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Mathematics required course 1

1 set

(about 4 hours)

(1) refers to collection and sorting.

(1) For example, understand the meaning of the set and the "ownership" relationship between the elements of experience and the set.

(2) Being able to choose a specific problem described by natural language, graphic language and set language (enumeration description method) and feel the significance and function of set language.

(2) The basic relationship between sets.

① By understanding the same meaning between sets, a subset of a given set can be identified.

(2) the specific situation, understand the significance of empty sets.

(3) Basic operation of set

① The intersection of two sets of understanding and meaning will ask the intersection of two simple sets.

② A subset of the complement set of a given set in the understood sense will need a subset of the given complement set.

Venn diagram (3) can be used to express the relationship between sets and the role of intuitive icons in operating experience.

Function concept and basic elementary function I

(about 32 hours)

(1) function

(1) Learn more about the mathematical model in which its functions describe the dependence between variables, and collect corresponding language functions according to the characteristics of learning. The function formed by the elements of the conceptual understanding of the corresponding relationship characteristic function will seek the definition domain and scope of a simple function; Understand the concept of mapping.

(2) In the actual situation, select the functions expressed by appropriate methods (such as mirror method, method list and analysis method) as needed.

③ Understand simple piecewise function and simple application.

(4) Functions, especially quadratic functions, understand monotonicity, maximum (minimum) value, its geometric meaning, the combination of specific functions, and the meaning of parity.

⑤ Learn how to use images to understand and study the function (see example 1).

(2) Exponential function of

(1) (cell division, the decay of archaeological drug C residues in human body, etc. ), and understand the actual background of exponential function model.

② Understand the meaning of rational exponential power, understand the meaning of real exponential power through concrete examples, and master the power of calculation.

(3) Understand the concept and significance of exponential function, draw the image of specific exponential function with the help of calculator or computer, and explore and understand the monotonicity and special points of exponential function.

④ In the process of solving simple practical problems, empirical exponential function is an important function model (see Example 2).

(3) Logarithmic function

(1) Understand the concept and nature of the number of its operations, and know that the general logarithm of the basic formula becomes a natural logarithm, or is commonly used; Understand the history of the number of reading materials and simplify the operation.

(2) Intuitively understand the logarithmic function model, and show the relationship between quantities through concrete examples, initially understand the concept of logarithmic function, and experience the important function model of logarithmic function; With the help of a specific logarithmic function, an image is drawn by a calculator or computer to explore and understand the monotonicity and special points of the logarithmic function.

(3) exponential function and logarithmic function should know (A >；; 0, ≠ 1).

(4) the function of power

Understand the concept of power function with examples; Combine the role of image and the change of understanding.

(5) Functions and equations

(1) Combined with the image of quadratic function, judge the existence of the root of quadratic equation in one variable and the root of quantity, and understand the zero equation root connection of function.

(2) According to the image, approximate solution is obtained by dichotomy of calculator, and the specific function of this method is understood through the corresponding formula, which is a common method for approximate solution of the equation.

(6) Function model and its application

(1) A calculation tool for comparing the growth differences of exponential function, logarithmic function and power function; Combined examples appreciate the type significance of the quantitative growth function of linear growth and exponential explosion growth.

② Collected some models of common functions (exponential function, logarithmic function, power function and sub-function) in social life, for example, models widely used to understand functions.

(7) Internship work

According to the theme, collect data, or historical events (Kepler, Galileo, Descartes, Newton, Leibniz, Euler, etc.) that happened around the 7th century. ). In the form of teamwork, write an article on the formation, development or application of a function concept and communicate in class. See the requirements of mathematical culture for specific requirements.

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Mathematics compulsory 2

A preliminary study on 1 solid geometry

(about 18 hours)

(1) space geometry

(1) Using physical model and observing a lot of computer software of space graphics, we know that columns, cones, platforms and spheres are simple combinations, and we can use these features to describe the structural features of real life and simple objects.

It can be viewed through a simple combination of simple spatial patterns (rectangle, sphere, cylinder, cone, prism and (2) can be drawn) to identify the three-dimensional model of these three views, and use the production model of the used material (such as cardboard) to draw its direct view.

(3) By observing the visual images of the view space of these two methods (parallel projection and central projection), different performances are obtained.

(4) Some buildings that complete the visual diagram of the internship view, such as the lottery (there is no strict requirement on the size and lines on the basis of not affecting the graphic function).

⑤ Learn the calculation formulas of sphere, prism, pyramid, surface area and volume station (no memory formula is required).

