Teaching: Higher Education Press Sino-Vocational Education National Planning Textbook
"Mathematics" (improved version) Chapter IV Function
Teaching Teacher: Zhang Yingfang, Kunming City, Yunnan Province
First, the status of teaching content
This topic is to learn the image of the function and. On the basis of the nature, an exploration classroom teaching is conducted. In a resolution process of specific problems, students can submit knowledge from understanding knowledge to skilled application knowledge, so that they can identify the relationship between knowledge understanding and knowledge applications, and tighten the function knowledge before and after they have completed.
Second, the teaching objectives
The teaching goal is the starting point and destination of teaching, so according to the requirements of the syllabus, the teaching objectives are developed with the characteristics of students' cognitive points, psychological characteristics and this lesson:
1, knowledge target
By establishing a function model, you can solve a simple practical problem.
2, quality objectives
Cultivate students' initiative to participate, independent learning, and courage to explore the scientific attitude; improve students discover problems, analyze problems, and solve practical problems.
3, emotional target
Stimulate students' enthusiasm of learning mathematics, firmly overcome difficult confidence, and enjoy the successful joy of solving practical problems.
Third, students analysis
After learning the function and the segmentation function, students have a certain basis for the function knowledge, but they are only in a simple solution, and the function knowledge is used to solve the problem in practical life. It is the lack of students. To this end, the topics must be close to the actual situation, but also to closely combine mathematical exploration, in the practical application of knowledge, so that students can understand the beauty of mathematics.
Fourth, teaching focus, difficulties and key
According to the knowledge situation of the syllabus, teaching objectives and students have mastered, the status and role of this section is determined in the status and role of this section, the difficulties and key are as follows:
1. Key: Establish basic methods of mathematical models and improve the ability to solve practical problems.
2. Difficulties: It is necessary to pay attention to the deepness of the problem, and avoid the problem of problems; if you have to close your life, you have to closely combine mathematical exploration.
3. Key: Stimulate students think and promote knowledge migration.
V. Teaching method
Under the guidance of constructivism, the "task-driven teaching method" is adopted, so that students achieve the construction of knowledge in the exploration model of mission independent learning. This highlights the focus of this class. At the same time, "Task Drive Teaching Method '" is necessary to give students a full independent time and space, but also to provide students with the necessary support, so that students are always active in the process of investigation. Learning in exploration In the environment, vivid teaching scenarios open the students' vision, so that students can know in voluntary, happiness and positive environments, and truly enhancement, thus breaking through difficulties.
6. Teaching aid
Investigation form, computer and multimedia teaching courseware.
Seven, teaching process
(1) Pre-class survey, gain a perception
1. Determine the topic
In order to break through "It is necessary to close to life, but also closely combine mathematics to find out multiple problems in the student survey in the student, to get the topics of them are both interested, and have mathematical exploration values.
Class investigation table
Issue
Equal selection (please select)
(1) Phone billing method and call charge
(2) Price and profit
(3) Investment costs and benefits
(4) Product supply and demand
In the investigation of the issue, students' performances are always active and active, and the results of the survey is that most students are more interested in the billing method of mobile phones, and ultimately determine the phone bills and call charges as research topics.
2, actual investigation
Check the following questions in front of the class to check the information or in the field visit:
(1) Collect the type of mobile phone call card in China and its billing conditions;
(2) What is the first element you have to consider if you want to choose in numerous cards?
(3) Is the big sales of a large sales?
(4) What is the mobile phone bill you plan to pay monthly?
In the exploration study, the investigation is not only to complete the task, but more importantly, students have the process of exploring and thinking about knowledge. surface. How to choose the fiscal card in the market, how to choose? The student discussing in front of the class warmly without a unified result, and finally they hope to solve this problem with the teacher. (2) Inquiry of classroom teaching, teachers and students
1. Feedback from the results of the survey before class.
Perform in a question.
2. Design information resources, ask questions
(1) Determine research topics
Among the various mobile cards collected by the students, choose the current sales of Dangzhou Volkswagen, mobile 69 cards and mobile 29 cards as the research object of this class.
(2) Data processing
Remarks:
During the data processing, the variables considering the charging situation is the local call time, short message fee, long call fees, and roaming fees, etc., it is not considered in this research.
