Lecture note on the property theorem of isosceles triangle
Sun, a middle school student in Wangtan Town, said that textbooks
This lesson is based on students' mastery of the basic knowledge of general triangles and their preliminary reasoning and proof. It is responsible for training students to learn to analyze and prove ideas, and plays a very important role in cultivating students' logical reasoning ability. The property that the two bottom angles of an isosceles triangle are equal is one of the bases for later proving that the two angles are equal, and the property that the three main line segments on the bottom of an isosceles triangle overlap is an important basis for later proving that the two line segments are equal, the two angles are equal and the two straight lines are vertical, so it is in a very important position in the teaching materials.
Second, talk about teaching objectives
Knowledge and ability: explore and master the property theorem of isosceles triangle, and use it for related argumentation and calculation. Understand the connection between isosceles triangle and equilateral triangle property theorem.
Process and Method: Cultivate students' ability to abstract and summarize propositions, and gradually penetrate the basic thinking method of geometric proof: analysis and synthesis.
Emotion and attitude: guide students to rediscover the law and cultivate their spirit of being brave in practice and exploration. Strengthen students' awareness of mathematics application. Third,
Teaching emphases and difficulties
Emphasis: the property theorem of isosceles triangle. Difficulties: the application of the three-in-one property of isosceles triangle ⅳ. Speaking and learning methods
Classroom teaching should reflect the spirit of students' development, so I adopted the teaching mode of "open inquiry" in this class, trying to let students actively participate in the cognitive process from asking questions to solving problems, so that students can fully, fully and fully participate and truly establish their dominant position. But teachers only act as organizers, guides and collaborators in mathematics learning, and guide, teach and correct in time.
Five, the teaching process theory:
The students' learning process is actively constructed on the basis of the original cognition, so I divide the teaching process into the following five links according to the students' cognitive rules:
The design intention of teaching activities in the teaching process
1. Reviewing and thinking about the images of herringbone roofs displayed by computers, Q: 1. What geometry has the roof been designed? 2、
We all know that it is a special triangle, so what's so special about it? (The two waists are equal, which is an axisymmetric figure) 3. Which is its axis of symmetry?
The topic is based on the isosceles triangle in daily life, aiming at cultivating students' ability to abstract mathematical problems from practical problems. At the same time, creating a rich old knowledge environment will help students find the connection point between old and new knowledge, especially question 3, which is actually the foreshadowing of the trinity nature of isosceles triangle.
Besides these special points, does the isosceles triangle have other special properties? In this lesson, let's learn the properties of isosceles triangle (leading to the topic). Modern teaching theory holds that students should clearly know the goal and significance of exploration and make material and spiritual preparations for exploration before the formal discovery process.
Second, observe the expression of 1,
Observe and guess. Please take out the prepared isosceles triangle and fold your waist with the teacher as required to see what you find.
The teacher demonstrates the ABC superposition of isosceles triangles with multimedia courseware, so that students can think about what conclusions you can draw. 2、
After the students answered the discovery theorem, the teacher guided them and concluded them one by one in a standardized mathematical language, and got two property theorems: Theorem 1: The two bottom angles of an isosceles triangle are equal.
Theorem 2: The bisector of the top angle, the middle line and the high line of the isosceles triangle coincide.
By allowing students to operate, observe and guess, experience the process of knowledge occurrence and discovery, and transform knowledge into students' active acquisition of knowledge.
Learning content is no longer presented in the form of conclusions, but indirectly in the form of questions; The psychological mechanism of learning is not just assimilation, but adaptation.
Third, understanding inquiry 3, inquiry theorem 1, (group A answers, group B answers independently) Group A: 1, what are the two acute angles of an isosceles right triangle?
2. If the vertex angle of an isosceles triangle is 40 degrees, what is its vertex angle? 3. If the base angle of an isosceles triangle is 40 degrees, what is its base angle?
Group b: 1. If an internal angle of an isosceles triangle is 40 degrees, what are the other angles?
2. If an internal angle of an isosceles triangle is 120 degrees, what are the other angles? 3. If an inner angle is 60 degrees, what are the other angles?
(Group A answers independently, and Group B answers independently) This leads to an inference: all angles of an equilateral triangle are equal, and all angles are equal to 60.
Second, fill in the blanks according to the nature 2:
(1)∵AB=AC,AD⊥BC,∴, 2005.
(2)∵AB=AC,BD=CD,∴, 2005.
A
B D C
(3)∵AB=AC,∠ 1=∠2,∴，
. In order to further explore the theorem, the following exercises are designed: the overall design of Exercise 1 follows the principles of low starting point, small grading, large capacity and high density, and its purpose is to enable students to master the law of finding angles by applying the isosceles triangle property theorem 1 and the triangle interior angle theorem. However, teachers do not directly instill this law in students, but let students discover the law themselves in the practice process, so that students can acquire the thinking of exploring commonness from problems. From the perspective of cognitive structure, it is difficult for students to prove that the angles are equal, the line segments are equal or perpendicular to their original cognitive structure with the trinity nature, so it is necessary to construct a new cognitive structure, which is an "adaptation" process, so the following set of fill-in-the-blank questions are designed to help students carry out construction activities. At the same time, remind students that the application of nature should be based on isosceles triangle, as an auxiliary buffer for the teaching of Example 2, and play a role in dispersing difficulties.
