Leping city No.2 Middle School, Jiangxi, 333300.
In recent years, the mathematics test questions in the senior high school entrance examination are all based on the development of students. While examining students' basic mathematical literacy such as basic knowledge and skills, we should strengthen the examination of students' mathematical ability, highlight the thinking value of mathematics, and emphasize ability and conception.
Keywords: application consciousness; Model thinking; Inquiry; reason
First, highlight the model ideas of equations and inequalities, and examine students' application consciousness.
"Mathematics Curriculum Standard" emphasizes that starting from students' existing life experience, students can experience the process of abstracting practical problems into mathematical models, explaining and applying them. Equations and inequalities are both mathematical models to describe practical problems, and both are processes to mathematize practical problems. Strengthening the cultivation and examination of application consciousness is the need of education and teaching reform.
Example1(24 questions in the senior high school entrance examination in Jiangxi Province in 2006) Xiaojie went to the school cafeteria to buy rice, and saw as many people in windows A and B (let's say one person, a>8). He stood at the back of the queue in window A. After two minutes, he found that four people left the queue every minute in window A, six people left the queue every minute in window B, and the last queue in window B was added.
(1) At this time, if Xiaojie continues to queue up at window A, how long will it take him to get to window A (expressed by an algebraic expression with a)?
(2) At this time, if Xiaojie moves quickly from window A to the back of the queue in window B, the time to arrive at window B is less than the time to continue queuing at window A, and the value range of a is found (regardless of other factors).
Solution (1) What time does he continue to queue in the A window?
(points)
(2) From the meaning of the question, you can get
.
Get a > 20.
Therefore, the range of a is a >；; 20.
Comments on this topic are based on the problems around the students, and the background is familiar. People who are slow will naturally have "Do you want to change teams?" In this way, the question (2) is introduced into nature, the question setting is natural and smooth, and there is no sense of artificial carving. By establishing the inequality model and solving the inequality, we can get the judgment that is instructive to action and provide strong support for action decision-making, which is the application of mathematics. The test questions contain rich mathematical knowledge, and the ability to solve practical problems by using inequality knowledge is investigated, which fully shows the broad space of mathematical application. [ 1]
Enlightenment from teaching Mathematics teaching should make students pay attention to the mathematics around them, extract the background of mathematics application with social value, think independently, learn to observe things, explain phenomena, analyze and solve problems from the perspective of mathematics, and cultivate application consciousness.
2. Highlight the thought of function and examine the application of function in dynamic geometry problems.
Example 2(2007 Jiangxi Provincial Senior High School Entrance Examination 2 1) is shown in the figure. In the middle, if the moving point starts from the point and moves to the point in a straight line, the moving speed is 2 unit lengths per second. If the crossing point is the intersection point, let the moving time of the moving point be seconds and the length be.
(1) Find the functional relationship about and write the range of independent variables;
(2) When is the value, the area of has the maximum value, and what is the maximum value?
Solution (1),.
Again,,,,.
.
The range of independent variables is.
(2)
.
At that time, there was a maximum value of. (or use the vertex formula to find the maximum value)
The length of line segment, the area of figure and the movement time of point form the corresponding relationship of function, which not only examines the application of function in dynamic geometry problems, but also examines the modeling ideas of linear function and quadratic function and the maximum value of quadratic function. It is comprehensive and well realizes the examination of the application of linear function and quadratic function in the curriculum standard.
The revealing function of teaching is the core content of junior middle school mathematics, and it is also an important basic knowledge and important mathematical thought. The idea of quadratic function is not only the need for students to master mathematical knowledge, but also the ability that students must have in the follow-up study. The comprehensive degree and difficulty of many questions in the senior high school entrance examination are more difficult than those in the textbook. Therefore, in teaching, we should strengthen the training of relevant knowledge, pay attention to cultivating students' function thoughts, and make them accumulate experience in function application.
Third, highlight the process, pay attention to experience, and examine students' thinking ability.
Mathematics curriculum standards emphasize the experience of geometric modeling, discovery and exploration, and the cultivation of students' geometric intuition and spatial concepts, which embodies the natural transition from intuitive geometry, experimental geometry to inferential geometry. Mathematics curriculum reform regards "process" teaching as an important concept.