SIM card
Billing situation
Shenzhou Volkswagen Card
Mobile 29 card
Mobile 69 card
Men's rent package fee
no
29 yuan / month
69 yuan / month
Basic rate time limit
Local network call
0.3 yuan / minute
Containing 150 minutes of local
Network call time
Containing 508 minutes
Network call time
Timeout billing standard
no
Call and answer
0.3 yuan / minute
Call and pick up
0.2 yuan / minute;
(3) Propose a problem
If you choose in three cards, which card is the most affordable?
3, independent learning exploration, get guess
(1) Sub-group discussion learning the first group: Analysis of the advantages and disadvantages of Shenzhou Volkswagen counting and suitable people; the second group: analyze the advantages and disadvantages of mobile 29 card billing methods and suitable people; third group: analysis The advantages and disadvantages of the 69 card counting method and the appropriate population; the fourth group: proposed a preliminary option and explains the reasons.
In the construction of the construction of the teaching, the teacher puts forward the problems that inspire the thinking, first keep your own opinion, guiding students to form their own opinions. Every student has its own experience world, through the group discussion, exchanges, cooperative learning, so that students' learning autonomy is fully displayed.
(2) Group Cooperative Learning Results Report
During the entire report, you should listen to students 'speech and encourage,' Let each student are free, boldly participate in exploration and communication. Insight the origin of their thoughts, seeing its rationality, then providing corresponding guidance, guiding students to see
Contradictions with their views and facts.
The problems encountered by the solution to the actual problem are taken out by establishing a mathematical model to solve the problem.
Through the exchange of communications and learning, the students' interest is stimulated, and the classroom atmosphere is enthusiastic, and students have basically entered a good state of teaching with teachers.
4. Teachers and students have established mathematical models and analyze assessments.
First, the functions of the function and the knowledge of the function and the segmentation function have been used to list the parsing of the function.
(1) The monthly use time set by the mobile phone card is T (minutes), the monthly call fee is Y (yuan), the functional relationship between the three mobile cards in one month is:
Remarks: The call time t is an integer, resulting from the results obtained.
(2) Make a function image in the same coordinate system and obtain a key intersection of three images:
(3) Teachers and students joint analysis assessment
A, Shenzhou Volkswagen: When f∈ [0,96], the use of the lowest call fee, daily average call is 3 minutes;
B, mobile 29 card: When f∈ (96,283), the card is most affordable, daily average call is 3 ~ 9 minutes;
C, move 69 card: When f∈ (283, ∞), it is more than 9 minutes to daily.
Choose the above three mobile phone cards to study, the church students are not a simple set of results, more importantly, make students get a solution to the problem. Through the student's active exploration, from guessing argument, the students' thinking activities, let them experience the joy of success, and the enthusiasm of student learning is overflow. 5, independent strengthening exercises, consolidating new knowledge
Example: When the mobile phone call expenditure of the students in March is 83 yuan, the groups calculate how long the classmates in March in different charging methods.
The first group: Shenzhou Volkswagen card counting calculation: A: 0.3t = 83
T = 276 minutes
The second group: mobile 29 card counting calculations; answer: 29 0.3 (T1 150) = 83
T = 330 minutes
The third group: mobile 69 card billing method calculation; A: 69 0.2 (T-508) = 83
T = 578 minutes
The fourth group: summarize and explain the reason.
During the group exercise, the teacher passed the inspection. The problems existing in the student are plugged in and guided, and the typical errors are discussed. The design of this group is focused on enabling students to migrate knowledge to new scenarios. That is, if the cost is the same, the spending method is different, and the call time of various cards also has a big difference, and again combined with the image how to make the billing options.
6, the joint summary of teachers and students, research expansion
(1) Research development, guide students to summarize the mode of modeling:
Through the summary, once again clarified the teaching goals of this section, the students have achieved self-feedback on knowledge, the construction of knowledge experience, and finally form their own insights.
(2) Evaluation of learning process
The students 'evaluation and the teacher's evaluation of students have been composed of students' self-evaluation. The main contents of the evaluation are:
● Actively investigate, collect information;
● All groups unite collaborate, independent exploration learning process;
● Teachers and students cooperate to complete the process of modeling;
● Each group has independently learned, strengthens the process of practicing knowledge;
● The harvest of knowledge through the search of this class.
