Derived Courseware of Mathematics for Liberal Arts in Senior Two: Basis of Teaching Materials
The concept of derivative is the content of the first section of the third chapter of Mathematics Elective Course 2-2 in full-time ordinary high school textbooks published by Beijing Normal University.
Second, the design ideas
Teaching material analysis:
Derivative is an important part of calculus, which comes from the needs of production technology and natural science; At the same time, it also promoted the development of production technology and natural science. It is not only widely used in astronomy, physics and engineering technology, but also plays an important role in daily life and economy.
This section is divided into four parts, one is the slope of the tangent of a point on the curve; The second is the instantaneous velocity of a non-uniform linear moving object; The third is the definition of derivative; The fourth is the geometric meaning of derivative. The purpose of learning the slope and instantaneous velocity of tangent is to introduce the concept of derivative and its geometric meaning, so as to deepen the understanding of the concept of derivative.
design concept
Students-oriented, attach importance to the thinking process, attach importance to the formation process of mathematical concepts, stimulate students' interest in learning, and consciously cultivate students' learning perseverance. Let students learn interesting and useful mathematics, and fully reflect the application value, thinking value and humanistic value of mathematics.
Third, the teaching objectives
1, knowledge and skills target:
Through the analysis of two examples, through the formation process of the concept of derivative, we can understand the actual background of the concept of derivative, so as to master the concept of derivative.
Cultivate students' ability of observation, analysis, comparison and induction through hands-on calculation, and understand extreme thoughts.
2, process and method objectives:
Through the exploration of problems, we can realize approximation and analogy, and explore the unknown with the known, from special to general mathematical thinking methods.
3. Emotions, attitudes and values:
Through the study of the concept of derivative, we can understand and agree with the dialectical viewpoint of "unity of opposites between finite and infinite" and accept the positive attitude of dealing with mathematical problems with dialectical materialism of motion change.
Fourth, the focus of teaching
The formation process of the concept of derivative.
Difficulties in teaching verbs (abbreviation of verb)
Understanding of the concept of derivative.
Key and difficult breakthrough measures:
1. Sensible people and rational people.
In the creative situation: "two innovations" are exciting; Layer-by-layer inquiry: divided into three types of inquiry, step by step, line by line, forming a concept.
2, the combination of numbers and shapes, the combination of ancient and modern.
Traditional calculation data provide students with initial feelings and experiences; Modern multimedia technology shows the formation process of tangent and instantaneous velocity intuitively and vividly, and breaks through the difficulties.
3, practical, hierarchical improvement.
The effect of teaching students in accordance with their aptitude is achieved by using hierarchical training and hierarchical homework.
Sixth, teaching preparation.
Calculator, multimedia computer, courseware, etc.
Seven. teaching process
Combining the principles of acceptability and operability, the implementation of teaching objectives is integrated into the teaching process, and students are helped to actively construct the concept of derivative through the formation, development and application of deductive derivative.
Eight, teaching reflection
1, the "student-oriented" educational concept is the fundamental guiding ideology of teaching design.
Students learn through the process of "experience", "experience" and "feeling", and finally form concepts, which fully embodies the modern education concept of student-oriented; The hierarchical design of exercises and homework tries to meet the diverse learning needs and teach students in accordance with their aptitude. However, in the specific implementation, the grasp of discretion depends on the situation.
2. Effective decomposition strategy is adopted in the breakthrough of difficulties.
(1) The three types of macro-inquiry conform to students' cognitive laws;
(2) Explore four microscopic steps to effectively decompose and break through difficulties;
(3) The situation runs through, and interest accompanies learning;
(4) Make full use of modern multimedia technology, and combine graphics and shapes to decompose difficulties.
3. The form and content are unified and have strong operability.
In all kinds of inquiry, the form and content are harmonious and unified, and the teacher's guidance is in place in time, which has strong maneuverability.
Mathematics Derivative Courseware for Liberal Arts Students in Senior Two Part I: Analysis of Teaching Content
Derivative is one of the core concepts of calculus. Derivative is the abbreviation of derivative function, and its essence is function, which is actually WeChat business.
Derivative is not only mathematical knowledge, but also a kind of mathematical thought, which also includes the thinking method of function and limit. The core of this section is to describe the instantaneous change rate with the limit of the average change rate. From the requirements of curriculum standards and the compilation of teaching materials, the formal definition of limit is diluted, and derivative is not treated as a special limit, but directly reflects the idea and essence of derivative through examples. Therefore, it is the focus of this section to let students fully experience the "extreme process and research thinking method"
Derivative belongs to factual knowledge-the instantaneous rate of change of a function exists objectively. It is only one way for us to study derivatives to describe them with the limit of average change rate and express them with the formal limit symbol. Derivative provides an important method and means for studying variables and functions, and has the extraordinary ability to simplify complex problems into simple rules and steps. It is not only the most effective tool for learning elementary functions, but also the necessary foundation for learning calculus, and it is also a tool for learning various sciences. Riemann once said, "Only after the invention of calculus,
Variables and functions have an almost nonexistent practical background in nature and society, so senior high school students should learn derivatives and their applications no matter whether they enter university in the future, and use them to investigate and understand the changes of actual phenomena. It is no exaggeration to say that it is difficult for students to reach a higher level of thinking without studying calculus. In a sense, the research value of mathematical thinking method contained in derivative is much higher than that of its knowledge. Through the formation process of the concept of derivative in this course, we can understand the ideas of approximation, combination of numbers and shapes and function, and further understand the essence of mathematics.