Example 3 (24 questions in the senior high school entrance examination in Jiangxi Province in 2007) has 6 points in the same plane rectangular coordinate system:
, , .
(1) Draw the circumscribed circle ⊙P, and point out the positional relationship between the point and ⊙P;
(2) If a straight line moves axially, when it passes a point, let this straight line be.
(1) Judge the positional relationship between a straight line and ⊙P, and explain the reasons;
② Rotate the straight line clockwise around the point again, and set the straight line when it passes through the point.
Because. Find the area of the graph enclosed by the straight line and the lower arc of ⊙P (the result is reserved).
The solution (1) draws the figure ⊙P, from which we can see that the radius of ⊙P is, and.
Point to ≧P
(2) (1) The straight line is translated upwards 1 unit passing point, passing point,
, , .
.
Then, the straight line is tangent to ⊙ p.
(② , , .
.
, .
The graphic area surrounded by straight lines and bad arcs is.
Comment on this question based on grid, relying on rectangular coordinate system to present the content of the question, so the question feels plain and innovative, especially the question setting. On the one hand, it is based on the positional relationship between points and circles, the positional relationship between straight lines and circles, and the calculation related to circles. On the other hand, every time a question is asked, candidates should solve it in a compact and orderly manner. This kind of question is novel and unique. The reliability and validity of the test questions have been improved. Because of the novel presentation angle, the test questions are purely knowledge-based at first glance. In fact, it is a test of basic knowledge expression ability, mainly because the content of the test questions is concise and the graphics are hidden, which invisibly increases the thinking ability of the candidates, and of course the ability requirements are also strengthened. In addition, the operation scenes such as drawing, translation and rotation are inserted in series, which fully embodies the test of hands-on ability and spatial imagination ability. Furthermore, Pythagorean theorem and inverse theorem, congruence of triangle, related properties of circle, determination of tangent and related calculation of circle are all highly integrated with point D as the link point, and the related knowledge points of circle naturally merge together, so it is also an inspection of candidates' comprehensive ability. It conforms to>'s spirit of emphasizing experience, thinking about space and highlighting the process of graphics, and also conforms to the requirements of diluting the deductive proof of The Circle.
Teaching inspiration pays attention to exploring the essence and changing law of graphics and space in teaching, developing intuitive teaching of space concept and geometry, naturally guiding students to realize the necessity of proof in the process of intuitive discovery, inquiry communication and gradual thinking, paying attention to the formation and development of knowledge, the exploration process of problem-solving ideas and thinking, and strengthening process teaching in teaching, so as to truly pay equal attention to conclusion and process. Grasp the basic knowledge and skills in the teaching of "circle"
Highlight mathematical activities, pay attention to mathematical literacy, and examine inquiry and reasoning.
Mathematical activities refer to a series of activities such as observation and analysis, operation experiment, abstract generalization, induction and analogy, reasoning and calculation, guess and proof in order to solve a mathematical problem or solve practical problems with mathematics.
Mathematical literacy is mainly manifested in: having solid basic knowledge and skills, being able to use the learned mathematical knowledge flexibly to solve practical problems and mathematics itself; Be able to observe real life from a mathematical perspective and ask some mathematical questions; Ability to think and analyze problems with mathematical thinking mode; Can correctly understand the relationship between mathematics and social life and its function. [3]
Example 4 (In 2006, Jiangxi Province took the senior high school entrance examination for mathematics 23 questions? As shown in the figure, on trapezoidal paper.
In ABCD, AD‖BC, AD > CD, fold the paper along a straight line passing through point D, so that point C falls.
At point C' on AD, the crease DE passes through BC at point E and connects C' E..
(1) Verification: The quadrangle CDC'E is a diamond;
(2) If BC = CD+AD, try to judge the shape of quadrilateral ABED and prove it.
Solution (1) Proof: According to the meaning of the question, we can know that:
CD = C'D，∠C'DE =∠CDE，CE = C'E，
Before BC, ∴.
∴ ∠CDE =∠CED。 ∴ CD = CE。
∴ CD = C'D = C'E = CE。
A quadrilateral is a diamond.
(2) Answer: When BC = CD+AD, the quadrilateral ABED is a parallelogram.
Proof: From (1), we know that CE = CD.
