Mathematical language is an organic combination of natural language, symbolic language and image language.
Example 1. If a series is a natural number, then.
Analysis: All kinds of things add up.
Comments: The known condition is the recursive formula of sequence, and the narrative mode of this question adopts symbolic language, which has the characteristics of high school algebra; In addition, the problem-solving method is also a common method in the sequence problem, and it is the application of the whole idea. The score rate of this kind of test is extremely low because students can't understand the meaning of the topic or can't think of a solution. In the usual teaching, students should be properly exposed to topics expressed in concise symbolic language and some innovative problem-solving methods.
Example 2. (Suzhou senior high school entrance examination) As shown in the figure, if two square pieces of paper completely overlap, if the upper one rotates around the center of the square, the exposed area (s) of ⊿ABC changes with the rotation angle (). The following figure shows the relationship between s and ().
Analysis: Through calculation, it can be found that the area of ⊿ABC increased gradually in ~ and decreased gradually in ~. So I chose B.
Comments: The translation of image language and its narrative method in this question have the characteristics of high school mathematics. The main solution to this kind of problem is to find out the functional relationship between two variables, but the changing law between variables can be obtained through observation in this question. This kind of question type well embodies the concepts of "the basic law of graphic change process" and "the function is to describe the relationship between changing things" that the curriculum standard pays attention to.
2. Knowledge background permeates high school knowledge.
Some senior high school entrance examination questions are based on middle school mathematics knowledge, and the design comes directly from high school mathematics, with high school mathematics background.
Example 3. For positive number x, specify f(x)=, for example, f(3)=, f()=,
Calculate F ()+F ()+F ()+… F ()+F (1)+F (1)+F (2)+F (3)+…+F (2004)+F (2005).
Analysis: It is obviously impossible to solve it by substitution, but if we pay attention to duality and then construct duality, it is easy to know that = 1, and the result is 2006.
Comments: The expression of this function is the expression of high school function, and it is a transcendental function. Students are required to transfer knowledge principles and methods through homework and mathematical activities with an analytical attitude and an exploratory eye. The key to solve this kind of problem is to master new laws, and then use induction and analogy to solve the problem. This kind of examination aims at cultivating students' comprehensive ability to solve problems and is the embodiment of the concept of "students' sustainable development".
Example 4. (Xianning senior high school entrance examination) A group company decided to contract one of its subsidiaries for foreign investment. Two eligible enterprises, Party A and Party B, drew up profit plans as follows:
Profits will be paid once a year. The profit in the first year was 65,438+0,000 yuan, an increase of 65,438+0,000 yuan every year over the previous year.
B: The profit is paid every six months. The profit in the first six months was 3,000 yuan, and the profit in the last six months was 3,000 yuan more than that in the first half.
Which company do you think is more profitable if it is contracted for four years?
If the contract is annual, please indicate the total profits paid by the two enterprises (unit: ten thousand yuan) with the included algebraic expression.
Analysis: Through analysis, we can easily find that the difference between the profits paid each time and the profits paid last time is an equal constant. In addition to direct addition, the following formula can be derived to calculate their sum S (where the number represents the first number and the last number).
(1) (ten thousand yuan); (ten thousand yuan).
Contracting to enterprise B, the head office will make more profits.
(2);
Comments: This topic is based on the content of "arithmetic progression" summation formula in senior high school algebra. Students can explore the law and deduce the sum formula of arithmetic sequence, which will lay the foundation for studying senior high school mathematics in the future and play a beneficial role in connecting junior high school and senior high school mathematics knowledge. In fact, the content of arithmetic progression will appear from time to time in primary and junior high schools, so it is necessary to talk about some basic contents of arithmetic progression. It is also an excellent tradition of mathematics education in China to advocate supplementing some contents other than textbooks. Armed with better and more weighty knowledge can cultivate students' thinking more effectively and help to reduce their academic burden.
3. Inference permeates high school knowledge.
3. 1 strengthens the examination of reasonable reasoning.
Reasonable reasoning mainly includes rough evaluation, analogy and induction. Curriculum standards clearly point out that reasonable reasoning ability plays an irreplaceable role in scientific discovery and student development. Therefore, in the senior high school entrance examination, some unique questions are set to test students' reasonable reasoning ability.
Example 5. We extend the concept of similar shape to space: if two geometric bodies are not necessarily equal in size but have the same shape, they are called similar bodies, for example, cubes are all similar bodies. Please summarize the three main properties of similarity: ① _ _ _ _ _ _ _ _ _ _ _ _ _; ②_____________; ③________________.
Analysis: This is an analogy from a kind of thing (similar shape) to a similar kind of thing (similar shape), or from a low dimension (plane) to a high dimension (space). Through two cubes, it is not difficult to get three main properties of similar bodies by analogy: the ratio of all corresponding line segments (or arcs) of similar bodies is equal to the similarity ratio; The ratio of the surface areas of similar bodies is equal to the square of the similarity ratio; The volume ratio of similar objects is equal to the cube of the similarity ratio.
