The national unified entrance examination for colleges and universities is a selective examination attended by qualified high school graduates and candidates with the same academic ability. Colleges and universities choose the best candidates according to their scores and established enrollment plans. Therefore, the college entrance examination should have high reliability, validity, necessary discrimination and appropriate difficulty.
Two. Examination requirements
According to the requirements of ordinary colleges and universities for freshmen's cultural quality, as well as the teaching contents of compulsory courses and elective courses I in the Curriculum Plan of Full-time Ordinary Senior High School and the Mathematics Teaching Outline of Full-time Ordinary Senior High School promulgated by the Ministry of Education in 2002, the mathematics subjects in 20 1 1 National Unified Examination Outline for Enrollment of Ordinary Colleges and Universities (liberal arts) are taken as the proposition scope of mathematics subjects in the college entrance examination for literature and history.
In the mathematics examination, according to the principle of "while examining the basic knowledge, we should pay attention to the ability", establish the guiding ideology of the ability-based proposition, integrate the examination of knowledge, ability and quality, and comprehensively test the mathematics literacy of candidates.
Mathematics examination should play the role of mathematics as a basic subject, which not only examines the knowledge and methods of mathematics in middle schools, but also examines the potential of candidates for further study in colleges and universities.
First, the knowledge requirements, ability requirements and personality quality requirements of the examination content
1. knowledge requirements
Knowledge refers to the mathematical concepts, properties, laws, formulas, axioms, theorems and mathematical ideas and methods in the teaching content stipulated in the mathematics syllabus of full-time senior middle schools.
The requirements for knowledge are: understanding, understanding and mastering, flexible application and comprehensive application.
(1) Understanding: It is required to have a preliminary perceptual understanding of the meaning of the listed knowledge and its related background, and know what this knowledge content is, which can (or will) be identified in related issues.
(2) Understanding and mastery: It requires a profound theoretical understanding of the listed knowledge, the ability to explain, exemplify, deform and infer, and the use of knowledge to solve related problems.
(3) Flexible and comprehensive application: It requires the system to master the internal relations of knowledge and be able to use the listed knowledge to analyze and solve more complex or comprehensive problems.
2. Capability requirements
Ability refers to thinking ability, calculating ability, spatial imagination ability, practical ability and innovative consciousness.
(1) Thinking ability: observing, comparing, analyzing, synthesizing, abstracting and summarizing problems or data; Able to use analogy, induction and deduction for reasoning; Can be expressed logically and accurately.
Mathematics is the science of thinking, and thinking ability is the core of mathematics discipline ability. Mathematical thinking ability is based on mathematical knowledge, through spatial imagination, intuitive guess, inductive abstraction, symbolic representation, operational solution, deductive proof and pattern construction, thinking and judging the spatial form, quantitative relationship and mathematical model in objective things, forming and developing rational thinking and forming the subject of mathematical ability.
(2) Calculation ability: it can perform correct operation, deformation and data processing according to laws and formulas; According to the conditions and objectives of the problem, find and design a reasonable and simple operation mode; Can estimate and approximate data as needed.
Operational ability is the combination of thinking ability and operational skills. Operations include numerical calculation, estimation and approximate calculation, combination and decomposition of formulas, calculation and solution of geometric quantities of geometric figures, etc. Combat capability includes the thinking ability in a series of processes, such as analyzing operational conditions, exploring operational direction, selecting operational formulas, determining operational procedures, etc., and also includes the ability to adjust operations when encountering obstacles in operational implementation and the skills to carry out operations and calculations.
(3) Space imagination: being able to make correct graphics according to conditions and imagine intuitive images according to the graphics; Can correctly analyze the basic elements and their relationships in graphics; Can decompose, combine and transform graphics; Will use graphics and charts to vividly reveal the essence of the problem.
Spatial imagination ability is the ability to observe, analyze and abstract spatial form, which is mainly manifested in the ability to recognize, draw and imagine graphics. Drawing refers to the transformation of written language and symbolic language into graphic language, and the addition or various transformations of auxiliary graphics to graphics; Graphic imagination mainly includes pictographic and non-pictographic, which is a high-level symbol of spatial imagination.