The evaluation of practical exploration courses is different from the evaluation of the routine mathematical questions, and should pay more attention to students' practice and autonomous learning. Everyone has a very strong sense of honor and accomplishment. It needs to be recognized and self-personalization. It is necessary to give more encouragement. Through the evaluation of the learning process, the students are not only gain knowledge, but also Strengthen the courage and confidence of the future in the future.
7. After class practice activities, forming capacity
Through the post-class practice activities, students have triggered the more in-depth thinking of mathematics knowledge, so that they realize that mathematical knowledge comes from life, but also serves life. Fully experience the 来 去 去 龙 龙. The following two groups were required to complete the student class:
(1) Combined with the method of modeling, analyze the short message fee of Shenzhou Volkswagen, mobile 29 card, mobile 69 card, and the billing of long calls is given the choice.
(2) Through the study of this class, you can find other relevant problems in the actual life, and add it.
Conclusion:
The teaching purpose in the new teaching model changes. The class should give students three things: giving students knowledge, this is the most basic level; give students a method, method is more important than knowledge; give students a wide vision (ability, emotional business - non-intelligence factor, mainly manifestation , Willpower, initiative, creativity, etc., improve their ability to find problems, observe issues, and understand problems, so that they have more deeper and more creative understandings to the problem, reaching the highest level of teaching purposes. Second Oo for four years in September
typography design
Analysis and choice of mobile phone card boarding mode
Teaching case
Vocational High Mathematics (Basic Edition)
Chapter III Section 1
Mapping
Unit: Shijiazhuang City Information Management School
Name: Sun Ping
Time: 2 004.9.2 5
§3.1 mapping
First, teaching objectives
1. Students who have different knowledge bases and capacity levels can understand or understand the concept of mapping, and clearly mapping the meaning of the three elements and value domains.
2. Mainly cultivating students to summarize abstract abilities, analyze problems, depletion thinking and innovative thinking.
3. Through the group cooperation, independent construction, stimulate students' interest in learning, let the students experience the value of the map concept.
Second, the key points and difficulties of teaching
1. The focus of this lesson is the concept of mapping.
2. The difficulty of this lesson is to understand the concept of mapping.
Third, the teaching methodology and learning method analysis
This section adopts the "Guided, Reading, Discussion, Practice, Practice - Five-Class Teaching Law" and "Group Learning Law" to mobilize students 'self-learning enthusiasm, pay attention to cultivating students' independent exploration, summarizing abstraction, analyzing issues , Divergent thinking and innovative thinking, penetrate the teaching principles of teaching, so that all students can achieve their development in cooperative learning.
1. Self-learning, training ability: create problems in student life, let students observe thinking, finding the common point of the two instances, thus abstracting the definition of mapping, and cultivating students to explore, inquiry, and summarizing abstract. By let students use their own abilities, students are cultivating their thinking and innovative thinking.
2. Executive education, comprehensive development: teaching, such as "say", "talk about it", intend to make the classmates of the basic classmates can understand the concept of mapping; "practice one practice" to further think about how to put " Ask the square root "This correspondence into a map and" Try a Test "to make students speaking mappings in mathematical knowledge, you can give the basic classmates to further improve opportunities; in addition, in the process of teaching Always pay attention to the different basic students to answer questions about different difficulties. All of these teaching arrangements are to make all students learn.
3. Cooperative learning, joint progress: through the group discussion to "discussion", "trial", cultivate the good habits of students cooperative learning, through the exchanges and discussions between groups, more broadly mobilize all students to participate in the enthusiasm and Active, making students with different levels to make long-term remedies and make progress together.
Fourth, teaching aid
Multimedia Courseware
V. Teaching process and method
The concept of mapping is very important, but it is more abstract, it is a key point in this chapter and a difficult point. For such a concept, it is expected that only one or two examples, so that students fully understand is unrealistic, only by analyzing a large number of instances, gradually deepening, can understand its exact meaning. Therefore, three big links have been arranged in teaching, namely, initial concepts, further understanding concepts, internalized experience concepts, these three links are progressive. 1. Through "Take a look", "I want to think", stimulate students to enter the country, analyze the distribution machine bits and allocate dorms, summarize, then abstract the concept of mapping, and explicitly mapped, three elements In addition, through "reading one reading" and "say", students have a preliminary understanding of the concept of mapping, which focuses on the cultivation of students' observations, thinking, analyzing, summarizing, abstract.
