Draft lecture on vertical diameter theorem and its inference 1 experts and judges:
Hello! I am glad to have the opportunity to participate in this activity and get your guidance.
The topic of my speech is: Axisymmetry of a circle-vertical diameter theorem and its inference. It is the content of the second part of the first section of the twenty-fourth chapter of the ninth grade experimental textbook mathematics, which is perpendicular to the diameter of the string.
There are two classes in this part of the textbook. The first section focuses on the axial symmetry of the circle, and the second section focuses on the rotation invariance of the circle.
Based on my understanding of the textbook and the actual situation of the students in my class, I adjusted the content of one lesson on the symmetry of circles to two lessons. Today I'm going to talk about the first lesson-the vertical diameter theorem and its inference.
Next, I will elaborate from four aspects: teaching content, teaching objectives, teaching methods and means, and teaching process design.
First, the description of the teaching content
Only when teachers have a more accurate, profound and essential understanding of teaching materials and examine students' acceptance from the perspective of "if I were a student" can they handle teaching materials well.
The vertical diameter theorem and its derivation reflect the important properties of a circle, which is an important basis for proving the equality and vertical relationship between line segments and arcs, and provides an important basis for the calculation and drawing of a circle. Therefore, this part is the focus of learning, and the problems and conclusions of vertical diameter theorem and its derivation are complicated and confusing, so it is also the difficulty of learning.
In view of this understanding, through reading the textbook, I have determined the following teaching contents:
(1) Understand the symmetry of a circle.
(2) Clarify the vertical diameter theorem and the problems and conclusions derived from it. (3) The vertical diameter theorem and its inference are used for relevant calculation and proof.
(4) Learn the method of adding auxiliary lines related to the vertical diameter theorem.
Teaching emphasis: vertical diameter theorem and its inference
Teaching difficulty: the proof method of vertical diameter theorem, in which the symmetry of circle is the key to understand vertical diameter theorem.
Second, the establishment of teaching objectives
According to the specific content of this class and the actual situation of students, I have established the following teaching objectives:
1. Understand the symmetry of a circle through intuitive demonstration.
2. Master the vertical diameter theorem and its inference through "experiment-observation-guess-proof".
3. Use the vertical diameter theorem to solve the problems of proof, calculation and drawing. 4. Cultivate students' mathematical intuition and abstract generalization ability. Stimulate students' exploration spirit.
Third, the choice of teaching methods and means
In teaching methods: under the guidance of teachers, this class mainly adopts the methods of students' independent inquiry, group cooperative learning, hierarchical teaching and hierarchical evaluation.
In the teaching process, the idea of "experiment-observation-guess-proof-discussion-summary-application" is followed, so that students can rise from perceptual knowledge to rational knowledge and then to practical application. Follow the principle of "step by step", guide students to cooperate and explore in the form of group discussion on the basis of independent analysis and serious thinking, so as to solve problems and master methods. At the same time, considering the learning needs of students at different levels, we strive to make every student gain something in the setting of questions, examples and exercises.
In terms of teaching methods: I use intuitive demonstration teaching tools and computer-aided teaching to improve classroom teaching efficiency.
Fourthly, the design of teaching process.
1, adhere to the principle that students are the main body and teachers are the organizers, guides and collaborators of the teaching process.
2. Focusing on one purpose: implementing teaching objectives.
3. Highlight a feature: help students realize the transition from perceptual knowledge to rational knowledge through "experiment-observation-guess-proof-application"
4. Usage: With the aid of intuitive teaching AIDS and computer-aided teaching, students are inspired to discover theorems, thus abstracting and summarizing theorems.
5. Received an effect: Through the study of this lesson, students can understand the connotation of theorems and learn to use theorems to solve problems. At the same time, it integrates learning knowledge, cultivating ability and optimizing thinking quality.