Comments: This question requires students to analyze, compare and summarize. The whole problem-solving process is a process of exploring and forming new knowledge, which fully embodies the reasoning method from special to general.
Example 6. Define an "F" operation for positive integer n: ① When n is odd, the result is 3n+5; (2) When n is an even number, the result is (where k is an odd integer) and the operation is repeated. For example, if n = 26, then:
If n = 449, the result of the 449th "F operation" is _ _ _ _ _ _ _ _ _ _ _ _ _ _.
Analysis: According to the defined "F" operation, count a few steps: 449, and you will find the law, and the result is 8.
Comments: The so-called induction refers to the discovery of general laws through the observation and synthesis of special cases, which is an important means to discover and understand the laws. This topic also has the characteristics of algorithmic language and is related to information technology. Usually, teaching can be not limited to textbooks, and some inductive and analogical activities can be designed, so that students can experience an observation, experiment and other activities, and guess the conclusion of the general situation through the observation of a large number of special situations in the activities, so as to explore the inherent laws of things.
Example 7 (exam questions for self-enrollment classes in Qidong Middle School) Given S=, the integer part of S is _ _ _ _ _.
Analysis:
& gt
That is, the integer part of ∴S is 165.
Comments: Direct calculation is very complicated. If the scale method is used to estimate the range of S, the problem will be solved.
3.2 Strengthen the problem of "infiltration".
The so-called "infiltration" problem refers to the problems related to senior high school mathematics concepts. It can not only examine students' ability to understand and accept new knowledge, but also examine students' ability to adapt to new problems and use new knowledge to solve practical problems. Therefore, it is helpful for students to develop the habit of inquiry in the process of obtaining answers, and improve their self-study level and mathematics literacy.
Example 8 (Ezhou senior high school entrance examination) selects two representatives from three people, A and B, A and C, B and C, and abstracts them into a mathematical model: choose the combination of two elements from three elements, and record it as general, and choose the combination of three elements, and record it as. According to the above analysis, the different selection methods for selecting four representatives from six people are _ _
Analysis: According to the meaning of the combination in the question and its calculation formula, there are two kinds.
Comments: This topic is based on the content of "combination" in senior high school algebra. Students are required to learn the meaning of combination and its calculation formula through reading, and can solve related problems. It can not only examine students' ability to understand and apply the new knowledge of "combination", but also exercise students' self-study ability to acquire knowledge, thus helping students to develop the habit of exploration in the process of obtaining answers and improving their self-study level and mathematical literacy. This topic enables students to "learn" and "learn"
3.3 Algebraic reasoning and high school mathematics integration
Algebraic reasoning problems have always been attached great importance in the senior high school entrance examination. In recent years, there have also been many algebraic reasoning problems with high viewpoints and novel questions.
Example 9 (Jingmen City Middle School Examination) Known real number,, satisfies,
Find the value of.
Solution 1: From the known ①, ②,
Substitute ① into ② to get ③. According to ① and ③, Equation ④ has two real roots, namely sum, which will be substituted into ④, namely.
.
Solution 2: Setting ①
②. Substitute ① into ②. Tidy up. Substitute the values of and into ① at the same time.
Comments: The first solution adopts the construction method and approximation method, and the second solution adopts the mean method of substitution, both of which are creative solutions. The content of this question is not beyond the scope of middle school textbooks, but its form and method examine the students' logical thinking ability at a higher level, and it is designed by the proposer using algebraic reasoning methods in high school mathematics. When reviewing the senior high school entrance examination, we should review the construction method as a special topic, so that students can master the problem-solving method systematically.
We should pay attention to the following two aspects when dealing with the senior high school entrance examination questions: first, the starting point of the questions is high, but the placement is low, that is, although the design of the questions comes from senior high school mathematics, the solution is the mathematical knowledge learned in junior high school, rather than introducing senior high school mathematics into the senior high school entrance examination; Second, the test questions are helpful to distinguish the abilities of candidates, which will appear in the future senior high school entrance examination. In the review, we should strengthen the "double basics", guide students to build a knowledge network, improve students' adaptability and innovation ability, so as to better meet the requirements of the new curriculum entrance examination. Besides, because? There is no deep connection between junior high school curriculum standard textbooks and senior high school textbooks. Therefore, it is necessary to supplement the content of junior high school integration, broaden students' horizons and improve their thinking and innovation ability. At the same time, in order to adapt to the future high school study, it is necessary to cultivate students' good habits of loving reading, learning, being good at seeking knowledge, being brave in innovation and acquiring new knowledge independently, so as to improve students' scientific literacy in learning mathematics.
