The lecture on "Geometric Meaning of Derivatives" is 1. The content of the class I am talking about is the first lesson in the third quarter of Chapter 2-2, an elective course in B version of mathematics teaching in senior high school. As for the teaching practice of this class, I will introduce my teaching ideas from the following eight aspects: (1) Talking about the outline; Say textbooks; Talk about learning; Oral teaching method; Speaking and learning methods; Talking about the teaching process; Say blackboard design; Self-evaluation and reflection.
First of all, talk about the syllabus
Because derivative is one of the core concepts of calculus, it provides an effective tool for studying the properties of functions. In recent years, the college entrance examination has strengthened the examination of derivatives, which is not only reflected in the problem-solving tools, but also paid attention to the examination of thinking orientation. It is like a leaping dragon and a phoenix, which subtly changes our thinking habits. The guidance of mathematical thought and the infiltration of dialectical thought are helpful for us to establish a scientific thinking orientation. For this reason, the geometric meaning of derivative is the only required standard named "understanding" in the whole derivative and its application part, and it is also the highest standard in this part of cognitive field, which shows its status and significance.
Second, talk about teaching materials.
The textbook starts with the idea of combining numbers and shapes, that is, secant, and defines tangent with intuitive "approximation" method to get the geometric meaning of derivative. Students form a complete concept through observation, thinking, discovery, induction and application, and dialectical thinking can penetrate, which is conducive to students' understanding and mastery of knowledge. There is not much knowledge in this section, but in the teaching practice of this section, we should highlight its key role in the past (further understanding the definition of derivative and discussing the changing speed of function value) and in the future (as the most effective tool to study the monotonicity of function and solve the extreme value and maximum value of function).
Third, talk about learning.
Through the study of the definition of average change rate and derivative of function in the first two sections, students have a preliminary understanding of derivative, but because the definition of derivative is abstract, students still have some difficulties in cognition. This section vividly shows the definitions of average change rate and derivative (instantaneous change rate) of a function through dynamic courseware demonstration, and explores the relationship between tangent slope (absolute value and steepness of slope) and function image trend (absolute value and change speed of derivative), thus becoming the most effective tool for studying function monotonicity, solving function extreme value and maximum value, and discussing the change speed of function value. Stimulate students' interest in learning, improve students' ability to explore and solve problems independently, combine numbers with shapes, and use knowledge flexibly.
According to the requirements of the above syllabus, teaching materials and cognition, based on students' cognitive level, the teaching objectives, key points and difficulties are set, and the teaching objectives are given from four levels: memorizing, understanding, mastering and using. Teaching focuses on the cultivation of non-intellectual factors, and the teaching difficulty lies in thinking ability.
Teaching goal: to understand the geometric meaning of derivative and find the tangent equation of curve.
Teaching emphasis: master the solution of a certain point and the tangent problem of a certain point.
Teaching difficulties: let students learn through observation, thinking and discovery, and summarize and inspire students to study problems.
Fourth, the teaching method of speaking.
Preparing lessons well provides a source of motivation for promoting the formation of students' way of thinking.
Multimedia-assisted teaching, through the dynamic demonstration of geometry sketchpad, can give full play to its characteristics of quickness, vividness and image, and let students discover the law through group discussion without asking questions, which is more conducive to the breakthrough of difficulties. Let students experience the process of "observing, thinking, discovering, summarizing and inspiring students to study". According to the conclusions of each group, teachers guide students to understand the geometric meaning of derivatives with approximate thinking methods, and at the same time try their best to summarize and pave the way for the change speed of monotonicity, extreme value and function value. Teach students the methods and basis of thinking, so that students can truly become the main body of teaching.
Verb (abbreviation of verb) and learning methods
Let students participate in teaching activities through group discussion, promote cooperative learning and communication among students, * * * discuss problems together, explore ways to solve problems, produce interactive effects, improve students' sense of cooperation, and * * * accomplish teaching objectives together.
Sixth, talk about the teaching process.
(a) review and introduction
Review the definition of average change rate of function and its geometric significance; The definition of derivative and its physical meaning lay the scene of analogy transfer. What is the geometric meaning of derivative?
(B) the process of exploring the geometric meaning of derivatives
1, the definition of tangent
Using the dynamic relationship between the tangent and secant of a circle, the tangent definition of a general curve is given in time (avoiding defining it from the common points).
2. Observe the relationship between secant and tangent dynamically.
By demonstrating the dynamic change trend of secant, it provides a platform for students to observe and think, guides students to do the same analysis, and intuitively obtains the definition of tangent. By approximation, the straight line whose secant tends to a certain position is defined as tangent, which makes students realize that this definition is applicable to all kinds of curves and embodies the intuitive essence of tangent, thus summarizing the geometric meaning of derivative. Here, the teacher should guide the students to draw the conclusion that the curve and the tangent of the curve can have more than 1 common points at a certain point. Lines and curves
When there is only one common point, it is not necessarily the tangent of the curve.
3. The application is embodied by an example, and the solving steps are summarized.
Seven, say blackboard writing design
Theme:
Review: Example 1. Find the tangent of the specified point.
Exercise:
Geometric meaning:
Example 2. Find the tangent of the specified point.
Understanding of tangent:
Example 3: Explore the slope of the known tangent and find the tangent equation.
Summary:
Homework:
Eight, self-evaluation and reflection
In the teaching process of this class, students' observation ability, analytical thinking ability, understanding and induction ability and combination of numbers and shapes are trained and tested. Pay attention to cooperation and exchange, summarize and affirm the achievements of students in each group in time, and let students get a sense of accomplishment. Paying attention to "double basics" and giving consideration to improvement not only points out the direction for students to continue their research after class, but also paves the way for future study and stimulates students' interest in exploring new knowledge.