Legal learning guidance:
Hands-on operation, observation and speculation, exchange and discussion, analysis and reasoning, induction and summary, so that students can actively participate in the process, exchange and interact.
The teaching process of this lesson includes:
Bring forth the old and bring forth the new, guide exploration-hands-on operation, observation and conjecture-guide argumentation, draw conclusions-practice more, evaluate at different levels-reflect on summary and assignment.
(1) Quote the classics to guide the inquiry.
Human's understanding of things mostly follows the rising process from perceptual knowledge to rational knowledge and from old knowledge to new knowledge. Therefore, I first guide students to review the old knowledge related to the new knowledge in this lesson, and show the following two questions:
(1) What is an axisymmetric figure?
(2) Observe which of the following figures is axisymmetric? And point out the number of symmetry axes.
The purpose of the first question is to arouse students' memory and clarify the concept of axisymmetric graphics. Then select several common geometric figures for students to judge, among which parallelogram strengthens the understanding of axisymmetric figures from the opposite side. The second group is about the axisymmetric graphics of logo patterns, so that students can know that there are axisymmetric graphics around us anytime and anywhere. At this time, students can give several practical examples to stimulate interest.
Then show the circle and ask: Is the circle an axisymmetric figure?
How many axes of symmetry does it have?
Where is the axis of symmetry?
Then students can fold round pieces of paper,
The teacher's projection demonstration is very clear:
A circle is an axisymmetric figure. It has countless axes of symmetry, and every straight line passing through the center of the circle is its axis of symmetry.
In this way, by creating problem situations, stimulating students' curiosity and bringing forth the new, the theme of this lesson-the symmetry of the circle is brought out.
(2) Hands-on operation, observation and guess
Let the students draw, fold, observe and guess on the circular paper prepared in advance as required. I draw a chord AB of ⊙ O.
Two. Draw a vertical line where AB intersects O at C and D, and the vertical foot is E. 。
Question 1: How many straight lines pass through point O and are perpendicular to AB? (Give reasons)
Design intent: Make it clear that there is only one straight line perpendicular to the chord.
Question 2: What are the other properties of diameter CD? (projection)
1. Guide the students to fold the ⊙O paper in half along the diameter CD, observe the overlapping part and guess the conclusion.
2. Conjecture of group exchange.
3, the teacher's projection demonstration and students * * * appreciate the conjecture conclusion.
Design intention: To strengthen the thinking quality of students' hands and brains by mobilizing their various sensory functions. At the same time, it paves the way for proving the vertical diameter theorem by "superposition method"
(3) Guide the argument and draw a conclusion.
After teachers and students come to the conclusion of conjecture, teachers question whether the result of conjecture is correct, and only by proving can students' active thinking be pulled back from experimental conjecture to strict proof of conjecture. Teaching arrangement:
The teacher's projection after students answer the known and verified questions.
Then guide the students to start with the symmetry of the circle. After discussing the connection between OA and OB, as long as it can be proved that the diameter CD is both the symmetry axis of the isosceles triangle OAB and the symmetry axis of the circle, the conclusion can be proved by the symmetry of the circle. Then ask the students to try to describe the process of the teacher's blackboard proof.
Then the content of vertical diameter theorem is summarized. And guide students to analyze the problems and conclusions of the theorem. Explain the two conditions for knowing the topic, and three conclusions can be drawn.
The true or false question is displayed.
(1) The diameter passing through the center bisects the chord (×)
(2) The line perpendicular to the chord bisects the chord (×)
(3) In ⊙ O, if OE⊥ chord AE is in E, AE=BE(√).
The key to guiding group discussion and allowing arguments is to let students explain their reasons and give counterexamples. After exchanging discussions and unifying thoughts, teachers should make full use of the evaluation mechanism to encourage students and emphasize the symmetry of the circle of the vertical diameter theorem-the vertical diameter theorem and two conditions in its inference are indispensable. At the same time, it is explained that the "diameter" in the condition of vertical diameter theorem refers to a straight line passing through the center of the circle, but when this condition is applied, it can be concluded that the chord is bisected and no diameter is needed, such as radius, the distance from the center of the circle to the chord, etc.