2. Through "talking about", "practice one training" and "discussion", students will further understand the concept of mapping in active participation - active reactions, and clarify the meaning of mapping. On the basis of the previous link, the understanding of mapping conceptively advanced a layer · This link focused on the observation of students, discovered problems, analyzed problems, and solving the problem of problem capability.
3. Through "Try a try, and let students summarize, summarize the intelligence points, not only make students feel the value of mapping in life and mathematics, so that the experiential goals can be upgraded to experience value, but also make students By summarizing the instance, sum up, the understanding of the map concept is rising to a new understanding of the height. At the same time, students who have different levels of students will raise the mappings in daily life or mathematical learning, which can stimulate students. Learning interest, greatly mobilizing students 'enthusiasm and participation. This link focuses on the cultivation of students' divergent thinking and innovative thinking.
(See the next page of the specific teaching process and method)
Six, board design
Teaching process
First, import
Take a look
Example 1 The situation of the assignment of the computer in the computer training. (See courseware)
Example 2 The situation of the classmates allocated the hostel when they were enrolled. (See courseware)
Think
What is the common in the same example?
(Each classmate has a position (dormitory), and only one unit (dormitory), that is, each element in the previous collection has the elements in the latter collection and it corresponds to it, and only one element and it corresponds to it. )
This abstracts the definition of mapping.
Second, new lesson
Read one reading
Definition of the mapping: Sets A and B are two sets, if there is a method F, so that each element A in the collection A, all determined element B in the collection B corresponds to it, and that f is A to B One map, remember
f: a → b
A | → B
Where b is called a like f, A is called B in f
Original icon. A The Icon Symbol F (A) under f is represented, so that the mapping F can also be recorded.
f (a) = b, a∈a.
The substance of the mapping is: one correspondence between the two collections, this correspondence method corresponds to each element in the first collection to the unique element in the second collection.
f: A → B has "directionality". Tell a a definition domain that maps F, called B.
One mapping f: a → b is defined by domain, accompanying and corresponding
Thoughts and method
Use students to create problems in the usual life, "Guide" into new classes, easy to cause students' interest in learning, so that students quickly entered into the country, so that students find two instances of common.
When looking for two instances, under the inspiration of teachers, most students should be able to summarize the abstracts of the two instances, and at this time the teacher gives the definition of the map. The definition of mapping is displayed, not allowed to write only mapping notes and key keywords on the blackboard to increase efficiency.
Require students to read "Read" definitions carefully, find the keywords in the definition, and analyze their meaning, reveal the essence of mapping, and reach the purpose of preliminary understanding of mapping definitions.
Here, students are looking for keywords in definition - analysis of their meaning - revealing the essence of definition, is a common method of learning concepts, allowing a better classmate to try.
The teacher points to disclose the essence of the definition of the map, can help students understand the definition of the map and thus this to determine the mapping method.
Put the concept of defining domains and accompanying domains in the mapping definition, which is more natural, and can be convenient for later descriptions.
The rules consist, they are called the three elements of the map.
Say
Which collection of the two examples is to which collection, indicating what the three elements are respectively, and say other pictures and original icons.
This map is "assigned dormitory", 6 L 8 dormitory has 8 original icons, leading: 'Comment 1 mapping can "one-to-one", "more pairs", can not "a pair".
Talk about
The following four correspondence, which forms a map is 1, 4, what is the three elements of them? 2, 8 is not mapping, why?
From this to the second point of comment on mapping:
Comment 2 Definition f (a) one {f (a) | a∈A}
It is obvious that F (a) is obviously f (a)
B.
And see the value domain of the first, 4 small questions.
Practice one practice
1. Judgment: "Square" is f (x) = x2, x ∈R is not a real set R to a mapping? What is its three elements?
Yes. Define the domain, accompanying is R, and the correspondence method is "Square".
It can make students who have a poor base, the purpose is
Guide students to initially understand the concept of mapping, understand mapping
Composition of elements, understanding and originality of the exact meaning, and
This naturally leads to the commentary 1.
The title is used in the topic to indicate the collection and corresponding, and it is straightforward.