(4) Practical ability: being able to comprehensively apply the learned mathematical knowledge, ideas and methods to solve problems, including solving simple mathematical problems in related disciplines, production and life; Be able to understand the materials stated in the question, summarize, sort out and classify the information provided, abstract the actual problem into a mathematical problem and establish a mathematical model; Can apply relevant mathematical methods to solve problems and verify them, and can correctly express and explain them in mathematical language.
Practical ability is the ability to mathematize objective things. The main process is to refine the relevant quantitative relations, conceive a mathematical model, turn practical problems into mathematical problems and solve them according to the real life background.
(5) Innovative consciousness: for novel information, situations and problems, choose effective methods and means to analyze information, comprehensively and flexibly use the learned mathematical knowledge, ideas and methods, conduct independent thinking, exploration and research, put forward ideas to solve problems, and creatively solve problems.
Innovative consciousness is the advanced expression of rational thinking. Observing, guessing, abstracting, summarizing and proving mathematical problems is an important way to find and solve problems. The higher the degree of transfer, combination and integration of mathematical knowledge, the stronger the sense of innovation.
3. Personality quality requirements
Personality quality refers to the individual feelings, attitudes and values of candidates. Candidates are required to have a certain mathematical vision, understand the scientific value and humanistic value of mathematics, advocate the rational spirit of mathematics, form the habit of prudent thinking and appreciate the aesthetic significance of mathematics.
Candidates are required to overcome their nervousness, take the test with a peaceful mind, control the test time reasonably, answer the test questions with a scientific attitude of seeking truth from facts, establish confidence in overcoming difficulties, and embody the spirit of perseverance.
Second, the examination requirements
The systematicness and rigor of mathematics discipline determine the profound internal relationship between mathematical knowledge, including the vertical relationship between each part of knowledge in their respective development process and the horizontal relationship between each part of knowledge. We should be good at grasping these relations in essence, and then construct the structural framework of mathematics test papers through classification, combing and synthesis.
(1) The examination of the basic knowledge of mathematics should be comprehensive and focused, and the key contents supporting the subject knowledge system should account for a large proportion, which constitutes the main body of the mathematics examination paper. It should pay attention to the internal relations of disciplines and the comprehensiveness of knowledge, and not deliberately pursue the coverage of knowledge. We should consider the problem from the overall height of the subject and the height of thinking value, design test questions at the intersection of knowledge networks, and make the examination of basic mathematics knowledge reach the necessary depth.
(2) The examination of mathematical thinking method is an abstract and generalized examination of mathematical knowledge at a higher level, which must be combined with mathematical knowledge to reflect the examinee's understanding of mathematical thinking method; Starting from the overall significance and ideological value of the subject, we should attach importance to general methods and downplay special skills, and effectively test candidates' mastery of mathematical ideas and methods contained in middle school mathematics knowledge.
(3) The examination of mathematical ability emphasizes "thinking with ability", that is, taking mathematical knowledge as the carrier, starting from the problem, grasping the overall meaning of the topic, organizing materials with a unified mathematical point of view, and paying attention to the understanding and application of knowledge, especially the comprehensive and flexible application, in order to test the ability of candidates to transfer knowledge to different situations, so as to test the breadth and depth of candidates' individual rational thinking and the potential for further study.
The examination of ability takes thinking ability as the core, comprehensively examines various abilities, emphasizes comprehensiveness and application, and conforms to the reality of candidates. The examination of thinking ability runs through the whole volume, focusing on the examination of rational thinking, emphasizing the scientific, rigorous and abstract thinking. The examination of computing ability is mainly the examination of arithmetic and logical reasoning, mainly algebraic operation, estimation and simplification. The examination of spatial imagination ability is mainly reflected in the mutual transformation of written language, symbolic language and graphic language, and in the recognition, understanding and processing of graphics. The examination should combine computing ability and logical thinking ability.