Then ask: If the two conditions in the topic are changed to "the diameter bisects the chord", can you draw three other conclusions? Example 1 Introduction of Nature in Teaching:
Example 1: It is known: As shown in the figure, in ⊙O, the diameter CD intersects with the chord AB in E, and AE=BE.
Verification: CD⊥AB,
Through the teacher's guidance and group discussion and analysis, the inference of the vertical diameter theorem is proved: the diameter of the bisector (non-diameter) is perpendicular to the chord and the two arcs opposite to the bisector. Make students realize that one of the two conditions set in the theorem can be exchanged with one of the three conclusions, and the other three conclusions can also be drawn. Then play the group discussion questions again.
Group discussion: Are the following propositions correct? Give reasons
1. The perpendicular bisector of the chord passes through the center of the circle and bisects the two arcs opposite to the chord. (√)
2. Divide the diameter of an arc opposite to the chord, divide the chord vertically, and divide the other arc opposite to the chord (√).
Further strengthen the initial understanding just now, and then sum up the law: five conditions, knowing two and pushing three. In the whole process, teachers should promptly guide students to analyze and discuss through drawing, explain the reasons and distinguish right from wrong, so as to effectively break through difficulties and highlight key points.
O
(D) Multi-practice, hierarchical evaluation
Example 2: As shown in ⊙O, the length of the chord AB is 8cm, and the distance from the center O to AB is 3cm. Find the radius of ⊙ O ..
1, intention of topic selection
At this point, students should master the basic knowledge of vertical diameter theorem and its inference, so as to make students by going up one flight of stairs and better implement the knowledge points. I arranged example 2, trying to make students clear through this example: when solving the problems about chord, radius (diameter) and the distance from the center of the circle to the chord, the vertical diameter theorem and Pythagorean theorem are usually combined. Achieve the purpose of one-stop communication. And paved the way for the teaching of example 3.
2. Teaching arrangements
First, the idea of solving the problem: this question reminds students to examine the meaning of the question and think about how to construct the radius of the circle and the distance from the center O to the chord ab. After individuals think independently and build graphs, discuss in groups. Finally, each group sent representatives to show their learning achievements and explain the reasons, and the teacher gave guidance, and finally projected the complete problem-solving steps. Ii. Reflection and expansion: Question: How many theorems did you use in solving this problem?
Through discussion, students realize that when solving problems about chord, radius (diameter) and the distance from the center of the circle to the chord, the vertical diameter theorem and Pythagorean theorem are usually combined by constructing a right triangle.
Then strike while the iron is hot, and further consolidate the results just discussed through three exercises with different difficulties.
A group has a chord length of 8cm and a diameter of 10cm, so the distance from the center to the chord is (3) cm. In group B, chord CD = 24, and the distance from the center of the circle to chord CD is 5, so the diameter of circle O is (26). If AB is the diameter of the circle O, the string CD⊥AB is in E, and AE = 65438 is in Group C, then CD = (16) III is graded: students' cognitive level is different, so I consciously divide the questions into three groups, A, B and C, in which Group A is for students with learning difficulties; Most students in group b should master the questions; Group c is a little more difficult, but it's not impossible for people with a little brains. It is aimed at upper-middle class students.
It should be noted that students can get full marks for each set of questions correctly. At this time, the teacher will patrol the guidance and judge the students who have finished each group in time. No matter who does the right question, they can score the students in this group. This arrangement enables students of different levels to learn something and stimulate their enthusiasm for learning.
Then each group asked the representative to explain the idea of solving the problem. After preheating, for example 3:
Example 3: Given the diameter ⊙O is 4cm and the chord AB=, find the degree of ∠OAB.