Simple view, small difficulty, can be arranged
Reply to make it an example of simply an example
Emission concept.
In addition, let students look carefully 1 and 4.
Shoot, see what they have, in mapping 1, second
Element 6 in a collection is not original, and in the mapping 4, each element in the second collection has an idea, thereby ie the concept of the mapped value field, and is analyzed by the definition of the mapping value domain. The value domain is a subset of the accompanying.
Let students look at some of the specific real numbers, by special
Take a general, to inspire students' ideas. Tight buckle
The keyword defined in the definition is "each", "the only determined" in B is determined. This question is to answer medium students to answer.
2. Judgment: "Square root" is not a map of the real set R to itself? Why?
Not. Because the load is like a square root, there is no square root in the definition of "each" element in the definition, there is a corresponding element in B; and the positive real number, such as two square meters, not satisfying each element of defined A in B in B There is a "unique determination" element with it.
3. Can you modify this example and get a mapping?
The corresponding method is changed to "seeking a square root", and the previous collection is changed to non-negative set R , and "calculating square root" is a mapping of the non-loaded set R to real set R, where the accompanying domain can be changed to R . Discussion
Observing the corresponding relationship given in September 2004, you can find some mappings and point out their definition domains, accompanying
Is the law?
Try a try
You can live in your daily life or have learned
Do you think of some instances do you think is a mapping?
Requirements: Each mapping that will be written on paper in the format given by the teacher, and can clearly indicate what the three elements of each mapping are.
The students can be properly judged by observing the square root of the specific implementation under the inspiration of the previous question. Be sure to let the students "tell" the reason for the map, if necessary, "1 square root is"? Or "Is there a square root in the real number of concentrations?" This topic has to answer medium students to answer.
This question is the introduction of the last question, can trigger students to think deeply, and further deepen their understanding of the concept of mapping. This question is to answer the better classmates to answer.
Teachers can lead the students together to make a certain preparation for the study of the students behind the students in the blackboard.
This part first allows the team to discuss, the teacher gives the appropriate tips to let students learn and understand the concept of mapping. In addition, there is also this question to inspire students' thinking, as the next student, the transition of mapping instances.
First group discussion, then use the physical projector to play the results of each group of discussions, let the exemplification of a group selection representative to speak on the podium, explain the results of this group and point out the three elements of each mapping, other teams judg this result is it right or not. Encourage groups to brainstorm, and raise as much example.
Arrange this teaching link is to enable students to transition from compliance to initiative, and learn from each other, fully discuss, and have deepened students' understanding of the concept, experience the value of mapping concept, and cultivated students to divergent thinking and innovative thinking The ability, while you can also make the slower classmates help each other between the team members
Third, small knot
This section learns the concept of mapping and several issues that need attention.
Mapping is an abstraction of many phenomena in daily life. Today, in the rapid development of science and technology, mapping has become one of the most basic and important concepts in modern mathematics. This chapter will continue to study the function is actually a special mapping. Therefore, learn the mapping allows us to better understand the concept of functions, so learn the knowledge of functions. From the next class, we will study the issues related to the function.
Fourth, homework
1. Must do: P80 A group 1, 2, 3;
Choose: B Subscription 1.
2. Through the group discussion, each person will at least three mappings. (Requirements: Don't repeat with examples on or books)
3. Thinking: P80 A Subscription 4 and B Subscription 2. (Ask next section)
Come.
Let the students give an example of mapping in life, students can think about many instances of the front side, should be said that the difficulty is not difficult. However, if the students give the mapping examples in mathematical knowledge, the difficulty will be large, because here should involve some concepts and operations in mathematics, that is, there must be certain mathematics literacy, and this is just a majority of professional students. More lacking, such as topics, many students do not distinguish between "square roots" and "arithmetic square roots". Therefore, here is the example of mapping mapping in mathematics, so that students will further experience the application value of mapping in mathematics, but there is a difficulty. Teachers can interpret examples in detail, pioneering students' thinking, and give certain tips in patrols, and make a speech that many students have more instances, through analysis, further inspiring student thinking, and preparing for students. Under the teacher's tips, let students summarize, summarize the intention of knowledge, which can make students further understand the knowledge of the knowledge, and more firmly learn from the beliefs of students, but also examine students to the knowledge of this lesson, and Students can cultivate students' ability to summarize summaries and language.