(2) the positional relationship between points, lines and surfaces

(1) and cuboid model, intuitively know and understand the positional relationship between points, lines and surfaces in space, and understand the positional relationship between abstract spaces, lines and surfaces. The following can be used as axioms and theorems for reasoning.

Axiom 1: If two points are on a straight line on a plane, then this straight line is on this plane.

Axiom 2: Are there three on a straight line and only on one plane?

Axiom 3: If two non-overlapping planes have the same point, then they have one and only one on a common dotted line.

Axiom 4: Two lines parallel to the same line are parallel.

Theorem: If two angles of two sides correspond to parallel spaces respectively, then the two angles are equal or complementary.

(2) Starting from the above definitions, axioms and theorems of solid geometry, through intuitive perception and operation, we can confirm speculative arguments, and recognize and understand the judgment of parallelism and verticality of lines and planes in space.

After the operation is confirmed, the following judgment theorems are summarized.

◆ Plane A is parallel to the straight line and the straight line on this plane, so the straight line is parallel to the plane.

◆ A straight line intersecting two parallel planes in a plane. These two planes are parallel.

◆ Two intersecting lines in the plane are perpendicular to the straight line, when the straight line is perpendicular to the plane.

◆ The plane perpendicular to the other plane is perpendicular to the two planes.

Operation confirmation, summed up with the following property theorem, proved this point.

◆ The intersection of a plane and a straight line parallel to the plane is parallel to the straight line, the straight line and the straight line plane that exceeds any one of them.

The two planes of are parallel to one plane, then any straight lines generated by this are parallel to each other and intersect with the intersection of these two planes.

◆ Two straight lines that are vertically parallel on the same plane.

◆ The two planes are perpendicular, and the line at the intersection of the lines perpendicular to the other plane is perpendicular to the plane.

③ Some simple propositions can be proved, and the spatial relationship between the conclusions can be used.

Preliminary study on plane analytic geometry

(about 18 hours)

Straight line with equation (1)

In the plane rectangular coordinate system, combined with specific graphics, the geometric elements are explored to determine the straight line position.

② Understand the concepts of inclination angle and slope straight line, and describe the main process of finding the slope of slope straight line through two points with the experience of algebraic method.

③ Two straight lines with parallel or vertical slopes can be determined.

(4) Determine the geometric elements of straight line position, explore and master linear equations (slope point, double point, general), and appreciate several forms of diagonal functions.

⑤ The solution of the equation can be used to find the coordinates of the intersection of two straight lines.

⑥ Explore and master the two-point formula and the distance formula from point to straight line, and the distance between two parallel lines will be found.

Circle sum equation (2)

(1) Review, determine the geometrical features of the circle in the rectangular coordinate system, and explore and master the standard equation and general equation of the circle.

(2) According to the given equation of straight line and circle, judge the positional relationship between straight line and circle, and circle and circle.

③ The equations of lines and circles can be used to solve some simple problems.

(3) In the process of learning, what is the initial plane analytic geometry and the idea of appreciation? Solving geometric problems by algebraic method.

(4) Spatial Cartesian coordinate system

(1) Through the specific situation, I feel it is necessary to establish a spatial rectangular coordinate system to understand the position depiction points in the spatial rectangular coordinate system.

(2) Explore and draw the spatial distance formula between two points through the vertex coordinates of a special cuboid (each side is parallel to the axis).

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Mathematics compulsory course 3

1。 Preliminary algorithm

(about 12 hour)

(1), block diagram

① the idea of solving specific problems (such as solving bilinear equations) through the process and steps of analysis? The algorithm of the algorithm understands its meaning.

(2) Explore and express the design block diagram of experience in the process of solving problems through simulation operation. The process of solving specific problems (such as solving ternary linear equations) and the three basic logical structures of the block diagram: order, conditional branching and circular understanding.

The basic idea of? In the process, (2) Basic algorithm statements: Incorporate the experience of specific problems in block diagram program statements, and understand several basic algorithm statements-input statements, output statements, assignment statements, conditional statements, and loops, so as to better understand the algorithm.

(3) Take reading China's ancient mathematical algorithms as an example to appreciate the contribution of China's ancient mathematics to the development of world mathematics.

2 Statistical data

(about 16 hours)

(1) random sampling

① Statistical problems in real life or other disciplines have certain value.

② Understand the necessity and importance of random sampling in combination with specific practical problems.

③ Learn to take samples from the population, such as simple random sampling, stratified sampling and systematic sampling, to understand the process of participating in solving statistical problems.

④ Collecting data through testing, data access and designing questionnaires.