(4) The examination of practical ability mainly adopts the form of solving application problems. The proposition should adhere to the principle of "close to life, fair background and control difficulty", and the design of test questions should conform to the reality of mathematics teaching in middle schools in China, taking into account the age characteristics and practical experience of candidates, so that the difficulty of mathematics application problems can meet the level of candidates.
(5) The examination of innovative consciousness is an examination of advanced rational thinking. In the examination, we should create relatively novel question situations, construct mathematical questions with certain depth and breadth, pay attention to the diversity of questions and reflect the divergence of thinking. We should carefully design problems, examine the main contents of mathematics and reflect the quality of mathematics. Test questions that reflect the movement changes of numbers and shapes; Research-oriented, exploratory and open questions.
On the basis of examining the basic knowledge, mathematics subject proposition pays attention to the examination of mathematical thinking methods, the examination of mathematical ability, the display of mathematical science and humanistic value, the consideration of the foundation, comprehensiveness and reality of test questions, the hierarchy among test questions, the reasonable stipulation of comprehensive degree and the examination from multiple angles and levels, and strives to meet the requirements of comprehensive examination of mathematical comprehensive literacy.
Ⅲ. Examination contents
1. plane vector
Examination content:
Vector. Addition and subtraction of vectors. The product of real numbers and vectors. Coordinate representation of plane vector. The fixed point of a line segment. The product of plane vectors. The distance between two points on a plane. Translation.
Examination requirements:
(1) Understand the concept of vector, master the geometric representation of vector, and understand the concept of * * * line vector.
(2) Master the addition and subtraction of vectors.
(3) Grasp the product of real number and vector, and understand the necessary and sufficient conditions of two vector lines.
(4) Understand the basic theorem of plane vector, understand the coordinate concept of plane vector, and master the coordinate operation of plane vector.
(5) Grasp the quantitative product of plane vector and its geometric meaning, understand that the quantitative product of plane vector can deal with problems related to length, angle and verticality, and grasp the conditions of vector verticality.
(6) Master the distance formula between two points on the plane and the coordinate formula of the bisector and midpoint of the line segment, and skillfully use it. Master the translation formula.
2. Set and simple logic
Examination content:
Set, subset, complement, intersection and union.
Logical connectives. Four propositions. Sufficient conditions and necessary conditions.
Examination requirements:
(1) Understand the concepts of set, subset, complement set, intersection set and union set, understand the meanings of empty set and complete set, understand the meanings of ownership, inclusion and equality, master relevant terms and symbols, and correctly use them to represent some simple sets.
(2) Understand the meanings of logical conjunctions "OR", "Qi" and "Fei", understand the four propositions and their relationships, and grasp the significance of sufficient conditions, necessary conditions and necessary and sufficient conditions.
3. Function
Examination content:
Mapping, function, monotonicity and parity of function.
Inverse function. The relationship between function images of reciprocal functions.
Extension of the concept of exponent. Operational properties of rational exponential powers. Exponential function.
Logarithm Operational properties of logarithm. Logarithmic function.
Application of functions.
Examination requirements:
(1) Understand the concepts of mapping and function.
(2) Understand the concepts of monotonicity and parity of functions, and master the judgment methods of monotonicity and parity of some simple functions.
(3) Understanding the concept of inverse function and the relationship between function images which are mutually inverse functions, we will find the inverse functions of some simple functions.
(4) Understand the concept of fractional exponential power, master the operational properties of rational exponential power, and master the concept, image and properties of exponential function.
(5) Understand the concept of logarithm and master the operational nature of logarithm; Master the concept, image and properties of logarithmic function.
(6) We can use the properties of function, exponential function and logarithmic function to solve some simple practical problems.
4. Inequality
Examination content:
Inequality. Basic properties of inequality. Proof of inequality. The solution of inequality. Inequalities with absolute values.
Examination requirements:
The properties of (1) understanding inequality and its proof.
(2) Grasp the theorem that the arithmetic mean of two (not extended to three) positive numbers is not less than its geometric mean, and simply apply it.