Summary can be said by a classmate, if it is not comprehensive, it can be added by other students.
Teachers' emphasis on this section and attracting both students to optimize the knowledge learned by this class, forming value orientation in the transition of sensible to rationality, while achieving a role in achieving.
Job 2 is intended to further expand students' thinking and deepen their understanding of the concept. And make students progress together in mutual help.
Speaking of the draft case
Changsha City Financial Vocational Secondary Professional School Tang Songlin
Today, my teaching theme is "three-vertical line theorem", which is a very important theorem for a three-dimensional geometry, which is a bridge that the space vertical problem is converted to a flat vertical problem. For the teaching of this class, my overall design idea can be summarized as two sentences, namely:
Interest import, autonomous exploration, overall construction, and promote the formation of students' thinking process:
Specific intuitive, analysis and summary, difficult to easily, promote the improvement of student mathematical ability.
Specifically:
I introduced a topic from the design principle of the cutter knife in the actual life, and stimulated the students' interest in the way of life. Next, through the two operational experiments, students will do their own, to find problems, find the laws of the problem, deepen students' understanding of theorem. I am divided into four sectors - guess, card, cutution, use, overall construction, let students form a whole and comprehensive understanding of theoretical learning. I strive to make specific intuitiveness, cultivate students' space imagination; analysis and summary, it is also from beginning to end through the whole class, fully reflecting the logic charm of mathematics; it is difficult to easily, seize the simple process behind complex problems, Reduces the difficulty of studying students, and improve their interest in learning.
Below I will detail the judges and peers in the concert and ideas in this class: first is "guess", "guess" is imported in practical problems, in two experiments, in research Bold conjecture during the process. The specific steps are as follows: By discussing the design principle of the cutter to introduce the topic, excitation students learn the interest of the three-way line theorem. An operational experiment, let the students independently explore, guessing the three-vertex line theorem "guess" is the introduction stage of the topic: Interest is to open students' enthusiasm, and practice is a catalyst for students to learn desire, so the introduction must be from At first, I grasp the heart, and interest and practice are the best way to invest in the classroom. This class starts from the principle of cutting knife, through two experimental discovery rules, fully mobilized from the beginning Student's learning enthusiasm. Then, "Certificate", first, the conclusions of the experiment are known, try to find the way to find the conjecture through the reverse analysis method - proof to get a three-vertened line agency - echo before and after, using a three-vertical line to understand the cutter Design principle, the entire proof process reflects the teaching ideas of the instructions and guiding the discovery.
"Classification" is analyzed in several cases in the role of theorem and the theorem application. It is divided into such cases: analysis 1: What is three vertical lines - analysis 2: The five positions of the straight line in the plane; and explain the position of the certificate and the straight line - Analysis 3: 3: 3: 3: 3: 3: 3: 3: 3: 3: The application and plane of the three-vertical line theorem The location of the placement is not related; this teaching process requires the preparation of teaching teachers to prepare to be meticulous, consider the problem to be comprehensive.
Finally, "use", "use" is a few steps that specify the application, how to clarify the position of the three-line line in a specific problem. In the specific teaching, I introduced the basic steps of the three vertical lines of the three-vertical lines: 2; three certificates - simplify graphics conditions in the process of the issues, grasp the problem fundamental - follow the procedure step Guide students to proof for specific problems. This is to say teachers in the process of appging ~ must pay attention to guiding students to seize the key to grab the problem through the complex conditions background.
Certificate, cut, use is a test stage of the topic, and this phase of teaching can also summarize the above four sentences - independent exploration, cooperative learning, excitation; abstract specific, phase L mutual conversion, raising a sense of recognition; Analysis of induction, reverse thinking, strengthening the formation of mathematical thinking; paying attention to summary, commutation is easy, and promoting the improvement of mathematical solutions. Classification from the discovery problem to design problems, analyze problems, solve problems, application problems, from beginning to end, students are in the protagonist, I just played a guide role, the teacher's status is also turned out by the original "Chief" . During the in-depth teaching analysis of layers, the students fully demonstrated the logic charm of mathematics, further improved their interest in mathematics.