(2) Sample estimation population

① To express the sample data by appreciating the significance and function of distribution through examples, we should learn to organize the frequency distribution table, draw the histogram, frequency, line graph and stem leaf diagram of frequency distribution (see example 1), and understand their characteristics.

(2) Understand the significance and function of standard deviation of sample data through examples, and learn to calculate standard deviation data.

③ Be able to select samples, extract reasonable basic numerical features (such as mean and standard deviation) from sample data, and give reasonable explanations.

④ The general idea of sample estimation is to solve statistical problems. Further understanding is that the sample frequency distribution will be used to estimate the overall distribution, and the samples of basic digital features will be used to estimate the randomness of the sample frequency distribution of the overall basic digital features.

⑤ We will use the idea of random sampling and sample estimation to solve some simple practical problems and provide some basic data for rational decision-making. Through analysis, we will understand the role of statistics and understand the difference between statistical thinking and deterministic thinking.

(6) Preliminary evaluation of the consciousness of data processing.

(3) Relevant variables

(1) By collecting the scatter plot of data of two related variables in practical problems, the scatter plot intuitively understands the relationship between variables.

② Different estimation methods describe the process of linear correlation between two variables. Know this idea? The linear regression equation of the least square method (see Example 2) can be established according to the linear regression equation of the coefficients in the given formula.

3。 possibility

(about 8 hours)

(1), understand the uncertainty of random events and the stability of frequency, and understand the meaning of probability and the difference between frequency and probability.

(2) Understand the probability of two examples of mutually exclusive events's addition formula.

(3) Using the probability formula of classical probability to calculate the probability of some random events in the basic enumeration method.

(4) Understand the meaning of random number, and use the usual means of simulation (including random number generated by calculator) (see Example 3) to preliminarily experience the estimation of geometric probability.

(5) By reading the materials, we can understand the cognitive process of human beings to random phenomena.

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Mathematics compulsory 4

1 trigonometric function

(about 16 hours)

(1) At any angle, radian

Any concept of angle and radian, radian angle is mutual understanding.

(2) Trigonometric function

(1) Understand the definition of trigonometric functions (sine, cosine and tangent) at any angle in the unit circle.

② The unit circle of the inductive formula (sine, cosine and tangent) of trigonometric function system can draw an image and understand the periodic characteristics of trigonometric function.

(3) Understand the image with the properties of sine function, cosine function and tangent function (such as monotonicity, maximum and minimum value, intersection with X axis, point, etc.). ).

(4) Understand the basic relationship of trigonometric functions with the same angle:

⑤ Understand its practical significance with specific examples; Depicting a calculator or computer image is helpful to observe the functional influence of image changes of parameters a and ω.

⑥ trigonometric function solves simple practical problems and realizes the important function model of trigonometric function describing periodic phenomena.

2 plane vector

(about 12 hour)

The Background and Basic Concepts of (1) Plane Vector

Through the analysis of force and force, we can understand the actual background of vector, the same meaning of plane vector and vector, and the geometry of vector.

Vector Linear Operation (2)

(1) main vector addition and subtraction, and understand its geometric meaning.

(2) Master the vector multiplication operation and understand its geometric meaning and the meaning of two vector lines.

③ Understand the properties and geometric significance of linear operators of vectors.

(3) Basic coordinate theorem of plane vector

(1) Understand the basic theorem of plane vector and its significance.

② Master the orthogonal decomposition of plane vectors and their coordinates.

③ Vector addition, subtraction, multiplication, division and telephone number on the coordinate plane.

④ Understand the straight line conditions represented by the coordinates of plane vectors.

(4) Plane vector diagram

① The meaning of "force" in physical understanding, and the plane vector diagram of its physical meaning as an example.

② The scalar product vector projection relation of empirical plane vector.

(3) The coordinate expression of the number calculated by the scalar product and the plane cross product of the master station.

(4) The number of scenes that can be used represents the included angle between two vectors, and the scalar product determines the vertical relationship between the two planes of the vector.

(five) the carrier of the application

The empirical vector method solves some simple plane geometric problems, mechanical problems and other practical problems, and the empirical vector deals with geometric problems and physical problems, such as tools, computing power and the ability to solve practical problems.

Transformation of trigonometric functions.

(about 8 hours)

The experience of (1) comes from the cosine formula of cross product angle, so as to better understand the function of vector method.

(2) From the difference of the emergent angle of cosine formula of complementary angle, the difference of sine, cosine and tangent formulas, and the formulas of sine, cosine and tangent, we can understand their internal relations.

