High school mathematics teaching plan design I
Teaching objectives
1。 Make students master concepts, images and properties.
(1) What kind of function can be judged according to the definition? Understand the rationality of cardinality restriction and define the domain clearly.
(2) Under the guidance of basic properties, the image drawn by the list tracking method can be identified from both numbers and shapes.
(3) We can compare the sizes of some powers by using the properties of, and we can draw a shape image by using the new image.
2。 Through the study of the essence of conceptual image, students' ability of observation, analysis and induction is cultivated, and the thinking method of combining numbers and shapes is further realized.
3。 Through research, students can realize the application value of mathematics and stimulate their interest in learning mathematics. Make students good at finding and solving problems from mathematics in real life.
Teaching suggestion
Textbook analysis
(1) is based on students' systematic study of the concept of function and their basic mastery of the nature of function. It is one of the important basic elementary functions. As a common function, it is not only the first application of the concept and properties of functions, but also the basis for learning logarithmic functions in the future. At the same time, it is widely used in life and production practice, and should be studied emphatically.
(2) The teaching focus of this section is to grasp the image and essence on the basis of understanding the definition. The difficulty lies in distinguishing the change of function value when the radix is sum.
(3) It is a function that students are completely unfamiliar with. How to make a systematic theoretical study of such a function is an important problem for students. Therefore, it is important to get the corresponding conclusions from the research process, but it is more important to understand the methods of systematically studying a class of functions. Therefore, students should be specially allowed to experience research methods in teaching so that they can transfer to other functions.
Teaching suggestion
(1) According to the textbook, the definition of "about" is a formal definition, that is, the characteristics of analytic expressions must be what they are, and there can be no difference, for example, and so on.
(2) Understanding and understanding the restrictive conditions of cardinality is also an important part of understanding. If possible, try to let the students study the limit and index of the base number by themselves, and the teacher will supplement or explain it with concrete examples, because understanding this condition is not only related to the understanding and classification of the nature, but also related to the understanding of the base number in the later logarithmic function learning, so we must really understand its origin.
Regarding the drawing of images, although the method of drawing points by list is adopted, we should avoid blind list calculation before drawing points in specific teaching, and also avoid blindly connecting points into lines. The list should be listed in key places, and the main points should be connected in appropriate places. Therefore, before drawing points with list, we should briefly discuss the properties of functions. After we have a general understanding of the existing scope, general characteristics and changing trend of the image to be drawn, we can draw points with list calculation as a guide to get an image.
Example of instructional design
The subject or problem of study.
Teaching objectives
1。 The definition, preliminary image, nature and simple application of understanding.
2。 Through the study of images and properties, students' ability of observation, analysis and induction is cultivated, and the thinking method of combining numbers and shapes is further realized.
3。 Through research, students can master the basic methods of function research and stimulate their interest in learning.
Teaching emphases and difficulties
The key point is to understand the definition and grasp the image and essence.
The difficulty lies in understanding the influence of cardinality on function values.
training/teaching aid
projector
teaching method
Heuristic discussion research formula
teaching process
One. Introduce a new course
We have studied exponential operation before. On this basis, today we are going to study a new common function--.
1。 6。 (blackboard writing)
This kind of function is mainly introduced because it is the need in real life. For example, let's look at the following questions:
Question 1: A cell _, from 1 _2, 2 _4, ... After such a cell _ times, the sum of the number of cells forms a functional relationship. Can you write the functional relationship between and?
Student answer: The relationship between and can be expressed as.
Question 2: There is a rope with a length of 1 meter. Cut off half of the rope for the first time, and cut off the remaining half of the rope for the second time ... After cutting twice, the remaining length of the rope is 100 m. Try to write the functional relationship between and.
Answer by the students.
In the above two examples, we can see that these two functions are different from those we studied before. They are in the form of power, and the independent variable is in the exponential position, so we call the function like shape.