1. Topic selection intention: On the basis of consolidating the results of Example 2, Example 3 is given, which connects the knowledge of solving right triangle and vertical diameter theorem, so that the knowledge can be integrated-you have me and I have you.
2. Teaching arrangement:
I solve the problem: Q: Can the problem of finding an angle be solved by solving the problem of a right triangle? Students will naturally think of constructing right-angled triangles and then making correct auxiliary lines. Then calculate the degree of acute angle by using the trigonometric function value of special angle. After the students show their grades, the teacher shows the complete problem-solving format and asks: Are there any other methods to solve problems? At this time, the symmetry of the circle may be obtained by some students and can be used in right triangle. If a right-angled side is equal to half of the hypotenuse, then the diagonal of this right-angled side is 30. Teachers should give full affirmation and encouraging evaluation. And then pass on a proof question,
Exercise: As shown in the figure, in two concentric circles with O as the center, the chord AB of the big circle intersects the small circle at points C and D. Verify: AC=BD.
Consolidate the practice of vertical diameter theorem and auxiliary line again.
Two. Reflection and expansion: When solving the problem of chords in a circle, it is often necessary to take "the diameter perpendicular to the chords" as an auxiliary line. In fact, it is often only necessary to make the vertical section of the chord from the center of the circle.
(5) reflect on the summary and homework.
This session is mainly for students to talk about the harvest and experience of this class. I will add as appropriate. Then, the layered homework is arranged according to the level of students. In this way, the enthusiasm of students is mobilized to the maximum extent, so that students at different levels can gain something and develop and improve on the original basis.
The above is my explanation of this course. Please correct me if there is anything wrong. Thank you!
Lecture notes on vertical diameter theorem and its inference II. Experts and judges:
Hello! I am glad to have the opportunity to participate in this activity and get your guidance. I said the topic of the class is: the sag in the circle.
Inference of path theorem. It is the first section of chapter 24 of the ninth grade of the nine-year compulsory education people's education edition. The second part of this content textbook arranges two classes. The first section focuses on the symmetry and vertical diameter theorem of a circle, and the second section focuses on the inference of the vertical diameter theorem. Combined with my understanding of teaching materials and the actual situation of students in our class, this paper explains from four aspects: teaching content, teaching objectives, teaching methods and means and teaching process design.
First, the description of the teaching content
Only when teachers have a more accurate, profound and essential understanding of teaching materials can they look at learning from the perspective of "if I were a student"
Students' acceptance can handle teaching materials well. At the same time, the vertical diameter theorem and its inference reflect the important properties of the circle, which is an important basis for proving the equality and vertical relationship between line segments and arcs, and also provides an important basis for the calculation and drawing of the circle, so this part of the content is the focus of learning, and at the same time, it is also a difficult point for learning because of its complicated topic setting and conclusion. In view of this understanding, through reading the textbook, I have determined the following teaching process:
First, wonderful introduction, second, practical exploration, third, simple application, fourth, classroom testing.
Second, the teaching objectives:
1. After exploring and proving the derivation of the vertical diameter theorem by using the symmetry of the circle, master the vertical diameter theorem and derivation; And can use the vertical diameter theorem to solve the related calculation and proof problems;
2. In the process of research, further experience the method of "guess-experiment-proof-induction-application"; 3. Let students actively participate in the experiment and experience the vertical diameter theorem, which is an important embodiment of the symmetry of a circle.
4. Through reasoning and discussion, gradually cultivate students' ability to observe, compare, analyze, find and summarize problems, and promote the development and improvement of students' creative thinking level.
Teaching emphasis: make students master the vertical diameter theorem and its inference, and remember the information in the vertical diameter theorem and its inference.
Teaching difficulties: exploring and proving the inference of vertical diameter theorem can apply vertical diameter theorem and inference to simple calculation or proof.
Teaching tool: self-made learning tool card courseware III. Teaching process: 1. Stimulate interest introduction.
