High School Mathematics

High school mathematics is a subject studied by high school students across the country. Including "sets and functions", "trigonometric functions", "inequalities", "series", "three-dimensional geometry", "plane analytic geometry" and other parts of the high school mathematics is divided into two main parts of algebra and geometry. Algebra is divided into two main parts: algebra and geometry. Algebra is mainly primary functions, quadratic functions, inverse proportional functions and trigonometric functions. Geometry is divided into plane analytic geometry and three-dimensional geometry.

I. Sets

(1) the meaning and representation of the set

1 through the example, to understand the meaning of the set, and appreciate the elements and the set of "belonging to" the relationship.

2 can choose natural language, graphical language, set language (enumeration or description) to describe different specific problems, feel the meaning and role of set language.

(2) Basic Relationships between Sets

1 Understand the meaning of containment and equivalence between sets, and be able to recognize subsets of a given set.

2 Understand the meaning of full and empty sets in a specific context.

(3) Basic Operations on Sets

1 Understand the meaning of concatenation and intersection of two sets and can find the concatenation and intersection of two simple sets.

2 Understand the meaning of complement of a subset of a given set and can find the complement of a given subset.

3 Be able to use Venn diagrams to express relationships and operations on sets, and appreciate the role of visual diagrams in understanding abstract concepts.

The concept of function and basic elementary functions:

(1) function

1 further appreciate that function is an important mathematical model for describing the dependence between variables, on the basis of which to learn to use the language of sets and correspondences to portray functions, and to appreciate the role of correspondences in portraying the concept of function; to understand the elements of a function, and to find the domain and value domains of simple functions; to understand mappings, and to understand the role of mapping. The concept of mapping.

2 In real-world situations, they can choose appropriate methods (e.g., graphical, tabular, analytical) to represent functions according to different needs.

3 Understand simple segmented functions and be able to apply them in a simple way.

4 Through the functions that have been learned, especially the quadratic function, understand the monotonicity of the function, the maximum (small) value and its geometric significance; combined with specific functions, understand the meaning of parity.

5 Learn to use the graph of a function to understand and study the nature of the function (see Example 1).

(2) exponential functions

1 (cell division, decay of C used in archaeology, changes in drug residues in the body, etc.), to understand the practical context of the exponential function model.

2 Understand the meaning of rational exponential powers, understand the significance of exponential powers of real numbers through specific examples, and master the operation of powers.

3 Understand the concept and meaning of exponential functions, be able to graph specific exponential functions with the aid of a calculator or computer, and explore and understand the monotonicity and special points of exponential functions.

4 In the process of solving simple practical problems, appreciate that exponential functions are an important class of function models.

(3) logarithmic function

1 understand the concept of logarithm and its arithmetic properties, and know that the base-exchange formula can be converted from general logarithm to natural logarithm or common logarithm; through the reading materials, to understand the history of the generation of logarithms and the role of simplifying arithmetic.

2 through specific examples, intuitive understanding of the logarithmic function model portrayed by the quantitative relationships, a preliminary understanding of the concept of logarithmic function, the logarithmic function is a class of important function model; can use a calculator or computer to draw a specific logarithmic function graph, explore and understand the monotonicity of the logarithmic function and the special point.

3 know that the exponential function and the logarithmic function are inverse functions of each other (a>0, a ≠ 1).

(4) Power Functions

Through examples, understand the concept of power functions; combined with the graph of the function, to understand their changes.

(5) Functions and equations

1 Combined with the graph of a quadratic function, determine the existence of the roots of a quadratic equation and the number of roots, so as to understand the connection between the zeros of a function and the roots of an equation.

2 According to the graph of a specific function, to be able to use a calculator to find the approximate solution of the corresponding equation by the bisection method, and to understand that this method is a commonly used method to find the approximate solution of an equation.

(6) Functional models and their applications

1 Using computational tools, to compare the exponential function, logarithmic function, and power function growth differences; combined with examples to appreciate the meaning of the growth of different types of functions such as straight line rise, exponential explosion, logarithmic growth.

2 collect some examples of function models (exponential function, logarithmic function, power function, segmented function, etc.) that are commonly used in social life, and understand the wide application of function models.

Second, trigonometric functions

(1) arbitrary angles, radians

Understand the concept of arbitrary angles and radian system, can carry out radian and angle reciprocal.

(2) Trigonometric Functions

1 Understand the definition of trigonometric functions (sine, cosine, tangent) of arbitrary angles with the help of unit circle.

2 With the help of trigonometric lines in the unit circle derive the induced formulas (sine, cosine, tangent), draw the graphs and understand the periodicity of trigonometric functions.

3 with the help of graphs to understand the sine function, cosine function in , tangent function on the properties (such as monotonicity, maximum and minimum values, graphs and x-axis intersection, etc.).

4 understand the basic equation of the trigonometric function of the same angle:

5 combined with specific examples, to understand the practical significance of the; can use a calculator or computer to draw the graph, observe the parameters of A, ω, the effect of the function on the graph of change.

6 will use trigonometric functions to solve some simple practical problems, and realize that trigonometric functions are an important function model for describing the phenomenon of periodic changes.

Third, series

(1) the concept of series and simple representation

Understand the concept of series and several simple representation (list, graph, general formula), understand that the series is a special function.

(2) Equal Difference and Equal Ratio Series

1 Understand the concept of equal difference and equal ratio series.

2 Explore and master the general formula and the formula for the sum of the first n terms of an equal-difference series and an equal-ratio series.