Four. Application and improvement example: as shown in the figure, the top corner of a house.
∠ BAC = 120, pass through the column AD⊥BC of roof A, and the room AB=AC, and find the degrees of ∠ B, ∠ C and ∠ CAD on the top frame.
Example 1: Prove that the bisectors of the two base angles of an isosceles triangle are equal.
A
Ed
B.C.
Because this is a geometric proposition described in written language, teachers and students discuss it together and divide the problem-solving process into the following steps:
(1) according to the proposition to draw the corresponding graphics, and marked with letters; (2) draw a conclusion by analyzing the topic, translate the proposition into geometric symbol language, and write what is known and verified.
(3) Explore the method of proof to inspire students to think from two aspects: "knowing" and "verifying". Start with what is known:
What is AB=AC related to?
B: BD and CE are the bisectors of △ABC. What's your connection with them?
C: What will A and B associate?
D: What are A, B and C associated with?
E: What does D have to do with it?
Starting from verification: What are the common methods to prove that two line segments are equal? (congruent triangles). Which two triangles are these two line segments in? Are these two triangles congruent? How to prove it?
This lesson abstracts geometric problems from the herringbone structure of residential buildings, and draws conclusions through exploration and practical activities. Here, the conclusion is applied to practice, thus solving practical problems in herringbone structure. This not only echoes before and after, but also embodies the idea that "mathematics comes from life and is applied to life", which is conducive to strengthening students' awareness of mathematics application.
The teaching of "proof" focuses on the experience of the basic methods and processes of proof, rather than the pursuit of the number of propositions and the skills of proof. Therefore, in the teaching of example 1, the purpose is to let students determine the learning tasks and steps and fully mobilize their learning enthusiasm.
Analysis and synthesis are the basic mathematical thinking methods, and students are required to think from two aspects. But it is more difficult for students who have just come into contact with demonstration geometry. Therefore, teachers will ask a series of questions to guide students to associate.
This topic is to prove that the two bisectors of the angle are equal through the congruence of the triangle, but for congruent triangles, △ABD and △ACE, △BCE and △CBD use two pairs of elements, namely, the public side and the public angle. Therefore, in the teaching process, we will make full use of this point, organize students to explore and prove different viewpoints, and make appropriate comparisons and discussions, which will help broaden students' horizons.
Four. Application and Improvement Example 2: Known: As shown in the figure, △
A
O
B D C O
In ABC, AB=AC, o is a point in △ABC, ob = oc, and the extension line of ao passes through BC and D.
Verification: BD=CD, AD⊥BC
Thinking: (1) What's the special conclusion of this question?
What is the difference? -Prove two conclusions
(2) How will you come to these two conclusions? -Separate certification or simultaneous certification
(3) Which is simpler? what to use
What nature?
On this basis, ask students to find their own ideas to solve problems according to the thinking method of the example 1, and discuss them in groups.
Variant extension:
(1) As shown in the figure, in Example 2, if point O is a point other than △ABC and AO connects BC and D, how to verify it?
(2) What if point O is on BC?
Through the study of example 1, students have a certain reasoning foundation, so they should be allowed to discover their own ideas of proving problems, so as to learn new methods of learning mathematics and gradually internalize them into their own experience. At the same time, it also embodies the learning style of independent exploration, cooperation and exchange.
Here, students are specially asked to experience the process of graphic transformation through variants. They feel that under certain conditions, graphic transformation will not change the essence of graphics. Finally, move point O to BC, so that students can experience the process from general to special.
Think about it: remember that the midpoint of the bottom of an isosceles right-angled triangular ruler is, then hang a plumb from the vertex and put the triangular ruler on the beam. If the suspension line passes through point m, it can be determined that the light beam is horizontal. Why? Further highlighting the key points and difficulties through thinking is also conducive to guiding students to observe and analyze real life by using mathematical thinking mode and enhancing their awareness of applied mathematics.
Verb (abbreviation of verb) experience and experience
Through the research of this class today, I have made it clear that my gains and feelings are as follows
I still have questions.
. Students are required to summarize according to this model, cultivate students' good habit of learning-summarizing-learning-reflecting, and at the same time gain the happiness of success through self-evaluation and improve students' self-confidence in learning.
Sixth, homework
(1) The corresponding job in the workbook. (2) It is known that D and E are on the BC side of △ABC, AB=AC, AD=AE, which proves that BD=CE( 1) further consolidates and improves the learned knowledge, (2) gives timely feedback, checks for leaks and fills gaps, and (3) shows hierarchy and openness.
Sixth, evaluation.
The concept of modern mathematics teaching requires students to change from "learning" to "learning". This course consciously creates a relatively free space from theorem discovery to theorem application, allowing students to actively observe, guess, discover and verify, and actively start, talk and think, so that students can form methods while learning knowledge. The whole teaching process highlights three points:
1. Pay attention to students' participation in the process of knowledge formation and experience the fun of applying mathematical knowledge to solve simple problems. 2. Pay attention to the interaction and cooperation between teachers and students to improve together.
3. Pay attention to the unity of knowledge and ability, so that students can master methods and use them flexibly while acquiring knowledge.