1, video "broken glass" (design intent: 1 let students enjoy the fun of music, 2 introduce the broken glass scene needed for teaching) 2. Broken glass in life. This picture is a picture of the decoration effect of a room, which captures the hearts of modern students during teaching. Suppose the picture is a few years later, you will get a reward from a company. Design intention: to stimulate students' interest in learning and use it as a clue to guide students to bring mathematics knowledge into life. ) Second, practical exploration
Activity 1: retell the vertical diameter theorem, tell the conditions and conclusions in the theorem, and transform the theorem into a known verified form with graphics. (Design intent: 1 Make students more familiar with the conditions and conclusions of the vertical diameter theorem and lay the foundation for exploring the inference of the vertical diameter theorem)
Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord, 1, the two arcs perpendicular to the chord passing through the center of the circle bisects the chord 1 bisects the chord, bisects the chord, bisects the chord, and bisects the optimal arc activity of the chord 2.
1. Conditions and conclusions in the observation theorem (design intent: leading to conjecture and inference) 2. Conjecture, experiment and proof form the first inference of the vertical diameter theorem.
(1), Guess: Is the straight line passing through the center of the circle and bisecting the chord necessarily perpendicular to the lower arc of the chord bisecting the chord?
The best arc to bisect the chord.
(2) Experiment: The inference of the vertical diameter theorem is obtained by origami (the diameter of the bisector is perpendicular to the chord,
And bisect the two arcs opposite to the chord, but the bisected chord cannot be the diameter)
(3) Proof: How to prove this proposition is true? Write the known and verified according to the proposition: as shown in the figure, the known CD is a string with a diameter of ⊙O, AB is ⊙O, AE=BE is verified: AB⊥CD.
AD=BD
AC=BC
(Design intention: To familiarize students with the inquiry process of mathematical knowledge)
3. Guess: Can any two of the five pieces of information be known and get the remaining three pieces?
Experiment: using the learning tool card in your hand, through origami and other activities, you can draw the conclusion that you know two things and push three things (note: find the one you suspect most)
(Design intent: Let students explore all the inferences by themselves, so that they can clearly understand and not doubt in future applications.
The application of knowledge can be inferred from two aspects to cultivate students' sense of teamwork and resource enjoyment. 4. Inductive arrangement (design intent: make students familiar with the inquiry results of this lesson) ① Passing through the center of the circle ② perpendicular to the chord.
③ bisecting the chord (if the condition is met, the bisected chord cannot be the diameter, otherwise it will not be established) ④ The optimal arc of bisecting the chord.
(5) The chord bisecting the arc.
Third, it is simple to use.
Activity 1. Fill in the blanks according to the pictures: in VIII.
Oh,
(1) If MN⊥AB and MN are diameters, then _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _;
(2) If AC = BC, MN is the diameter, and AB is not the diameter, then _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _; (3) If MN⊥AB and AC = BC, then _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _;
(4) If AM=BM, MN is the diameter, and _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ refers to
(Design intention: simply apply the vertical diameter theorem and inference to realize the exercise)
Activity 2: Memory Competition
As shown in the figure, in ⊙o, if the radius is R, the distance from O to AB is OD=d and BD=a, what is the relationship among them?
r2
=a2
+b2
(Design Intention: Review Pythagorean Theorem)
Tip: If two quantities in Rt△ are known, you can use Pythagorean theorem to find the third quantity. Don't forget the chord AB=2a.
Activity 3: Ability Competition (Design Intention: Recall the first class.
The exercises of right triangle, radius, half chord, chord center distance, Pythagorean theorem and auxiliary line formed by the experience of doing problems lay the foundation for solving the practical problems left over before class. )
1, in ⊙O, oc is perpendicular to the chord AB, AB = 8, OC = 3, then AC =, OA =. 2.In ⊙O, OC bisects the chord AB, AB = 16,
OA = 10, then ∠ OCA = 0, OC =.