1. Language narrative permeates high school knowledge.
Mathematical language is an organic combination of natural language, symbolic language and image language.
Example 1. If a series is a natural number, then.
Analysis: All kinds of things add up.
Comments: The known condition is the recursive formula of sequence, and the narrative mode of this question adopts symbolic language, which has the characteristics of high school algebra; In addition, the problem-solving method is also a common method in the sequence problem, and it is the application of the whole idea. The score rate of this kind of test is extremely low because students can't understand the meaning of the topic or can't think of a solution. In the usual teaching, students should be properly exposed to topics expressed in concise symbolic language and some innovative problem-solving methods.
Example 2. (Suzhou senior high school entrance examination) As shown in the figure, if two square pieces of paper completely overlap, if the upper one rotates around the center of the square, the exposed area (s) of ⊿ABC changes with the rotation angle (). The following figure shows the relationship between s and ().
Analysis: Through calculation, it can be found that the area of ⊿ABC increased gradually in ~ and decreased gradually in ~. So I chose B.
Comments: The translation of image language and its narrative method in this question have the characteristics of high school mathematics. The main solution to this kind of problem is to find out the functional relationship between two variables, but the changing law between variables can be obtained through observation in this question. This kind of question type well embodies the concepts of "the basic law of graphic change process" and "the function is to describe the relationship between changing things" that the curriculum standard pays attention to.
2. Knowledge background permeates high school knowledge.
Some senior high school entrance examination questions are based on middle school mathematics knowledge, and the design comes directly from high school mathematics, with high school mathematics background.
Example 3. For positive number x, specify f(x)=, for example, f(3)=, f()=,
Calculate F ()+F ()+F ()+… F ()+F (1)+F (1)+F (2)+F (3)+…+F (2004)+F (2005).
Analysis: It is obviously impossible to solve it by substitution, but if we pay attention to duality and then construct duality, it is easy to know that = 1, and the result is 2006.
Comments: The expression of this function is the expression of high school function, and it is a transcendental function. Students are required to transfer knowledge principles and methods through homework and mathematical activities with an analytical attitude and an exploratory eye. The key to solve this kind of problem is to master new laws, and then use induction and analogy to solve the problem. This kind of examination aims at cultivating students' comprehensive ability to solve problems and is the embodiment of the concept of "students' sustainable development".
Example 4. (Xianning senior high school entrance examination) A group company decided to contract one of its subsidiaries for foreign investment. Two eligible enterprises, Party A and Party B, drew up profit plans as follows:
Profits will be paid once a year. The profit in the first year was 65,438+0,000 yuan, an increase of 65,438+0,000 yuan every year over the previous year.
B: The profit is paid every six months. The profit in the first six months was 3,000 yuan, and the profit in the last six months was 3,000 yuan more than that in the first half.
Which company do you think is more profitable if it is contracted for four years?
If the contract is annual, please indicate the total profits paid by the two enterprises (unit: ten thousand yuan) with the included algebraic expression.
Analysis: Through analysis, we can easily find that the difference between the profits paid each time and the profits paid last time is an equal constant. In addition to direct addition, the following formula can be derived to calculate their sum S (where the number represents the first number and the last number).
(1) (ten thousand yuan); (ten thousand yuan).
Contracting to enterprise B, the head office will make more profits.
(2);
Comments: This topic is based on the content of "arithmetic progression" summation formula in senior high school algebra. Students can explore the law and deduce the sum formula of arithmetic sequence, which will lay the foundation for studying senior high school mathematics in the future and play a beneficial role in connecting junior high school and senior high school mathematics knowledge. In fact, the content of arithmetic progression will appear from time to time in primary and junior high schools, so it is necessary to talk about some basic contents of arithmetic progression. It is also an excellent tradition of mathematics education in China to advocate supplementing some contents other than textbooks. Armed with better and more weighty knowledge can cultivate students' thinking more effectively and help to reduce their academic burden.
3. Inference permeates high school knowledge.
3. 1 strengthens the examination of reasonable reasoning.
Reasonable reasoning mainly includes rough evaluation, analogy and induction. Curriculum standards clearly point out that reasonable reasoning ability plays an irreplaceable role in scientific discovery and student development. Therefore, in the senior high school entrance examination, some unique questions are set to test students' reasonable reasoning ability.
Example 5. We extend the concept of similar shape to space: if two geometric bodies are not necessarily equal in size but have the same shape, they are called similar bodies, for example, cubes are all similar bodies. Please summarize the three main properties of similarity: ① _ _ _ _ _ _ _ _ _ _ _ _ _; ②_____________; ③________________.
Analysis: This is an analogy from a kind of thing (similar shape) to a similar kind of thing (similar shape), or from a low dimension (plane) to a high dimension (space). Through two cubes, it is not difficult to get three main properties of similar bodies by analogy: the ratio of all corresponding line segments (or arcs) of similar bodies is equal to the similarity ratio; The ratio of the surface areas of similar bodies is equal to the square of the similarity ratio; The volume ratio of similar objects is equal to the cube of the similarity ratio.