∵ BC = CD + AD，∴ AD = BE。
And \ad‖be, ∴ quadrilateral Abed is a parallelogram.
Overlapping evaluation is one of the practical problems, which reflects the examination of students' ability of "operation-discovery-guess-proof" from the experimental operation. The knowledge involved includes basic knowledge such as parallelogram, rhombus and trapezoid, and the main knowledge of this chapter is examined.
Question (2) is an inquiry question, which examines students' inquiry ability.
Enlightenment from teaching: Using origami and other operational behavior carriers to write test questions can not only examine students' mastery of basic knowledge such as rectangle, trapezoid and square, but also examine mathematical thinking methods such as the combination of numbers and shapes. In teaching, we should make full use of the intuition and operability of learning tools, master basic graphics, guide students to do more work, observe more and analyze more, and grasp the essence of the problem.
Example 5 (Jiangxi Mathematics Examination in 2007 10) is shown in the figure, so it can be seen that,
The point is on the side, and the quadrilateral is a rectangle. Please draw it with a scaleless ruler.
Bisector (please keep drawing traces).
solve
On the surface, this topic is a drawing problem. In fact, the core is to investigate the analysis and application of the essence of graphics and the drawing method of geometry.
The understanding of tools reflects the application and naturalness of geometric knowledge in practice, and has high reliability and validity.
Enlightenment of Teaching In the usual teaching process, we should try our best to avoid rote learning and mechanical exercises, actively guide students to put their energy into the analysis of problems and phenomena, understand the essence of problems, and teach mathematics vividly.
Highlight the application of statistical concepts and probability, and examine students' information consciousness.
Mathematics Curriculum Standard emphasizes the formation and development of students' statistical concepts and probability consciousness, and pays attention to understanding statistics and probability and their influence on decision-making in real situations. The study of statistics and probability is not specific knowledge, laws and rules, but the study of processes, ideas and concepts. It pays attention to the background of the problem and the application of probability statistics in social life and science, rather than treating some contents as pure calculation.
Example 6 (In 2006, there were 22 math questions in the senior high school entrance examination in Jiangxi Province? A stationery store sells only three kinds of pens, A, B and C. The following table and statistical chart are the profits and sales of these three kinds of pens last month respectively.
Sales statistics of a, b and c fountain pens.
A, b, c three profit statistics.
Pen model A B C
Profit per branch (RMB) 0.6 0.5 1.2
(1) Calculate the profits of these three fountain pens in our store last month, and show the profit distribution with a fan-shaped statistical chart;
(2) If the store plans to buy 600 pens of these three models next month, combined with the sales situation last month, how many pens of A, B and C do you think have the highest total profit? What is the total profit at this time?
The profit of (1)A fountain pen is 0.6×300 = 180 yuan.
The profit of type B pen is 0.5×600 = 300 yuan.
The profit of type C fountain pen is1.2×100 =120 (yuan).
Department statistics as shown in the figure:
(2) Input 300 A-type pens, 200 B-type pens and 100 C-type pens, with the highest total profit.
At this time, the total profit is 300× 0.60+200× 0.50+100×1.20 = 400 yuan.
With the help of data from the actual background of students' lives, comments reflect the important role of statistical knowledge in practice, emphasizing that the examination of statistical knowledge should generally be combined with the characteristics of examination methods in the actual background; It is generally followed to examine students' ability to reasonably obtain data information and process data information from statistical tables and charts, and to infiltrate students' ability to make scientific decisions by using statistical data on this premise.
The model of "obtaining information-processing information-scientific application" has good validity, reliability and popularization.
Teaching enlightenment of common statistical charts in students' life. Understanding the information in the chart is the basic quality that citizens should have, and it is also one of the new contents of the new curriculum. Usually, students should be taught with relevant data from newspapers and schools, so that they can truly appreciate the usefulness of learning mathematics. In teaching, students can be trained to make reasonable judgments and predictions based on statistical results, experience the role of statistics in decision-making, find relevant information according to problems, obtain data information, and express their views on some data in daily life.
Judging from the mathematical contents of the above six examples, there are equations, inequalities, linear functions, quadratic functions, quadrangles, circles, statistics and so on. What are the core contents of junior high school mathematics? The thinking methods involved are equation thinking, function thinking, combination of numbers and shapes, statistical thinking and so on.