(3) Mastering analysis, synthesis and comparison to prove simple inequalities.
(4) Master the solution of simple inequality.
(5) Understand the inequality │ A │-│ B │≤A+B │≤A │+│ B │.
5. Trigonometric function
Examination content:
Popularization of the concept of angle. Curvature system.
Trigonometric function at any angle. The trigonometric function line in the unit circle. The basic relations of trigonometric functions with the same angle are: sin2α+cos2α= 1, sinα/cosα=tanα, tanα cotα = 1. Inductive formulas of sine and cosine.
Sine, cosine and tangent of sum and difference of two angles. Sine, cosine and tangent of a double angle.
Images and properties of sine function and cosine function. Periodic function. The image of function y=Asin(ωx+φ). Images and properties of tangent function. Find the angle with the known trigonometric function value.
Sine theorem. Cosine theorem. Solution of oblique triangle.
Examination requirements:
(1) Understand the concept of arbitrary angle and the meaning of radian, and correctly convert radian and angle.
(2) Understand the definition of sine, cosine and tangent at any angle. Understand the definitions of cotangent, secant and cotangent; Master the basic relationship between trigonometric functions and angles, master the inductive formulas of sine and cosine, and understand the significance of periodic function and minimum positive period.
(3) Master the sine, cosine and tangent formulas of the sum and difference of two angles; Master the sine, cosine and tangent formulas of double angles.
(4) The trigonometric formula can be used correctly to simplify, evaluate and prove the identities of simple trigonometric functions.
(5) Understand the images and properties of sine function, cosine function and tangent function, and draw the graphs of sine function, cosine function and function y=Asin(ωx+φ) with the "five-point method" to understand the physical meanings of a, ω and φ.
(6) The angle will be obtained from the known trigonometric function values and represented by the symbols arcsinx, arccosx and arctanx.
(7) Master sine theorem and cosine theorem, and use them to solve oblique triangles.
6. Series
Examination content:
Sequence.
Arithmetic progression and his general formula. Arithmetic progression's first n-sum formula.
Geometric series and its general formula. The first n-sum formula of geometric series.
Examination requirements:
(1) Understand the concept of sequence, understand the meaning of general term formula of sequence, and understand recursive formula is a way to give sequence, and write the first few terms of sequence according to recursive formula.
(2) Understand the concept of arithmetic progression, master arithmetic progression's general formula and the first n summation formulas, and solve simple practical problems.
(3) Understand the concept of geometric series, master the general formula of geometric series and the first n summation formulas, and solve simple practical problems.
7. Equations of straight lines and circles
Examination content:
The inclination and slope of a straight line, the point inclination and two-point equation of a straight line equation. General formula of linear equation.
The condition that two straight lines are parallel and perpendicular. The intersection of two straight lines. Distance from point to straight line.
The plane region is represented by binary linear inequality. Simple linear programming problem.
Concepts of curves and equations. The curve equation is listed by known conditions.
Standard equation and general equation of a circle. Parametric equation of a circle.
Examination requirements:
(1) Understand the concepts of inclination angle and slope of a straight line, master the slope formula of a straight line passing through two points, master the point inclination formula, two-point formula and general formula of a straight line equation, and skillfully solve the straight line equation according to conditions.
(2) Knowing the condition that two straight lines are parallel and vertical, the angle formed by two straight lines and the distance formula from point to straight line, we can judge the positional relationship of two straight lines according to the equation of straight lines.
(3) Understand that binary linear inequalities represent plane regions.
(4) Understand the significance of linear programming and apply it simply.
(5) Understand the basic ideas of analytic geometry and coordinate method.
(6) Master the standard equation and general equation of a circle and understand the concept of parametric equation. Understand the parametric equation of a circle.
8. Conic curve equation
Examination content:
Ellipse and its standard equation. Simple geometric properties of ellipse. Parametric equation of ellipse.
Hyperbola and its standard equation. Simple geometric properties of hyperbola.
Parabola and its standard equation. Simple geometric properties of parabola.
