First,? gather
The Meaning and Representation of (1) Set
① Understand the meaning of set and the "subordinate" relationship between elements and set through examples.
② We can choose natural language, graphic language and assembly language (enumeration or description) to describe different specific problems and feel the significance and function of assembly language.
(2) the basic relationship between sets
① By understanding the meaning of inclusion and equality between sets, we can identify a subset of a given set.
② Understand the meaning of complete works and empty sets in specific situations.
(3) Basic operations of sets
① To understand the meaning of union and intersection of two sets, we require union and intersection of two simple sets.
② Understanding the meaning of the complement set of a subset in a given set will lead to the complement set of a given subset.
(3) venn diagram can be used to express the relations and operations of sets, and the role of intuitive graphs in understanding abstract concepts can be realized.
Function concept and basic elementary function;
(1) function
① Further understand that function is an important mathematical model to describe the dependence between variables, and on this basis, learn to describe functions with sets and corresponding languages, and understand the role of correspondence in describing the concept of functions; Knowing the elements that make up a function, we can find the definition and value range of some simple functions; Understand the concept of mapping.
② In actual situations, appropriate methods (such as image method, list method and analysis method) will be selected according to different needs to express functions.
③ Understand the simple piecewise function and apply it simply.
④ Understand the monotonicity, maximum (minimum) value and its geometric significance of the function through the learned function, especially the quadratic function; Understand the meaning of parity with specific functions.
⑤ Learn to use function images to understand and study the properties of functions (see example 1).
(2) Exponential function
(1) (cell division, the decay of archaeological C, the change of drug 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 operation of power.
③ To understand the concept and significance of exponential function, we can 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, I realized that exponential function is an important function model.
(3) Logarithmic function
(1) Understand the concept of logarithm and its operational properties, and know that general logarithm can be converted into natural logarithm or ordinary logarithm by changing the base formula; By reading the materials, we can understand the history of logarithm and its role in simplifying operations.
② Through concrete examples, we can intuitively understand the quantitative relationship described by the logarithmic function model, preliminarily understand the concept of logarithmic function, and realize that logarithmic function is an important function model; With the help of calculator or computer, we can draw images of specific logarithmic functions and explore and understand the monotonicity and special points of logarithmic functions.
③ Know that exponential function and logarithmic function are reciprocal functions (A >；); 0，a≠ 1)。
(4) Power function
Understand the concept of power function through examples; Combine the images of functions to understand their changes.
(5) Functions and equations
① Combining the image of quadratic function, we can judge the existence and number of roots of quadratic equation in one variable, so as to understand the relationship between zero point of function and roots of equation.
(2) According to the image of a specific function, it is a common method to find the approximate solution of the corresponding equation by dichotomy with the help of a calculator.
(6) Function model and its application
① Compare the growth differences of exponential function, logarithmic function and power function with calculation tools; Combined with examples, we can understand the meaning of growth of different function types such as linear rise, exponential explosion and logarithmic growth.
② Collect some examples of function models (exponential function, logarithmic function, power function, piecewise function, etc. ) It is often used in social life to understand the wide application of functional models.
Second, trigonometric functions
(1) Any angle and radian
Understand the concept of arbitrary angle and radian system, and realize the conversion between radian and angle.
(2) Trigonometric function
① Understand the definition of trigonometric functions (sine, cosine and tangent) with the help of the unit circle.
② Derive inductive formulas (sine, cosine and tangent) with the help of trigonometric function lines in the unit circle, and draw pictures to understand the periodicity of trigonometric functions.
③ Understand the properties of sine function, cosine function and tangent function (such as monotonicity, maximum and minimum value, image intersecting with X axis, etc.). ) with the help of images.
④ Understand the basic relationship of trigonometric functions with the same angle:
⑤ Understand the practical significance with concrete examples; With the help of the image drawn by calculator or computer, we can observe the influence of parameters a and ω on the change of function image.
⑥ trigonometric function can be used to solve some simple practical problems, and it is recognized that trigonometric function is an important function model to describe periodic changes.
Third, the order
The Concept and Simple Representation of (1) Sequence
Understand the concept of sequence and several simple expressions (list, image, general formula), and understand that sequence is a special function.
(2) arithmetic progression and geometric progression
(1) Understand the concepts of arithmetic progression and geometric progression.
② Explore and master the general formula of arithmetic progression and geometric progression and the formula of the sum of the first n items.
(3) In specific problem situations, we can find the arithmetic relationship or proportional relationship of the sequence, and use relevant knowledge to solve corresponding problems (see example 1).
④ Understand the relationship between arithmetic progression and geometric progression, linear function and exponential function.
Fourth, inequality.
(1) inequality relation
Feel a lot of unequal relations between the real world and daily life, and understand the actual background (group) of inequality.
(2) One-dimensional quadratic inequality
① Experience the process of abstracting a quadratic inequality model from the actual situation.
(2) Understand the relationship between unary quadratic inequality and corresponding functions and equations through function images.
