Mathematics compulsory 2

A preliminary study on 1. solid geometry

(about 18 class hours)

(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 two sides of two angles in space are parallel to each other, 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.

◆ If the straight line out of the plane is parallel to the straight line in the plane, the straight line is parallel to the plane.

◆ Two intersecting straight lines in one plane are parallel to another plane, so the two planes are parallel.

◆ If a straight line is perpendicular to two intersecting straight lines in the plane, the straight line is perpendicular to the plane.

◆ If one plane intersects the perpendicular of another 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, the intersection line between any plane passing through the straight line and the plane is parallel to the straight line.

◆ If two planes are parallel, the intersection lines obtained by the intersection of any plane and these two planes are parallel to each other.

◆ Two straight lines perpendicular to the same plane are parallel.

◆ If two planes are perpendicular, the straight line perpendicular to the intersection line in one plane is perpendicular to the other plane.

③ We can use the conclusions to prove some simple propositions of spatial relationship.

2. Analysis of Plane Analytic Geometry

(about 18 class hours)

(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.

Analytic geometry

1, straight line

Linear equation of the distance between two points and a fixed fractional point

|AB|=| |

|P 1P2|=

y-y 1=k(x-x 1)

y=kx+b

2. Conic curve

Circular ellipse

The standard equation (x-a) 2+(y-b) 2 = R2.

The center of the circle is (a, b) and the radius is r.

The general equation x2+y2+dx+ey+f = 0.

Where the center of the circle is (),

Radius r

(1) Use the distance d from the center of the circle to the straight line and the radius r of the circle to judge or use the discriminant to judge the positional relationship between the straight line and the circle.

(2) Use the sum and difference of center distance d and radius to judge the positional relationship between two circles.

hyperbola

Focus F 1 (-c, 0), F2(c, 0)

(a，b>0，b2=c2-a2)

weird

collinearity equation

The focal radius | mf 1 | = ex0+a, | mf2 | = ex0-a parabola y2 = 2px (p > 0).

Focus f

collinearity equation

translation of axes

Here (h, k) is the coordinate of the origin of the new coordinate system in the original coordinate system.

[Edit this paragraph] Mathematics required course 3

1. Preliminary algorithm

(about 12 class hours)

The meaning of (1) algorithm, program block diagram.

(1) by analyzing the process and steps to solve specific problems (such as solving binary linear equations, etc.). ), we can understand the idea and significance of the algorithm.

② Through imitation, operation and exploration, experience the process of expressing and solving problems by designing program block diagram. In the process of solving specific problems (such as solving ternary linear equations, etc. ), understand the three basic logical structures of program block diagram: sequence, conditional branch and loop.

(2) Basic algorithm statements: Through the process of transforming the program block diagram of specific problems into program statements, we can understand several basic algorithm statements-input statements, output statements, assignment statements, conditional statements and loop statements, and further understand the basic idea of the algorithm.

(3) By reading the algorithm cases in ancient mathematics in China, we can understand the contribution of ancient mathematics in China to the development of mathematics in the world.

2. Statistics

(about 16 class hours)

(1) random sampling

(1) can raise some valuable statistical questions from real life or other disciplines.

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

③ In the process of solving statistical problems, learn to use simple random sampling method to extract samples from the population; Through case study, we can understand the methods of stratified sampling and systematic sampling.

④ Data can be collected through experiments, consulting materials and designing questionnaires.

(2) estimate the population with samples

① Understand the significance and function of distribution through examples. In the process of representing sample data, learn to list the frequency distribution table, draw the frequency distribution histogram, frequency line diagram and stem leaf diagram (see example 1), and understand their respective characteristics.

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

③ We can reasonably select samples according to the needs of practical problems, extract basic numerical features (such as mean and standard deviation) from sample data, and make reasonable explanations.

④ In the process of solving statistical problems, we will further understand the idea of estimating the population with samples. We will estimate the population distribution with the frequency distribution of samples and estimate the basic digital characteristics of the population with the basic digital characteristics of samples. Understand the randomness and numerical characteristics of sample frequency distribution.

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

⑥ Form a preliminary evaluation consciousness of data processing.

(3) Correlation of variables

① Make a scatter plot by collecting the data of two related variables in the real question, and use the scatter plot to intuitively understand the correlation between variables.

② Experiencing the process of describing the linear correlation of two variables with different estimation methods. Knowing the idea of least square method, we can establish a linear regression equation according to the given coefficient formula of linear regression equation (see Example 2).

3. Possibility

(about 8 class hours)

(1) Understand the uncertainty and frequency stability of random events in specific situations, and further understand the meaning of probability and the difference between frequency and probability.

(2) Understand two mutually exclusive events's probability addition formulas through examples.

(3) Through examples, we can understand the classical probability and its probability calculation formula, and use enumeration method to calculate the number of basic events and the probability of some random events.

(4) Knowing the meaning of random numbers, we can use simulation methods (including random numbers generated by calculators for simulation) to estimate the probability and get a preliminary understanding of the meaning of geometric probability (see Example 3).

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

~ I hope I can help you . .