Lecture Notes on Geometric Meaning of Derivative 2 I. Textbook:
1, the position and function of teaching materials
Derivative is one of the core concepts of calculus, which provides an effective method for studying functions. In previous courses, students have fully understood the concept of derivative. From the perspective of shape, starting with secant, the textbook of this lesson defines tangent with intuitive "approximation" method, and obtains the geometric meaning of derivative, which is more conducive to students' understanding of the essential connotation of derivative concept. In this course, you can use the geometric sketchpad for animation demonstration. Let students form a complete concept through observation, thinking, discovery, thinking and application. Through the study of this section, students can better understand that derivative is the most effective tool to study the monotonicity and change speed of function, and it is the key content of this chapter.
2. Emphasis, difficulty and key of teaching.
Teaching emphasis: the geometric meaning of derivative, the solution of tangent equation and the thinking method of "combination of numbers and shapes, approximation".
Teaching difficulty: understanding the essential connotation of derivative geometric meaning
1) Approximation method used in secant to tangent;
2) Understand the concept of derivative and relate all kinds of meanings. For example, the derivative reflects the change speed of the function f(x) near the X point, and the derivative is the slope of the tangent at a certain point on the curve, and so on.
Second, tell the teaching objectives:
According to the requirements of the new curriculum standards and students' cognitive level, the teaching objectives are determined as follows:
1, knowledge and skills:
By exploring the geometric meaning of derivative through experiments and understanding the concept of tangent of curve at one point, we will find the tangent equation of unary function at one point.
Process and method:
Through the formation process of tangent definition, cultivate students' thinking ability of analysis, abstraction and generalization; Understand the idea and connotation of derivative, and improve the knowledge and understanding of tangent.
Through the concrete application of approximate thought and the combination of number and shape, students can realize the transfer of thinking mode and understand the scientific thinking method.
3, emotional attitudes and values:
Mathematical thoughts such as infiltration and approximation, combination of numbers and shapes, and direct substitution of music stimulate students' interest in learning, guide students to understand the dialectical relationship between special and general, finite and infinite, quantitative change and qualitative change, feel the unified beauty of mathematics, and realize the application value of mathematics.
Third, talk about teaching methods and learning methods.
For a straight line, its derivative is its slope, so students will naturally think about whether the derivative has special geometric significance in the function image. Moreover, students have just learned the concept of curve tangent. Based on the above analysis, I have determined the following teaching methods:
Teaching method: Introduce this lesson from the definition of the tangent of a circle, and then guide students to discuss the definition of the tangent of a general curve. Through the animation demonstration of the geometric sketchpad, we can get the definition of "approximation" method of curve tangent, and also get the geometric meaning of derivative and the mathematical idea of intuitive perception of "approximation" through the experimental observation of the geometric sketchpad. Therefore, I use the combination of experimental observation, inquiry research teaching and information technology-assisted teaching to highlight key points and break through difficulties.
Learning style: In order to give full play to students' subjective initiative and improve their comprehensive ability, this class adopts independent, cooperative and inquiry learning style.
Teaching AIDS: geometric drawing board, slides
Fourth, talk about teaching procedures.
1, create a situation
Student Activities-Question Series
Question 1 How do we judge whether a straight line is a secant or a tangent of a circle in plane geometry?
Question 2: Is the straight line L tangent to the curve C?
(1) and (2) and the positional relationship between straight line and hyperbola.
Question 3: So how do you define the tangent of a general curve?
Design intention: construct cognitive conflict through analogy.
Student activities-review
Definition of derivative
Design intention: pave the way for this class from two aspects: theory and knowledge base.
Step 2 explore knowledge
Student activities-experimental exploration
Ask one; What are the steps to find the derivative?
Step 1: Find the average change rate; Step 2: When approaching 0, the constant of the infinite approach of the average change rate is.
Design intention: This is to describe the derivative from the perspective of "number" and prepare for exploring the geometric meaning of derivative.
Question 2; Can you tell me what the average change rate means with pictures? Please draw it in the function image.
Design intention: The average change rate obtained through students' hands-on practice represents the slope of secant PQ.
Question 3; Can you describe the change of secant PQ in the process? Please draw it in the image.
Design intention: the process described from the angles of "number" and "shape" respectively. From a numerical point of view, q (); From the shape point of view, in this process, point Q approaches point P infinitely, and secant PQ approaches a certain position. The straight line at this position is called the tangent of the curve.
Inquiry 1: Students observe the changing trend of secant through the demonstration of geometric sketchpad, and the teacher guides to give the definition of tangent of general curve.
Design intention: with the help of multimedia teaching, guide students to discover the geometric meaning of derivatives, make problems intuitive and easily break through difficulties; In this process, students can experience approximate thinking methods. It can strengthen students' understanding of the concept of logarithmic derivative.
Question 4; Can you generalize the geometric meaning of the derivative of the function at from the above process?
Design intention: guide students to find and say: PQ tangent Pt is being cut, so PQ tangent PT is being cut.
The slope of the tangent. Therefore, = slope of tangent point.
Teaching evaluation of verbs (abbreviation of verb)
1. Evaluate students' learning process by whether they actively participate in activities and whether they can cooperate with others to explore;
2. Evaluate students' learning ability through their choice of methods;
3. Evaluate students' learning effect through exercises and homework.
4. In teaching, students learn as researchers, and in the process of solving problems, their understanding of knowledge changes from vague to clear, from intuitive perception to accurate mastery through their own experience;
5. The design goal of this lesson is to make students understand the basic idea of calculus, feel the unity of approximation and precision, the unity of motion and stillness, and feel the change from quantitative to qualitative. I hope to use this lesson to penetrate the essence of dialectical thought.