(3) Can solve the unary quadratic inequality, and try to design the program block diagram for the given unary quadratic inequality.
(3) Binary linear inequalities and simple linear programming problems.
① A set of binary linear inequalities is abstracted from the actual situation.
(2) Understand the geometric meaning of binary linear inequalities, and express binary linear inequalities in groups according to plane regions.
③ Some simple binary linear programming problems are abstracted from the actual situation and can be solved.
(4) Basic inequality:
① Explore and understand the process of proving basic inequalities.
② Basic inequalities can be used to solve simple maximum (minimum) problems.
V. Preliminary solid geometry
(1) space geometry
① Using physical models and computer software to observe a large number of spatial graphics, we can understand the structural characteristics of columns, cones, platforms, balls and their simple combinations, and can use these characteristics to describe the structure of simple objects in real life.
(2) Can draw three views of simple space graphics (simple combination of cuboid, sphere, cylinder, cone, prism, etc.). ), can identify the three-dimensional model represented by the above three views, can make models with materials (such as cardboard), and can draw their own front views by oblique double-sided method.
③ By observing the views and straight views drawn by two methods (parallel projection and central projection), we can understand the different representations of spatial graphics.
(4) Complete the internship, such as drawing some views and front views of buildings (the requirements for size and lines are not strict without affecting the graphic characteristics).
⑤ Understand the formulas for calculating the surface area and volume of spheres, prisms, pyramids and platforms (no need to memorize formulas).
(2) the positional relationship between points, lines and surfaces
(1) With the help of the cuboid model, on the basis of intuitive knowledge and understanding of the positional relationship between points, lines and surfaces in space, the definition of the positional relationship between lines and surfaces in space is abstracted, and the following axioms and theorems that can be used as the basis of reasoning are understood.
Axiom 1: If two points on a straight line are on a plane, then the straight line is on this plane.
Axiom 2: When three points that are not on a straight line intersect, there is one and only one plane.
Axiom 3: If two non-coincident planes have a common point, then they have one and only one common straight line passing through the point.
Axiom 4: Two lines parallel to the same line are parallel.
Theorem: If the two sides of two angles in space are parallel, then the two angles are equal or complementary.
② Based on the above-mentioned definitions, axioms and theorems of solid geometry, we can know and understand the related properties and judgments of parallelism and verticality of straight lines and planes in space through intuitive perception, operational confirmation and speculative argumentation.
Operation confirmation, summed up the following judgment theorem.
A straight line out of the plane is parallel to a straight line in the plane, then the straight line is parallel to the plane.
Two intersecting straight lines in one plane are parallel to the other plane, so the two planes are parallel.
A straight line is perpendicular to two intersecting straight lines on a plane, so this straight line is perpendicular to the plane.
When one plane intersects the perpendicular of the other plane, the two planes are perpendicular.
The operation is confirmed, and the following property theorems are summarized and proved.
If a straight line is parallel to a plane, then the intersection of any plane passing through this straight line and this plane is parallel to this straight line.
If two planes are parallel, the intersection line obtained by the intersection of any plane and these two planes is parallel to each other.
Two straight lines perpendicular to the same plane are parallel.
If two planes are perpendicular, a straight line perpendicular to the intersection in one plane is perpendicular to the other plane.
(3) Some simple propositions of spatial relationship can be proved by using the obtained conclusions.
A preliminary study on plane analytic geometry;
(1) row sum equation
(1) In the plane rectangular coordinate system, combined with specific graphics, the geometric characteristics of determining the position of a straight line are explored.
② Understand the concepts of inclination angle and slope of a straight line, experience the process of describing the slope of a straight line by algebraic method, and master the calculation formula of the slope of a straight line passing through two points.
③ Two straight lines can be judged to be parallel or vertical according to their slopes.
(4) According to the geometric characteristics of determining the position of a straight line, explore and master several forms of linear equation (point oblique, two points, general), and understand the relationship between oblique line and linear function.
⑤ The coordinates of the intersection of two straight lines can be obtained by solving the equation.
⑥ Explore and master the distance formula between two points and the distance formula from point to straight line, and find the distance between two parallel straight lines.
(2) Circle sum equation
(1) review and determine the geometric characteristics of the circle, explore and master the standard equation and general equation of the circle in the plane rectangular coordinate system.
② According to the given equation of straight line and circle, we can judge the positional relationship between straight line and circle and between them.
③ Some simple problems can be solved by equations of straight lines and circles.
(3) During the initial study of plane analytic geometry, I realized the idea of using algebraic method to deal with geometric problems.
(4) Spatial Cartesian coordinate system
(1) Through specific situations, feel the necessity of establishing a spatial rectangular coordinate system, understand the spatial rectangular coordinate system, and describe the position of points by using the spatial rectangular coordinate system.
② By representing the coordinates of the vertices of a special cuboid (each side is parallel to the coordinate axis), the distance formula between two points in space is explored.