Experience summary: When solving the problem of line segment in a circle, it is often necessary to find a right triangle consisting of radius, half chord and chord center distance, and apply Pythagorean theorem in it.
3 Known: As shown in the figure, if you make a concentric circle with O as the center, the chord AB of the intersecting circle is at point C and point D. If CD = 6 and AB = 8, then AC = _ _ _ _ _ _ _ _ _
10
A vertical line passing through the center of a circle as a chord.
Tip: When solving the string problem in a circle, usually
16
Fourth, classroom test (design intent: test students' mastery of this class)
They are AB, the midpoint of the chord AB, AB=4m, CD =1m.. What is the length of radius OD?
Share the results
Experience summary: When solving the problem of line segments in a circle, it is often necessary to use the center of the circle as the vertical line of the chord or connect the center of the circle with the endpoint of the chord.
Make a right triangle consisting of radius, half chord and chord center distance as an auxiliary line.
Knowing two pushes three.
① Through the center of the circle ② perpendicular to the chord.
(3) bisecting the chord (when making conditions, the bisecting chord cannot be the diameter, otherwise it will not be established)
(4) bisect the upper arc of the chord (5) bisect the lower arc of the chord.
Sixth, homework
1 As shown in the figure, the railway MN meets the highway PQ at point O, and there is a residential building at point ∠QON = 30O A, with AO=200m. If the train runs, the noise will affect the surrounding areas within the range of 150m, then when the train runs along the ON direction on the MN railway, will the residential buildings be affected by the noise? If the train is traveling at a speed of 25 meters,
2 As shown in the figure, the curve of an expressway is a round chord (that is, point O in the figure is the center of ⊙O, where CD=600m, E is the upper point, OE⊥CD, the vertical foot is F, and EF=90m). Find the radius of this detour.
Lecture notes on vertical diameter theorem and its inference 3 i. teaching material analysis
1, content status: from the knowledge system, the vertical diameter theorem is the content of the third chapter of the ninth grade (the first volume) of the new curriculum standard for compulsory education. It is a process in which students study the special centrosymmetric graphic circle after learning the rotation and centrosymmetric, and it is a new exploration of the basic properties of the circle after learning the basic concepts of the circle. It is one of the compulsory test sites for the senior high school entrance examination.
2. Learning objectives:
(1) Explore the vertical diameter theorem by using the symmetry of a circle. (2) Can use the vertical diameter theorem to solve problems. (3) wholeheartedly and carefully.
3. Key points and difficulties:
Learning emphasis: exploration and application of vertical diameter theorem. Difficulties in learning: using the vertical diameter theorem to solve problems.
Second, the analysis of learning situation
1. Psychological characteristics of students: After entering the third grade, students are active in thinking, eager for knowledge, curious about exploring problems, and competitive in class. Compared with before, they have a certain knowledge reserve, but their enthusiasm for learning has decreased and their self-awareness has increased.
2. Students' cognitive basis: Before studying this section, students have learned the basic concepts of circle, defined the basic concepts such as diameter and chord, solved problems by using the axisymmetric property, learned Pythagorean theorem, and have the basic ability to further learn vertical diameter theorem. 3. Students' experience basis: Students have made clear the learning procedures of the demonstration class in their previous study, and they can use the study plan to prepare for the demonstration.
Third, the analysis of teaching rules
Analysis of teaching methods: In view of students' cognitive level and psychological characteristics, in this class, I will guide students to make group presentations in the learning atmosphere of group cooperation, guide students to actively participate in teaching activities in an organized, purposeful and targeted manner, and encourage students to adopt learning methods of independent inquiry, cooperation and communication, so as to develop the habit of comprehensive and orderly thinking in the process of observation, thinking and application.