Second, the setting of teaching objectives
Knowledge and skills:
(1) Know the relationship between average change rate and instantaneous change rate; Can correctly distinguish the average rate of change and instantaneous rate of change; Will describe the actual background of the concept of derivative, know that the instantaneous rate of change is derivative, know the relationship between the derivative of a function at a certain point and the derivative function of a certain interval, and understand the idea and connotation of derivative.
(2) The derivative of a simple function at a certain point can be found according to the definition, and the basic steps of finding the derivative of a function at a certain point can be preliminarily summarized according to the definition.
Process and method:
(1) Let students observe and experience the "approaching" process from the average rate of change to the instantaneous rate of change through the dynamic demonstration with the geometric sketchpad, and experience the extreme thinking method.
(2) Through a series of independent and cooperative inquiry activities, perception describes the instantaneous change rate with the average change rate-infinite approach.
(3) Cultivate students' abilities of observation, analysis, comparison, induction and analogy through the abstract process of example-speed-change rate, and experience the method of studying problems from special to general.
Emotions, attitudes and values;
(1) Feel the function of derivative in solving practical problems and realize the function and value of derivative thought.
(2) Through a series of inquiry activities formed by derivative concepts, we can further understand the significance of cooperative learning and enhance students' awareness and ability of cooperative communication.
(3) Infiltrate patriotic education and stimulate students' patriotic enthusiasm by introducing the cases of winning gold medals in Olympic diving.
Third, the analysis of students' learning situation.
Students have mastered the average rate of change, the average speed and the instantaneous speed of the function in senior one physics, and accumulated a lot of experience about the rate of change of the function. In addition, senior two students are active in thinking and have the ability of induction, generalization, analogy and abstract thinking. As a new concept, derivative has a strong thirst for knowledge and a positive emotional attitude, which lays the foundation for the study of this course.
Because the instantaneous rate of change is that the derivative is "infinitely close" to the average rate of change, and "infinity" is abstract, it is the first contact for students. Students are required to have both intuitive perception ability and higher abstract thinking ability, which is the necessary cognitive basis of this section.
It is the first leap in this section to abstract examples into mathematical models from average speed and instantaneous speed to average change rate and instantaneous change rate. It is the second leap of thinking and understanding in this section to describe the instantaneous change rate from the average change rate by extreme thinking method. The first leap can be completed by students, and the second leap can be initially realized by students with the help of dynamic demonstration of geometric sketchpad, but for "
It is infinitely close to 0, but it can never be 0 ". Students can't do it independently or cooperatively, so teachers need to give full play to their leading role in this respect.
To sum up, the difficulty of this section is: the understanding of limit thought, using the limit of average change rate to describe the scientific nature of instantaneous change rate. Make more use of examples, reduce the degree of abstraction and strengthen the understanding of the process; Give students enough time to fully cooperate and communicate; Teachers should pay due attention to guidance and use "dynamic" to see "static".
Fourthly, the analysis of teaching strategies.
Teaching follows the "four masters" principle of "students as the main body, teachers as the leading factor, training as the main line and developing thinking as the main theme". With the appropriate series of activities as the link, we will create a space for students to explore independently, cooperate and communicate, guide students to experience the process of rediscovering mathematical knowledge, and let students acquire knowledge, develop their thinking and understand mathematics through participation.
Strengthen the understanding of the average change rate, consolidate the cognitive foundation, increase examples, and perceive from multiple models and angles, so that students can describe the thinking method of instantaneous change rate with the infinite approximation of the average change rate.
In the processing of knowledge content, we should dilute the incomprehensible limit thought, not pursue strict formalization, and highlight the thinking method of letting students experience infinite approximation intuitively.
According to the intuitive significance of the average change rate and students' thinking level, firstly, make full use of the intuitive display of the geometry sketchpad to strengthen the guidance of students' discovery learning; Secondly, on the basis of independent inquiry, let students fully carry out cooperative learning, discover the connotation of derivatives, understand mathematical thinking methods and experience the happiness of success; Third, in view of individual difficulties, teachers should give careful guidance to improve classroom efficiency.
Take the "strange average speed" as the problem situation, create cognitive conflicts and stimulate students' thirst for knowledge; * * Starting from the intuitive change of the average speed, four series of inquiry activities are designed, step by step, asking questions at different levels, guiding students to gradually abstract the formation of the derivative concept on the basis of full intuitive perception, so that students can fully experience the idea and method of "limit" in the formation of the derivative concept.
In view of the objective differences between students and the abstract degree of this section, students with good thinking level in each mathematics classroom study group are mainly used to help students with certain difficulties in this section, so that "students with learning difficulties" have better opportunities for display and communication in the group; Try to give more excellent students the opportunity to fully demonstrate in class; Teachers strengthen the feedback on students' autonomous learning and cooperative learning, and explain the problems in each group in detail, so as to improve the ability of all students to analyze and solve problems in the learning process to varying degrees.
Because this section is a new concept teaching, the key point is to let students experience the "extreme process and research thinking method", so we use the quadratic function that students are most familiar with.
As a model, feedback students' understanding of derivative concepts and research methods; According to the definition, the development level of students' thinking ability is fed back by induction and derivative methods.
Verb (abbreviation of verb) summary.