Comments: This question requires students to analyze, compare and summarize. The whole problem-solving process is a process of exploring and forming new knowledge, which fully embodies the reasoning method from special to general.
Example 6. Define an "F" operation for positive integer n: ① When n is odd, the result is 3n+5; (2) When n is an even number, the result is (where k is an odd integer) and the operation is repeated. For example, if n = 26, then:
If n = 449, the result of the 449th "F operation" is _ _ _ _ _ _ _ _ _ _ _ _ _ _.
Analysis: According to the defined "F" operation, count a few steps: 449, and you will find the law, and the result is 8.
Comments: The so-called induction refers to the discovery of general laws through the observation and synthesis of special situations, which is an important means to discover and understand the laws. This topic also has the characteristics of algorithmic language and is related to information technology. Usually, teaching can be not limited to textbooks, and some inductive and analogical activities can be designed, so that students can experience an observation, experiment and other activities, and guess the conclusion of the general situation through the observation of a large number of special situations in the activities, so as to explore the inherent laws of things.
Example 7 (exam questions for self-enrollment classes in Qidong Middle School) Given S=, the integer part of S is _ _ _ _ _.
Analysis:
& gt
That is, the integer part of ∴S is 165.
Comments: Direct calculation is very complicated. If the scale method is used to estimate the range of S, the problem will be solved.
3.2 Strengthen the problem of "infiltration".
The so-called "infiltration" problem refers to the problems related to senior high school mathematics concepts. It can not only examine students' ability to understand and accept new knowledge, but also examine students' ability to adapt to new problems and use new knowledge to solve practical problems. Therefore, it is helpful for students to develop the habit of inquiry in the process of obtaining answers, and improve their self-study level and mathematics literacy.
Example 8 (Ezhou senior high school entrance examination) selects two representatives from three people, A and B, A and C, B and C, and abstracts them into a mathematical model: choose the combination of two elements from three elements, and record it as general, and choose the combination of three elements, and record it as. According to the above analysis, the different selection methods for selecting four representatives from six people are _ _
Analysis: According to the meaning of the combination in the question and its calculation formula, there are two kinds.
Comments: This topic is based on the content of "combination" in senior high school algebra. Students are required to learn the meaning of combination and its calculation formula through reading, and can solve related problems. It can not only examine students' ability to understand and apply the new knowledge of "combination", but also exercise students' self-study ability to acquire knowledge, thus helping students to develop the habit of exploration in the process of obtaining answers and improving their self-study level and mathematical literacy. This topic enables students to "learn" and "learn"
3.3 Algebraic reasoning and high school mathematics integration
Algebraic reasoning problems have always been attached great importance in the senior high school entrance examination. In recent years, there have also been many algebraic reasoning problems with high viewpoints and novel questions.
Example 9 (Jingmen City Middle School Examination) Known real number,, satisfies,
Find the value of.
Solution 1: From the known ①, ②,
Substitute ① into ② to get ③. According to ① and ③, Equation ④ has two real roots, namely sum, which will be substituted into ④, namely.
.
Solution 2: Setting ①
②. Substitute ① into ②. Tidy up. Substitute the values of and into ① at the same time.
Comments: The first solution adopts the construction method and approximation method, and the second solution adopts the mean method of substitution, both of which are creative solutions. The content of this question is not beyond the scope of middle school textbooks, but its form and method examine the students' logical thinking ability at a higher level, and it is designed by the proposer using algebraic reasoning methods in high school mathematics. When reviewing the senior high school entrance examination, we should review the construction method as a special topic, so that students can master the problem-solving method systematically.
We should pay attention to the following two aspects when dealing with the senior high school entrance examination questions: first, the starting point of the questions is high, but the placement is low, that is, although the design of the questions comes from senior high school mathematics, the solution is the mathematical knowledge learned in junior high school, rather than introducing senior high school mathematics into the senior high school entrance examination; Second, the test questions are helpful to distinguish the abilities of candidates, which will appear in the future senior high school entrance examination. In the review, we should strengthen the "double basics", guide students to build a knowledge network, improve students' adaptability and innovation ability, so as to better meet the requirements of the new curriculum entrance examination. Besides, because? There is no deep connection between junior high school curriculum standard textbooks and senior high school textbooks. Therefore, it is necessary to supplement the content of junior high school integration, broaden students' horizons and improve their thinking and innovation ability. At the same time, in order to adapt to the future high school study, it is necessary to cultivate students' good habits of loving reading, learning, being good at learning, being diligent in thinking, being brave in innovation and acquiring new knowledge independently, so as to improve students' scientific literacy in learning mathematics. Only in this way can they be in an invincible position in future study.