.
rectangle
The opposite sides are parallel and equal.
All four corners are right angles.
Diagonal lines will bisect each other.
(4) The positional relationship between circles (concept, nature, theorem, positional relationship and calculation) includes three parts: the positional relationship between points and circles, the positional relationship between straight lines and circles, and the positional relationship between circles. In the "positional relationship between a point and a circle", the textbook first gives three different positional relationships between a point and a circle in combination with the shooting problem, then discusses the circle with three points, and introduces the reduction to absurdity in combination with "three points can't make a circle in a line". In the "positional relationship between straight line and circle", the textbook first discusses three positional relationships between straight line and circle, then focuses on the tangent of straight line and circle, gives the judgment theorem, property theorem and tangent length theorem of straight line and circle, and introduces the inscribed circle of triangle on this basis. In the "positional relationship between circles", the emphasis is on discussing the different positional relationships between circles. This section focuses on the positional relationship between a straight line and a circle. The judgment theorem, property theorem and tangent length theorem of tangent are commonly used theorems to study the related problems of straight line and circle, and are the key contents of this section. The idea of reduction to absurdity has penetrated in the previous chapter, and this section is formally proposed as indirect proof, which is still difficult for students to accept, so the teaching of reduction to absurdity is the difficulty of this section; In addition, the propositions and conclusions of tangent judgment theorem and property theorem are easily confused, and the proof of property theorem needs reduction to absurdity, so the teaching of these two theorems is also the difficulty of this section and this chapter.
Regular polygon is a special polygon, which has some properties similar to a circle. For example, a circle has unique symmetry. It is not only an axisymmetric figure and a centrally symmetric figure, but also a straight line with any diameter is its axis of symmetry. Any rotation angle around the center of the circle can coincide with the original figure. A regular polygon is also an axisymmetric figure, and a regular N-polygon has n symmetry axes. When n is an even number, it is also a central symmetric figure, and every rotation around the center can coincide with the original figure. It can be seen that regular polygons and circles have many internal relations. In addition, regular polygons are also widely used in production and life, so the content of "regular polygons and circles" is arranged in the textbook. On the basis of reviewing students' existing concepts of regular polygons, the textbook takes regular pentagons as an example to prove the method of finding regular pentagons by dividing the circumference equally. Then the related concepts of regular polygon, such as center, radius, central angle and vertex, are introduced, and the drawing method of regular polygon is further introduced. The calculation of regular polygons is the key content of this section. These calculations are the basic knowledge in geometry. To master them correctly, we should also comprehensively use the knowledge we have learned before, which is often used in production and life. The teaching difficulty of this section lies in students' acceptance and understanding of "n" in regular N polygons. Students are used to certain figures, such as triangles, quadrangles and circles, but they are usually not used to N-sided shapes. In order to reduce the difficulty, the proof and calculation problems involved in the textbook are all based on specific polygons. In teaching, we should pay attention to the conclusion and method of popularizing concrete graphics, so that students can realize the leap from concrete to abstract and from special to general, and improve their thinking ability.
The main contents of the next 24.4 sections of the textbook are some calculations related to circles, including "arc length and sector area" and "side area and total area of a cone". "Arc length and sector area" is derived from the formula of circumference length and area learned in primary school. Using these formulas, we can calculate the perimeter and area of some simple combined figures related to circles. Because the side development diagram of the cone is fan-shaped, the textbook then introduces the calculation of the side area and total area of the cone. These calculations are not only the basic calculations in geometry, but also often used in daily life. Using this knowledge can also solve some simple practical problems. The calculation of cone lateral area can also cultivate students' concept of space, so we should also pay attention to this part of teaching.
(C) course learning objectives
1. Understand the circle and its related concepts, understand the relationship between arc, chord and central angle, explore and understand the positional relationship between point and circle, straight line and circle, and circle and circle, explore and master the relationship between central angle and central angle, and the relative characteristics of central angle and diameter.
2. Understand the concept of tangent, explore and master the positional relationship between tangent and tangent radius, judge whether the straight line is the tangent of a circle, and draw the tangent of a circle through a point on the circle.
3. Understand the inner and outer center of the triangle and explore how to make a circle through one point, two points and three points that are not on a straight line.
4. Understand the concept of regular polygon and master the method of drawing inscribed circle of regular polygon with equal circumference; A cone that can calculate arc length, sector area, lateral area and total area.
5. Combining with the exploration and proof of related graphic properties, further cultivate students' rational reasoning ability, and develop students' logical thinking ability and expressive ability of reasoning and argumentation; Through the teaching of this chapter, we can further cultivate students' comprehensive ability to use knowledge and solve problems with what they have learned, and at the same time educate students on dialectical materialist world outlook.
(5) Graphic and transformation (graphic similarity, translation, rotation, axial symmetry and central symmetry) 1. Through concrete examples, we can understand axisymmetric and axisymmetric graphics, explore the basic properties of axisymmetric, and understand the property that the connecting lines of corresponding points are vertically bisected by axisymmetric;
2. Explore the axial symmetry relationship between simple graphs, and make one or two simple graphs with axial symmetry as required; Understand and appreciate the application of axial symmetry in real life, and can use axial symmetry for simple pattern design;
3. Understand the concept of vertical line in line segment, explore and master its properties; Understand the concepts of isosceles triangle and equilateral triangle, explore and master their properties and judgment methods;
4. Be able to use the knowledge learned in this chapter to explain the phenomena in life and solve simple practical problems, develop the concept of space in the process of observation, operation, imagination, demonstration and communication, and stimulate the interest in learning space and graphics.
The competition in junior high school mainly lies in algebra, circle and quadratic function.