I. Algorithms, Probability and Statistics
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 (for example, 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, I 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.
3. Probability (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.
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 linear regression equation according to the given coefficient formula of linear regression equation.
Two. Common logical terms
1。 Proposition and its relationship
① Understand the inverse proposition, negative proposition and negative proposition of a proposition.
② Understand the meanings of necessary conditions, sufficient conditions and necessary and sufficient conditions, and analyze the relationship among the four propositions.
(2) Simple logical connectives
Understand the meaning of "or", "and" and "not" through mathematical examples.
(3) Full name quantifiers and existential quantifiers
① Understand the meanings of full-name quantifiers and existential quantifiers through abundant examples in life and mathematics.
② Propositions containing quantifiers can be correctly denied.
3. Derivative and its application (about 16 class hours)
The Concept of (1) Derivative and Its Geometric Significance
(1) Through the analysis of a large number of examples, through the transition from the average rate of change to the instantaneous rate of change, we can understand the actual background of the concept of derivative, know that the instantaneous rate of change is a derivative, and understand the idea and connotation of derivative (see examples 2 and 3).
② Understand the geometric meaning of derivative intuitively through function images.
(2) the operation of derivative
(1) According to the definition of derivative, the derivatives of functions y=c, y=x, y=x2, y= 1/x can be obtained.
② We can use the derivative formula of basic elementary function and the four algorithms of derivative to find the derivative of simple function.
③ A derivative formula table can be used.
(3) The application of derivative in function research.
(1) Explore and understand the relationship between monotonicity and derivative of functions with examples (see Example 4); The monotonicity of function can be studied by derivative, and the monotone interval of polynomial function with no more than three degrees can be found.
② Understand the necessary and sufficient conditions for the function to obtain the extreme value at a certain point by combining the image of the function; The derivative will be used to find the maximum and minimum values of polynomial functions with no more than three degrees, and the maximum and minimum values of polynomial functions with no more than three degrees in a given interval. 2. Conic Curve and Equation (about 12 class hours)
(1) Understand the actual background of conic section and feel the role of conic section in depicting the real world and solving practical problems.
(2) Go through the process of abstracting an ellipse model from a specific situation (see example 1), and master the definition, standard equation and simple geometric properties of an ellipse.
(3) Understand the definitions, geometric figures and standard equations of parabola and hyperbola, and know their simple geometric properties.
(4) By studying conic curves and equations, we can further understand the idea of combining numbers with shapes.
(5) Understand the simple application of conic curve.
Three. Statistical cases (about 14 class hours)
Through typical cases, learn the following commonly used statistical methods, and can initially apply these methods to solve some practical problems.
① Explore typical cases (such as "Is lung cancer related to smoking?" ), you can understand the basic idea, method and preliminary application of independence test (only 2×2 contingency table is needed).
② Through the exploration of typical cases (such as "quality control" and "whether the new drug is effective"), we can understand the basic ideas, methods and preliminary applications of the actual inference principle and hypothesis testing (see example 1).
③ Explore typical cases (such as Insect Classification). ), you can understand the basic ideas, methods and preliminary applications of cluster analysis.
④ By exploring typical cases (such as "the relationship between people's weight and height"), we can further understand the basic ideas, methods and preliminary applications of regression.
2. Reasoning and proof (about 10 class hour)
(1) Rational reasoning and deductive reasoning
(1) Understand the meaning of sensible reasoning by combining the learned mathematical examples with real life examples. We can conduct simple reasoning through induction and analogy, and experience and understand the role of perceptual reasoning in mathematical discovery (see Examples 2 and 3).
(2) With examples in mathematics and life, understand the importance of deductive reasoning, master the basic methods of deductive reasoning, and apply them to some simple reasoning.
③ Understand the connection and difference between perceptual reasoning and deductive reasoning through concrete examples.
(2) Direct proof and indirect proof
① Understand the two basic methods of direct proof: analytical method and synthesis method, and combine the mathematical examples that have been learned; Understand the thinking process and characteristics of analytical methods and comprehensive methods.
(2) Understand a basic method of indirect proof-reduction to absurdity, combining with the mathematical examples that have been learned; Understand the thinking process and characteristics of reduction to absurdity.