Examination requirements:
(1) Master the definition, standard equation and simple geometric properties of ellipse, and understand the parameter equation of ellipse.
(2) Master the definition, standard equation and simple geometric properties of hyperbola.
(3) Master the definition, standard equation and simple geometric properties of parabola.
(4) Understand the preliminary application of conic curve.
Article 9(A). Straight line, plane and simple geometry (candidates can choose one of 9 (a) and 9(B))
Examination content:
Plane and its basic properties. Intuitive drawing method of plane graphics.
Parallel lines. The angle of a parallel side. The angle formed by lines of different planes. The common perpendicular of a straight line in different planes. Distance of straight lines on different planes.
Determination and properties of parallelism between straight line and plane, perpendicularity between straight line and plane, distance from point to plane, projection of oblique line on plane, angle between straight line and plane, triple verticality theorem and its inverse theorem.
Determination and properties of parallel planes. Distance between parallel planes. Dihedral angle and its plane angle. Determination and properties of the perpendicularity of two planes.
Polyhedron, regular polyhedron, prism, pyramid, sphere.
Examination requirements:
(1) To understand the basic properties of a plane, he can draw a vertical view of a horizontally placed plane figure by oblique sides. He can draw graphs of various positional relationships between two straight lines and a plane in space, and can imagine their positional relationships according to the graphs.
(2) Master the judgment theorem and property theorem of parallelism and verticality of two straight lines. Master the concepts of the angle and distance formed by two straight lines. For the distance of straight lines in different planes, we only need to calculate the distance given the common perpendicular.
(3) Mastering the concepts of judging theorem and property theorem of straight line parallel to plane, judging theorem and property theorem of straight line perpendicular to plane, projection of oblique line on plane, angle formed by straight line and plane, three perpendicular theorems and their inverse theorems.
(4) Master the judgment theorem and property theorem of two planes being parallel, the concepts of dihedral angle, its plane angle and the distance between two parallel planes, and the judgment theorem and property theorem of two planes being perpendicular.
(5) Simple problems can be proved by reduction to absurdity.
(6) Understand the concepts of polyhedron, convex polyhedron and regular polyhedron.
(7) Understand the concept and properties of prism and draw a straight prism.
(8) Understand the concept of the pyramid, master the nature of the regular pyramid, and draw a direct view of the regular pyramid.
(9) Understand the concept of the ball, master the properties of the ball, and master the surface area formula and volume formula of the ball.
Article 9 (b). Straight line, plane, simple geometry
Examination content:
Plane and its basic properties. Intuitive drawing method of plane graphics.
Parallel lines.
Determination and properties of parallelism between straight lines and planes, determination of verticality between straight lines and planes, triple verticality theorem and its inverse theorem.
The positional relationship between two planes.
Space vector and its addition, subtraction, multiplication and division. Coordinate representation of space vector. Quantity product of space vector.
Direction vector of straight line, angle formed by non-planar straight line, common perpendicular of non-planar straight line, and distance of non-planar straight line.
Verticality of straight line and plane, normal vector of plane, distance from point to plane, angle between straight line and plane, projection of vector on plane.
Determination and properties of parallel planes. Distance between parallel planes. Dihedral angle and its plane angle. Determination and properties of two planes being perpendicular.
Polyhedron Regular polyhedron Prism. Pyramid. Ball.
Examination requirements:
(1) To understand the basic properties of a plane, I can draw a vertical view of a horizontally placed plane figure by oblique survey. I can draw the graphs of various positional relationships between two straight lines, straight lines and planes in space. I can imagine their positional relationship according to the picture.
(2) Mastering the judging theorem and property theorem of parallel lines and planes, judging theorem of perpendicular lines and planes, three perpendicular lines theorem and its inverse theorem.
(3) Understand the concept of space vector and master the addition, subtraction, multiplication and division of space vector.
(4) Understand the basic theorem of space vector, understand the concept of space vector coordinates, and master the coordinate operation of space vector.
(5) Master the definition and properties of the product of space vector, the calculation formula of the product of space vector in rectangular coordinates, and the calculation formula of the distance between two points in space.