One. The concept of (blackboard writing)
1。 Definition: Function in the form of call. (blackboard writing)
The teacher will explain after giving the definition.
2。 Some notes (blackboard writing)
(1) Rights clause:
The teacher first asked the question: Why should the cardinality be greater than 0 and not equal to 1? If students find it difficult, they can divide the questions into what if? For example, at this time, the corresponding function value in the real number range does not exist.
If it is meaningless, then no matter what value it takes, it will always be 1, so there is no need to study it. In order to avoid the above situation, provisions and.
(2) the domain of blackboard writing
Teachers guide students to review the index range and find that the index can take rational numbers. At this point, the teacher can point out that when the exponent is irrational, it is also a definite real number. For irrational exponential power, it is suitable for the learned properties and algorithms of rational exponential power, so the exponential range is extended to the real number range, so the definition domain of is. Another reason for the expansion is to make it more representative and valuable.
(3) Whether it is true (blackboard writing)
Just now, we have understood the requirements of radix and exponent respectively. Let's understand it from a holistic perspective. By definition, we know what a function looks like. Please see if the following function.
( 1) , (2) , (3)
(4) , (5) 。
Students answer and explain the reasons. The teacher comments according to the situation, pointing out that only (1) and (3) are yes, of which (3) can be written as an exponential image.
Finally, remind students that the definition is a formal definition, which must be exactly the same in form, and then lead the problem to a deeper level, with the nature of the domain and function of the preliminary study. At this time, the key to research is to draw its image and then summarize nature in detail.
3。 Inductive attribute
How to draw a picture? Use the list to trace the discovery. The teacher is ready to clarify the nature, and then the students answer.
function
1。 Domain:
2。 Scope:
3。 Parity: It is neither odd function nor even function.
4。 Intercept: not on axis, 1 on axis.
For attributes 1 and 2, we can say together and ask what they do. (determining the approximate position of the image) should also be proved to the third article. For monotonicity, I suggest finding something special. Take a look before you come to a conclusion. The last one is also the basis for guiding the drawing of function images. (The image is located above the axis and does not intersect the axis. )
On this basis, teachers can guide students to list and draw points. When choosing points, students should also be reminded that the values should be positive and negative, because they are not symmetrical, and the number of points taken should not be too small, because of the ambiguity of monotonicity.
Here, teachers can draw points with computers and give ten sets of data, while students can draw points by themselves, at least six sets of data. When connecting points into a line, students must be reminded of the changing trend of the image (the smaller the image, the closer it is to the axis, and the larger the image, the faster it rises), so as to form a smooth curve.
Two. Images and attributes (blackboard writing)
1。 Image painting: List painting under the guidance of nature.
2。 Sketch:
After drawing the first image, ask the students if they need to draw the second image. Is it representative? (The condition that the teacher can prompt the cardinality is that the value can be divided into two sections. Let the students understand that they need to draw a second picture, so take this as an example.
At this time, the drawing method of its image should be left to students to choose, and students should realize that the drawing method of the points in the list is wrong, and the method of image transformation is simpler. That is = symmetrical with the image, and the image at this time already exists and has the conditions for transformation. Let the students do their own symmetry, and the teacher draws pictures with the help of a computer. The images are obtained in the same coordinate system.
Finally, ask the students if they need to draw again. (There may be two possibilities. If students don't think it is necessary to draw again, ask why and let them tell the nature. If they think it is necessary to draw again, the teacher can draw such an image with the computer for comparison, and then find it.
Because images are the characteristics of shapes, let's first look at their characteristics from a geometric point of view. Teachers can make the following list:
If the students can't say the above, the teacher can put forward the observation angle for the students to describe, and then let the students transform the geometric characteristics into the properties of the function, that is, describe it from the algebraic angle, and fill in the other part of the form.
After filling it out, ask the students to make another list like this example and fill in the corresponding contents. In order to further sort out the nature, teachers can propose to classify and sort out the nature of functions from another angle.
3。 Nature.