Analysis of learning methods: As a demonstration class, students will go through the process of defining goals, reviewing old knowledge, preparing to show, showing what they have learned, and consolidating and upgrading under the guidance of teachers, so as to cultivate students' good study habits of studying alone, thinking quietly, communicating effectively, actively cooperating and showing boldly.
Fourth, the teaching process and approximate time allocation (1) Clear objectives, (1 minute)
The goal is displayed on the blackboard, and the teacher guides the students to understand (2) review the past and learn the new (3 minutes)
Review the basic knowledge points by asking individual questions, so as to be fully prepared. (3) Assign tasks and prepare presentations (5 minutes).
The teacher assigned the exhibition task and guided the students to prepare for the exhibition. (4) Group presentation and variant training (20 minutes)
Students are grouped and presented in an orderly manner. Ask questions in the demonstration and do variant training. Participants are required to write standardized, complete process, loud voice, smooth expression and compact connection. (5) Summarize and organize the study plan (3 minutes)
Students will sort out the wrong questions, supplement the incomplete problem-solving process, and ask for a two-color pen. (6) Feedback detection, consolidation and improvement (12min)
Complete the feedback test part of the study plan and strive to complete it according to the class.
Fifth, the reflection on the diameter perpendicular to the chord after teaching is also called the perpendicular theorem, which is an important section of calculating the circle in junior high school. This lesson mainly goes through three links: the first link is to let students draw the conclusion that the circle is an axisymmetric figure by folding the self-made circular picture. Every straight line passing through the center of the circle is its axis of symmetry, and there are countless axes of symmetry. The second link is to let students get the content of the meridian streamer theorem through inquiry. The third link is to use the vertical meridian theorem to solve the calculation of related aspects. Among them, the second link is the focus of this class, and it is also a highlight of my class. Specifically, it goes through the following five steps:
(1) Let the students take out the circular picture in their hands and fold it in half to find the center of the circle. Students are very interested. Some students use two vertical diameters to fold to get the center of the circle, and some students use two inclined diameters to fold to get the center of the circle, but the methods are all very good. )
(2) Let one of the two mutually perpendicular diameters stay still, and the other diameter will be translated downwards to become an ordinary chord, still keeping a vertical relationship with the original diameter.
(3) Ask students to draw a line perpendicular to the diameter on their pictures, and ask them to fold the circular picture in half along the diameter. What conclusions will the students find? (bisect the chord, bisect the two arcs opposite the chord)
(4) Ask students under what conditions to reach these conclusions.
(5) Finally, guide the students to summarize the content of the vertical meridian theorem, and then the teacher will supplement, emphasize and write on the blackboard. Through this inquiry process, most students participate in the classroom, which cultivates students' operation ability and innovation ability, and also stimulates students' interest in inquiry. In this relaxed and happy activity, students have mastered the vertical diameter theorem and realized the effectiveness of teaching, which is what I think is the most successful place of this class.
Of course, the whole class also has many shortcomings. For example, the calculation of the vertical theorem is not properly arranged, which is embodied in the following aspects: (1) It is a little difficult for students to solve the Zhao Zhouqiao problem in the textbook as the first exercise, and some simple type problems should be solved first. For example, knowing the length of the chord and the distance from the center of the chord to find the radius of the circle, so that students can not only consolidate the vertical meridian theorem, but also experience the joy of success, and then it becomes a natural thing to deal with the Zhao Zhouqiao problem. (2) In the process of proving the bisector of the chord in the vertical meridian theorem, try to leave some time for students to write on the blackboard to avoid students' lack of initiative, and more students will participate in the class.
(3) Infiltrate some emotional education to students, and let them know that mathematics comes from life and is applied to life.
In short, students should be fully understood and studied in teaching design and classroom teaching. We should not only prepare teaching materials, but also prepare students. It is necessary to truly establish a teaching concept based on student development. Only in this way can we provide students with sufficient opportunities for teaching activities and communication, so that students can become masters of mathematics learning from simple knowledge recipients.