The principle of "module construction, overall review, key highlighting, rationalism, and scheduling", let students have formed a holistic understanding of the knowledge, the four teaching section, the difficult content is hit by ~ emphasize, refining teaching content Help them clarify learning ideas. This class, I think teaching focus is to understand the three-vertex line theorem and its role, master the basic steps of the three-line customs, and solve some simple practical problems. The difficulty is the discovery and certification process of the three-vertical line theorem; how to clarify the position of the three vertical lines in a specific problem is the key to the use of three vertical lines.
At the end of the subject, the first end echo was achieved, and the theorem was emphasized, and the content of the next class was taken out. The "three-vertical line theorem" is in a position in the textbook. The problem of how to prove how straight lines of the slash and the plane (especially excessive vertical); it is to lead the three-way line theorem. The end of the subject has to pay attention to the first end to form a system to form a system, deepen the memory of this lesson. Taking the fish, it is better to teach it, and finally I think that the teaching of this lesson I teach students to concrete and abstract conclusal ways method; the problem of solve the problem of exploration of the problem; grasp the fundamental Chemusting is easy to think; timely summary learning methods.
2004.9.23
National Secondary Vocational School Mathematics Teaching Observation Class
Teaching case
Question: three vertical line theorem
Teacher: Tang Songlin Unit: Changsha City Finance and Economics
Teaching purpose: 1. Enable students to understand and master the three-vertical progeny, and can use them to solve some basic problems.
2. Cultivate students' conjecture, argument, and application.
Teaching focus: three steps to use their certificate: must, two, three certificates.
Teaching Difficulties: Discovery and Argumentation of Triad Line Theorem. Teaching method: guide discovery.
Teaching Means: Adopt modern multimedia teaching methods.
Teaching process:
(1) The introduction of review and new lessons
Practical activities introduced topics: Requires students to cut a right-angled trapezoid into a rectangular shape using a cutter, proposing the problem of the working principle of the cutter, and exclusion the student's learning, and introduces the exploration and discovery process of the three-line theorem.
In order to learn this class, take students to review well-related knowledge points, and better recover students' memory of old knowledge, which is helpful to improve the learning efficiency of this class.
(Division) Question Review: 1. Straight line and plane vertical definition?
(Birth):. And to the student, the reverse of the straight line and the plane is also established. It actually gives an important way to prove two straight lines vertical (especially excessive vertical), that is, the line is vertical → line Line vertical.
2. Straight line and plane vertical decision theorem?
(Birth):. It is pointed out that this theorem is an important way to proof | linear and plane vertical.
3. What is the plane of the plane, and the shooting of the slash in the plane?
(Birth):. Top the method of taught students to find the shooting of the slash on the plane.
In the process of the question, the animation demonstration is performed in turn, and the content of the review is vividly displayed.
(2) Guess and discover
(Division): According to the straight line and plane vertical definition, we know that if the straight line is vertical, the straight line is vertical in all straight lines in the plane, then I can not tell, if the straight line is inclined with plane, it is in the plane All straight lines are not vertical?
The teacher guides the student movement to do experiments and test our guess:
[Experimental Requirements]: Take a long straight roll along the desktop, the straight angle of the right triangle is always tight, the triangular plate can slide or rotate along the rib, see if a position can be found, so that the triangular board is short The straight corner is just consistent with the desktop.
[experiment procedure]:
(This process uses computer abstraction)
[Experimental Conclusion]: We have found that the prizter, which is incorrect! The plane of the plane is vertical with the straight line in the plane, and there are countless numbers with its vertical lines in the plane. Vertical vertical Also have exoperable.
(Division): So how do you want to determine a straight line in the plane and the plane of the plane is vertical?
The teacher summarizes the analysis of the students' answer and guides students to do experiments (2):
[Experimental Requirements]: Place the rule on the table, let the short straight corner of the tricode on the desktop, stand upright, how to make the slope of the triangle and the ruler vertical?
[Experimental Process]: (This process uses computer abstraction)
[Experimental results]:
Conclusion One: When the short-straight angle of the triangle is vertical and the ruler is vertical, the slope of the triangular plate is also vertical.
Conclusion II: Three sides of abstraction, rule, and triangle, the desktop has become the lines and planes in our math. At this time: When A0⊥A, PO⊥A.
Conclusion 3: Transfer to a text: a straight line in the plane, if the shooting of a slash of the plane is vertical, it is also vertical with the slash.