(1) No matter what the value is, there is a domain, and the domain of is, too many.
(2) At that time, it was a increasing function in the defined domain, and at that time, it was a decreasing function.
(3) When,,,,.
After summing up, especially remind students to remember the image of the function. Charts allow you to read properties from them.
Three. Simple application (blackboard writing)
1。 Use monotonic ratio. (blackboard writing)
After studying the concept, image and properties of a kind of function, the most important thing is to use it to solve some simple problems. First, let's look at the following questions.
Example 1. Compare the sizes of the following groups.
(1) and; (2) and;
(3) and 1 (blackboard writing)
First, let the students observe the characteristics of two numbers. What are the similarities? Students pointed out that they have the same base but different indices. Then ask, according to this feature, what method is used to compare their sizes? Let the students associate and put forward the method of constructing function, that is, regard these two numbers as the function value of a function and compare the sizes by using its monotonicity. Then, taking the topic (1) as an example, the solution process is given.
Solution: It is playing an increasingly important role in the world, and
& lt。 (blackboard writing)
Finally, the teacher stressed that this process must be clearly written in three sentences:
(1) construct a function, and point out the monotone interval and the corresponding monotonicity of the function.
(2) Comparison of independent variables.
(3) Comparison of function values.
The process of the last two questions is brief. Ask the students to describe the process according to the question (1).
Example 2. Compare the sizes of the following groups.
(1) and; (2) and;
(3) and. (blackboard writing)
Ask the students to observe the difference between the number of groups in Example 2 and that in Example 1, and then think about the solution. Guide students to find that it can be written as (1), turn it into a problem with the same bottom, and then solve it by 1. For (2), it can be written or transformed into the same bottom problem, while (3) the previous method is not applicable. Consider a new transformation method to make students think. (Teachers can remind students that the function value is related to 1, and 1 can be used as a bridge. )
Finally, the students say > 1,< 1,>.
After the solution, the teacher summed up the method of comparing sizes.
(1) Method of constructor: Numbers are characterized by the same base and different fingers (including those that can be converted into the same base).
(2) Bridge comparison method: use the special number 1 or 0.
Three. Consolidation exercise
Exercise: Compare the sizes of the following groups (blackboard writing)
(1) and (2) and;
(3) and; (4) and. Short answer process
Four. summary
1。 The concept of
2。 Images and properties of
3。 Simple application
Five. blackboard-writing design
High school mathematics teaching plan design II.
oval
I. teaching material analysis
(A) the status and role of teaching materials
This section is another practical exercise of learning curves and equations by coordinate method after the equations of straight lines and circles. The study of ellipse can provide a basic model and theoretical basis for the study of hyperbola and parabola. Therefore, this lesson plays a connecting role and is one of the key contents of this chapter and this section.
(B) Teaching focus and difficulties
1. Teaching emphasis: the definition of ellipse and its standard equation.
2. Teaching difficulty: derivation of elliptic standard equation
(3) Three-dimensional target
1. Knowledge and skills: master the definition and standard equation of ellipse, clarify the concepts of focus and focal length, and understand the derivation of standard equation of ellipse.
2. Process and method: By guiding students to try to draw pictures, we can discover the formation process of ellipses, summarize the definition of ellipses, and cultivate students' ability to observe, discriminate, analogize and summarize problems.
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3. Emotion, attitude and values: Through active exploration, cooperative learning, mutual exchange and summing up knowledge, students can feel the joy of exploration and success and enhance their confidence in learning.
Second, teaching methods and means
Using heuristic teaching, classroom teaching adheres to the principle of taking teachers as the main body, students as the main body, thinking training as the main line and ability training as the main attack.
"It is better to teach people to fish than to teach people to fish." Students are required to experiment, explore independently, cooperate and communicate, abstract the definition of ellipse, and explore the standard equation of ellipse with coordinate method, so that the learning process of students becomes a "re-creation" process under the guidance of teachers.