(3) Using the above formula, a simple identity transformation (including the guiding area and the poor, the plot of the poor, the half-angle formula, but not requiring memory) is derived.

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Mathematics compulsory 5

solving triangle

(about 8 hours)

(1) By exploring the relationship between the sides and angles of any triangle, we can master the sine theorem and cosine theorem, and measure and solve some simple triangle problems.

(2) the ability to solve some practical problems related to measurement and geometry by using the knowledge and methods of sine and cosine laws.

series

(about 12 hour)

Concepts of (1) sequences and simple symbols

Understand the concept of this series and several simple representations (list, image, general formula). The products of this series have special functions.

(2) arithmetic progression, geometric sequence

(1) Understand the concepts of arithmetic progression and geometric progression.

② Explore and master the general formulas, the first n items and formulas of arithmetic progression and geometric progression.

(3) In the case of a specific problem, a series of operational relations or geometric relations have been found, and this problem can be solved with corresponding knowledge (see example 1).

④ Empirical arithmetic progression, geometric series and time function, exponential function.

3。 inequality

(about 16 hours)

Relationship range from (1)

Emotion exists in the real world and daily life, and comes from a large number of relationships and unequal (group) backgrounds.

(2) Quadratic inequality

The actual situation of process abstraction of quadratic inequality model ① Experience.

(2) Relate the function image about the quadratic inequality of one variable with the corresponding function equation.

③ When solving a given quadratic inequality, try to design a block diagram solution.

(3) Binary linear inequalities and simple linear programming problems.

Abstract binary inequality from the actual situation.

② Understand the geometric meaning of binary linear inequality, and the flat region represents binary inequality (see Example 2).

(3) Abstraction can solve some simple binary linear programming problems (see Example 3).

(4) Basic inequality:

① The basic process of exploring and understanding inequality.

② Basic inequalities will be used to solve simple maximum (minimum) values (see Example 4).

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Mathematics elective course

Elective course 2- 1

Logically speaking (about 8 hours)

(1) proposition and its relationship

(1) Understand the inverse proposition of the proposition, regardless of the inverse proposition of the proposition.

Understanding the necessary and sufficient conditions and the necessary and sufficient conditions is of great significance. The relationship between these four propositions is analyzed.

(2) Simple logical connection

Logical connection between the meanings of "or" and "not"

(3) Full name quantifiers and existential quantifiers

① Understand the meaning of universal quantifiers and existential quantifiers.

② The correct proposition contains a quantifier negation.

Quadratic curve and equation (16 hours)

(1) quadratic curve

① Understand the actual background of conic curve, describe the real world with empirical cone, and solve practical problems.

(2) from the experience to abstract the specific situation, the process of ellipse and parabola model, and master their own definitions, standard equations, geometric shapes and simple properties.

(3) Understand the definition and standard equation of hyperbolic geometry, and know the properties of hyperbola.

④ Coordinate method can be used to solve some simple geometric problems, conic curves (the positional relationship between straight lines and conic curves) and practical problems.

⑤ Through the study of conic curves, we can better understand the idea of combining numbers with shapes.

Curve and Equation (2)

Understand the corresponding relationship of curve equations and further feel the basic idea of combining numbers with shapes.

Space vector and three-dimensional geometry (about 12 hours)

The Space Vector of (1) and Its Operation

(1) Empirical vectors from two-dimensional space and business processes that drive them.

② Understand the basic theorem of space vector and its significance, and master the concept of orthogonal decomposition of space vector and its coordinates.

③ Linear operators of principal space vectors and their coordinates.

(4) The scalar product of the vector and the verticality of the vector line can be used to determine the scalar product of the main space vector and its coordinates.

(2) Application of space vector

① Understand the direction vector of a straight line and the normal vector of a plane.

② The relationship between line and offline, vertical and parallel lines, and plane and plane can be expressed by vector language.

③ Some relationships between straight lines that can be proved by vector method, plane position theorems (including three mutually perpendicular theorems) (see Example 1, Example 2, Example 3).

④ The angles of lines, lines, planes and planes can be vectors to solve the calculation problem. Empirical vector method plays an important role in studying geometric problems.

Reference case

Example 1. It is known that right-angled triangular prism ∠ ACB = 90, ∠ BAC = 30, and m is the midpoint of the prism. Prove:

Example 2. It is known that the rectangle ABCD is perpendicular to the rectangle ADEF, and the AD is on the same side, but they are not on the same plane. The points on the diagonal are m, n, BD, AE and.

Proof: MN∑ plane CDE.