3 To be able to discover the equidistant or isoperimetric relationship of the series in a specific problem situation and to be able to use the knowledge to solve the corresponding problems (see Example 1).

4 To appreciate the relationship between the equal-difference series, the equal-ratio series and the primary function, exponential function.

IV. Inequalities

(1) Inequality

Feel the existence of a large number of unequal relationships in the real world and daily life, and understand the practical background of inequalities (groups).

(2) Quadratic Inequalities

1 Experience the process of abstracting the model of quadratic inequalities from practical contexts.

2 Understand the connection between a quadratic inequality and the corresponding function and equation through function graphs.

3 To be able to solve quadratic inequalities and attempt to design a block diagram of a program to solve a given quadratic inequality.

(3) Sets of Binary Inequalities and Simple Linear Programming Problems

1 Abstract sets of binary inequalities from real-world situations.

2 Understand the geometric significance of binary inequalities and be able to represent groups of binary inequalities in a plane region.

3 Abstract some simple binary linear programming problems from real-world situations and be able to solve them (.

(4) Fundamental Inequalities:

1 Explore and understand the process of proving fundamental inequalities.

2 To solve simple maximum (small) value problems using basic inequalities.

V. Preliminary Three-dimensional Geometry

(1) Space Geometry

1 Observe a large number of spatial shapes using physical models and computer software to recognize the structural features of columns, cones, tables, spheres, and their simple composites, and to be able to use these features to describe the structure of simple objects in real life.

2 To be able to draw three views of simple space figures (simple combinations of rectangles, spheres, cylinders, cones, prisms, etc.), to recognize three-dimensional models represented by the above three views, to use materials (e.g., cardboard) to make models, and to draw their visualizations by the oblique bisector method.

3 Understand the different forms of representation of spatial figures by observing views and visualizations drawn in both methods (parallel projection and central projection).

4 to complete the internship assignments, such as drawing some of the views and visualization of the building (on the basis of not affecting the characteristics of the graphics, dimensions, lines, etc. are not strictly required).

5 Understand the formula for calculating the surface area and volume of a ball, prism, prismatic cone, table (not required to memorize the formula).

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

1 with the rectangular model, in the intuitive knowledge and understanding of space points, lines and surfaces on the basis of the positional relationship, the abstraction of the definition of space lines and surfaces positional relationship, and to understand the following can be used as a basis for reasoning axioms and theorems.

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

Axiom 2:There is one and only one plane through three points that are not on a line.

Axiom 3:If two planes that do not coincide have a common **** point, then they have and only one common **** line through that point.

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

Theorem:Two angles in space are equal or complementary if two sides of each angle correspond to parallel.

2 With the above definitions, axioms and theorems of three-dimensional geometry as the starting point, we can recognize and understand the properties and determinations of parallel and perpendicular lines in space through intuitive perception, operational confirmation, and discursive argumentation.

The operation confirms and summarizes the following theorems.

A line outside a plane is parallel to a line inside this plane.

Two intersecting lines in one plane are parallel to another plane, then the two planes are parallel.

A line that is perpendicular to two intersecting lines in a plane is perpendicular to that plane.

A plane is perpendicular to two planes if it crosses the perpendicular of another plane.

The operations are confirmed by generalizing the following property theorems and proving them.

If a line is parallel to a plane, then the line of intersection of any plane passing through the line with this plane is parallel to the line.

If two planes are parallel, then the lines of intersection of any one of the planes with the two planes are parallel to each other.

Two lines perpendicular to the same plane are parallel.

If two planes are perpendicular, then a line perpendicular to the line of intersection in one plane is perpendicular to the other plane.

3 Be able to use the conclusions obtained to prove some simple propositions about spatial location relationships.

Preliminary Plane Analytic Geometry:

(1) Straight lines and equations

1 Explore the geometrical elements that determine the position of a straight line in a planar right-angled coordinate system in the context of a specific figure.

2 Understand the concepts of angle of inclination and slope of a straight line, experience the process of inscribing the slope of a straight line algebraically, and master the formula for calculating the slope of a straight line passing through two points.

3 Be able to determine whether two lines are parallel or perpendicular based on slope.

4 Based on the geometric elements of determining the position of a line, explore and master several forms of the equation of a line (point-slope, two-point, and general), and appreciate the relationship between slope-intercept and primary functions.

5 Be able to solve systems of equations to find the coordinates of the intersection of two lines.

6 Explore and master the formula for the distance between two points, the formula for the distance from a point to a line, and can find the distance between two parallel lines.

(2) Circles and Equations

1 Recall the geometric elements that determine a circle, and explore and master the standard and general equations of a circle in a plane right-angle coordinate system.

2 Be able to determine the positional relationship between a line and a circle, and a circle and a circle, based on the equations of the given line and circle.

3 Be able to solve some simple problems using the equations of a line and a circle.

(3) In the process of learning the preliminary plane analytic geometry, experience the idea of using algebraic methods to deal with geometric problems.

(4) Spatial Cartesian Coordinate System

1 Through specific situations, feel the necessity of establishing a spatial Cartesian Coordinate System, understand the spatial Cartesian Coordinate System, and will use the spatial Cartesian Coordinate System to inscribe the position of points.

2 Explore and derive the formula for the distance between two points in space by representing the coordinates of the vertices of a special rectangular body (all the prongs are parallel to the coordinate axes respectively).