(6) Understand the concepts of the direction vector of a straight line, the normal vector of a plane and the projection of the vector on the plane.
(7) Master the concepts of angles and distances formed by straight lines, straight lines and planes, and planes and planes. For the distance of straight lines in different planes, we only need to calculate the distance given by the common perpendicular or the distance in the coordinate representation, master the property theorem of the perpendicularity between straight lines and planes, and master the judgment theorem and property theorem of the parallelism and perpendicularity between two planes.
(8) Understand the concepts of polyhedron, convex polyhedron and regular polyhedron.
(9) Understand the concept and properties of prism and draw a straight prism.
(10) Understand the concept of pyramids and master the properties of regular pyramids. You can draw a regular pyramid directly.
(1 1) Understand the concept of the ball, master the properties of the ball, and master the surface area formula and volume formula of the ball.
10. permutation, grouping and binomial theorem
Examination content:
Classification counting principle and step counting principle.
Arrange. Formula of permutation number.
Combination. Combination number formula. Two properties of combinatorial numbers.
Binomial theorem. Properties of binomial expansion.
Examination requirements:
(1) Master the principles of classified counting and step-by-step counting, and use them to analyze and solve some simple application problems.
(2) Understand the meaning of permutation, master the calculation formula of permutation number, and use it to solve some simple application problems.
(3) Understand the meaning of combination, master the formulas and properties of combination numbers, and use them to solve some simple application problems.
(4) Grasp the properties of binomial theorem and binomial expansion, and use them to calculate and prove some simple problems.
1 1. Possibility
Examination content:
Probability of random events. Probability of equal possibility events. Mutually exclusive events has a probability of occurrence. The probability of mutually independent events happening at the same time. Independent repeat test.
Examination requirements:
(1) It is meaningful to know the regularity and probability of random events.
(2) Knowing the significance of the probability of equal possibility events, we use the basic formula of permutation and combination to calculate the probability of some equal possibility events.
(3) In order to understand the meaning of mutually exclusive events and independent events, we will use mutually exclusive events's probability addition formula and independent event probability multiplication formula to calculate the probability of some events.
(4) Calculate the probability that the event happens exactly k times in n independent repeated tests.
12. Statistics
Examination content:
Sampling method. Estimation of population distribution.
Estimation of overall expectation and variance.
Examination requirements:
(1) Understand the significance of random sampling and stratified sampling, and use them to sample simple practical problems.
(2) The sample frequency distribution will be used to estimate the overall distribution.
(3) The sample will be used to estimate the overall expectation and variance.
13. derivative
Examination content:
Background of derivatives.
The concept of derivative.
Derivative of polynomial function.
Using derivative to study monotonicity and extremum of function, maximum and minimum of function.
Examination requirements:
(1) Understand the practical background of the concept of derivative.
(2) Understand the geometric meaning of derivatives.
(3) Mastering the derivative formulas of functions y=c(c is a constant) and y=xn(n∈N+), we can find the derivative of polynomial function.
(4) Understand the concepts of maxima, minima, minima and minima, and use derivatives to find maxima and minima of monotone intervals, maxima and minima of polynomial functions and closed intervals.
(5) Using derivatives to find the maximum and minimum of some simple practical problems.
Ⅳ. Examination form and examination paper structure
The examination takes the form of closed book and written test. The full score of the whole paper is 150, and the examination time is 120 minutes.
The whole paper includes the first volume and the second volume, the first volume is a multiple-choice question; Volume 2 is a multiple-choice question.
Examination papers generally include multiple-choice questions, fill-in-the-blank questions and solution questions. Multiple choice questions are single choice questions with one of four choices. Fill in the blanks only by filling in the results directly, without writing out the calculation process or derivation process; Solution questions include calculation questions, proof questions, application questions, etc. The solution should be written in words, calculation steps or derivation process.
The test paper should be composed of easy questions, medium problems and difficult problems, with appropriate overall difficulty, with medium problems as the main problem.