Example 3. Known unit cubes e and f are the midpoints of edges, respectively. Find:

(1) the angle formed by EF, (2) the angle formed by 2)AF and plane, and (3) the dihedral angle of size.

Elective course 2-2

Derivative and its application (about 24 hours)

The Derivative of the Concept of (1) and Its Geometric Significance

① By analyzing a large number of examples and the rate of change from the average to the instantaneous rate of change in this process, we can understand the actual background of the concept of derivative. The known instantaneous change rate is the idea and intention of derivative appreciation (take 1- 1 case 2, case 3).

(2) The function image intuitively understands the geometric meaning of the derivative.

(2) the operation of derivative financial instruments

① According to the definition of derivative and derivative of function.

Formula (2) can be used to find the derivative of a simple function by four algorithms given the derivative of a basic elementary function, and can be used to find a simple compound function (limited to tables).

③ The derivative formula table will be used.

(3) Derivative learning function

(1) refers to the monotonous relationship between geometric exploration and cognition, the function of derivatives (see elective course 1- 1 case 4), the monotonicity and monotone interval of derivative functions, and there are no more than three polynomial functions.

(2) Combining the function of the image, the necessary and sufficient condition extreme value of the function is closed at a certain point; No more than 3 times, no more than 3 times to realize the valuable derivatives and polynomial functions of the maximum and minimum values of interval polynomial functions; General method and effect of studying function properties by derivative.

(4) Examples of optimization problems in life.

For example, the experience of optimization problems such as maximum profit, minimum material consumption and maximum efficiency, and the role of derivatives in solving practical problems when taking 1- 1 (see Example 5).

(5) Basic theorems of definite integral and calculus

(1) The practical background of solving the trapezoidal area with curved sides, the variable force effect and understanding the definite integral, the basic idea of device geometry appreciation, the concept of definite integral, and a preliminary understanding? definite integral

② The relationship between time, speed and distance of a variable high-speed moving object in a certain period can intuitively understand the significance of the basic theorem of calculus (see example 1).

Reasoning and proof (about 8 hours)

(1) Rational reasoning and deductive reasoning

① Understand the significance of rational reasoning in mathematical discovery (see elective course 1-2 Case 2 Case 3). Use simple reasoning, induce analogical experience and understand reasoning.

② Appreciate the importance of deductive reasoning, master the basic models of deductive reasoning, and use them for some simple reasoning.

③ Understand the connection and difference between rational reasoning and deductive reasoning through concrete examples.

(2) Direct evidence and indirect evidence

① Understand two basic methods of direct proof: analytical synthesis method, and understand the thinking process and characteristics of analytical synthesis method.

② Understand the basic methods of indirect reduction to absurdity, the thinking process of understanding and the characteristics of reduction to absurdity.

(3) Mathematical induction

To understand the principle of mathematical induction, we can prove some simple mathematical propositions by mathematical induction.

(4) Mathematical culture

Through the introduction of situations (such as Euclid's Elements of Geometry, Marx's Capital, Jefferson's Declaration of Independence and Newton's Law), we can appreciate the axiomatic thought.

(2) The role of computer automatic reasoning in description and mathematical proof.

Various extensions of digital system introduction (about 4 hours)

(1), in order to understand the actual demand of the number of rows in the process of expanding the number of rows under the problem situation, the experience of internal contradictions in mathematics (the rules of calculating the number and equation theory), the role of feeling and the connection between human rational thinking and the real world.

(2) Understand the complexity and complexity of basic concepts and the necessary and sufficient conditions for equality.

(3) Understand the algebraic representation and geometric meaning of complex methods.

(4) The algebraic form of complex numbers has geometric meanings other than four operations and subtraction operations. .

Reference case

Example 1. According to the law of motion, we already know the derivative of time, that is, the speed of motion at a certain moment (that is, instantaneous speed or instantaneous rate of change). The overall change between these considerations. The interval is divided into n cells, and it can be assumed that the length between the cells is equal, and its length is. We assume that the rate of change is approximately constant, so we can say that at every small time interval,

X time rate of change.

From the first small range, that is, the assumed rate of change is about, so

Similarly, the second subinterval, that is, from the assumed rate of change, is about therefore.

Wait a minute. Approximations between all cells of the resulting changed position are added.

Total change of s

We can write the total change between two positions. On the other hand, when divided by infinity, okay? Tending to infinity, this type

The limit of definite integral, that is, the position between total changes. Therefore, we can draw the following conclusions:

That is, the definite integral of the total rate of change.

Especially, when the object is moving at a uniform speed, in real time,

When uniform acceleration is used for one purpose, that is, it is constant,