(3) Proof
(Division): Requires students to use (Figure 5) according to our experimental conclusions, design a problem, and then reverse the argument.
(Born): It is known: PA, P0 is the vertical line and slash of the plane α, respectively, and A0 is a shot of P0 on the plane α, A
α, a⊥ao.
(Division): Common ways is to prove that a straight line vertical is a plane where the line is located, now to prove A⊥PO, if PO
Plane PAO, then as long as the A vertical plane PAO can be proof. How to prove that a vertical plane PAO? Then prove that two intersecting straight lines in the vertical plane PAO can be proven.
Certification: (Teacher-Students Completed together).
Affirm our experimental conclusion is correct.
(Division): The proven of this proposition reflects the method of "line-up vertical-÷ line line vertical '", this method is very important. Everyone must pay enough attention. The above proposition reflects a "one straight line in the plane." The plane of the plane and the slash in this plane have a vertical relationship between these three. This is a famous three-vertical line theorem.
(4) Analysis Theorem
(Division): Typically read the theorem, at the same time, and put forward the following issues to discuss the students, and a "cross-section" word on the blackboard.
1 ° three-vertened line theorem is used to do?
(Explanation): Explanation of life: Cutting knife. It is mainly used in mathematics to prove that linear and straight lines, especially two exotic lines vertical.
2 ° Since a straight line in the plane is arbitrary, it has the following five basic situations in the plane: (Animation Display).
There are several situations in the 3 ° plane position relationship, in addition to such levels:
(Three vertical lines can be used): (Animation display).
The graphics of the above cases are often encountered during the certificate, and we should use the three-vertical line theorem. It is easy to neglect for the straight line of the plane, but the horizontal position is not horizontal. This is A difficulty of determining the three-vertical line relationship of the certificate, should give enough children.
(5) Theorem is used
It is known: the square ABCD-A1BLCLDL.
OK: (1) acl⊥bd; (2) ACL⊥B1C; (3) How many strips are vertically perpendicular to the straight line ACL on the square.
Teachers explain the first question:
The thinking process of using the three-vertical line theorem is:
"Must" a certain plane and one in the plane
A straight line and a flat slant line.
That is, it is clear that the two straight lines to prove vertically as a straight line in the plane? What is the plane? Which is a slanting of a plane.
"Two Find" finds this plane vertical line and the shooting of the slash on the plane.
"Three Certificate": One straight line in the plane and the shooting vertical.
By certain, I found that we found in this square, there are many lines that have no help to our certificate, purely interfered with lines, if we will
They all wipe them, see (Figure 17), then
Compare it with the previous basic situation,
We find that it is actually uniform (Figure 7)
of. (This process uses computer animation processing
Implementation) The procedure of the certificate is:
Determine the straight line BD, ACL is a straight line and slash within planar ABCD, and the line VLC is the vertical line of plane ABCD, and the AC is connected to the AC, and the line AC is a shooting in the plane ABCD. Because ABCD is square, AC⊥BD can be known from the three-vertical line theorem: ACL ⊥BD. Currency Guide Students Thinking the Second Q:
What is "a certain"? "Two Looking" is looking for? After "must", "two", delete some lines that have no help, it is actually a graphic, see ( Figure 18). Compared with the previous basic graphics, it is actually completely consistent with (Figure 12), if we convert it to the angle. Let the plane are in a horizontal state, then it is actually consistent with (Figure 7). How to prove ACI_J_B1C? The above computer animation process is as follows:
Teachers guide students to think about the third question: On the basis of the front two questions, use the definition of different straight lines to find six face-to-angles with ACL vertical: BD, D1B1, A1D, A1 B, D1 C , B1C. (Use animation to connect these lines sequentially.)
(6) small knot
"Guess", "certificate", "cross-section", "use", four teaching section is the main content of our whole class. These four words are discovered around the three-vertical line theorem, arguing, analyzing, and application. The following is a simple review of the key content in these four steps, helping students to learn about learning, so that students have a whole, comprehensive understanding of the content.
(7) homework
Through the conditions and conclusions of the exchange of three-vertical line, form a new proposition, requiring the student lesson to proof this new proposition according to the proof of the three-way line theorem today, and tell the student, this is the class less. The three-vertical line theorem to learn. The first end echoed, and the end of the end of the class.