Third, the teaching procedure.
1. Creating situations and understanding ellipses: Through experimental exploration, understanding ellipses leads to the teaching content of this lesson and stimulates students' curiosity.
2. Draw an ellipse: Draw a picture to give students an opportunity to operate and cooperate in learning, thus stimulating students' interest in learning.
3. Teacher's demonstration: Through the multimedia demonstration and the change of data, students can understand the forming process of ellipse more rationally.
4. Ellipse definition: Pay attention to the three conditions in the definition, so that students can better grasp the definition.
5. Derive the equation: Teachers guide students to simplify the complex, break through the difficulties, and get the standard equation of the ellipse with the focus on the X-axis and the Y-axis by using the graphics in students' hands, so as to re-understand the standard equation of the ellipse.
6. Example explanation: standardize students' problem-solving process through examples.
7. Consolidation exercise: Consolidate the teaching content of this lesson with a variety of questions.
8. Summary: Through summary, let students have a complete system of what they have learned, highlight key points, grasp key points, and cultivate students' generalization ability.
9. Homework after class: Design mandatory questions and multiple-choice questions for students of different levels.
10. blackboard writing design: the purpose is to outline the main line of the whole textbook, present a complete knowledge structure system and highlight key points, and use colors to increase the intensity of information and make it easy to master.
Fourthly, teaching evaluation.
This course implements the new curriculum concept, is student-oriented and starts from the training of students' thinking. By learning the definition of ellipse and its standard equation, students' original cognitive rules are activated, which lays the foundation for the optimization of knowledge structure.
Senior high school mathematics teaching plan design III
Subject: operation of exponent and exponent power
Class type: new teaching
Teaching methods: teaching and exploring.
Selection of Teaching Media: Multimedia Teaching
The Operation of Exponent and Exponential Power —— Learner Analysis;
1. demand analysis: before learning exponential function, students should master the operation of exponent and exponential power skillfully, and expand the range of exponential value to real number through this section, laying the foundation for learning exponential function.
2. Analysis of learning situation: I was exposed to the operation of positive exponential power in middle school, but it is far from enough for us to learn exponential function. Through this lesson, students can have a deeper understanding of the operation of exponential power.
The Operation of Exponent and Exponential Power —— Analysis of Learning Task;
1. teaching material analysis: This section contains many important mathematical thinking methods, such as generalization and approximation. The textbook pays full attention to the connection with practical problems, which embodies the importance of this section and the practical application value of mathematics.
2. Teaching emphasis: the concept of radical and the nature of n-th square root; The significance and operational properties of fractional exponential power: the mutual transformation of fractional exponential power and root.
3. Teaching difficulties: the nature of n-th root; Significance and operation of fractional exponential power.
The Operation of Exponent and Exponential Power: Clarification of Teaching Objectives;
1. Knowledge and skills: Understand the concept and nature of radical, master the operation of fractional exponential power, and be familiar with the reciprocity between fractional exponential power and radical.
2. Process and method: Through exploration and thinking, cultivate students' universal and approximate mathematical thinking methods, and improve students' knowledge transfer ability and active participation ability.
3. Emotion, attitude and values: In the teaching process, let students explore independently to deepen their understanding of N-degree root and fractional exponential power. Inquiry ability is an important aspect of learning mathematics, understanding mathematics and solving mathematical problems.
Teaching flow chart:
The Operation of Exponent and Exponential Power —— Teaching Process Design;
First, the new curriculum introduction:
(A) Introduction to the knowledge structure of this chapter
(B) the question raised
1. Question: When an organism dies, its original carbon 14 will decay according to a certain law, and it will decay to half of its original value every 5730 years. This time is called "half life". According to this law, people have come to the relationship between the content of P in organisms and the age of death T:
(1) After 5730 years of death, the value of carbon content P in organisms is
(2) After 5730×2 years of death, the P value of carbon content in organisms is
(3) After 6000 years of biological death, the value of carbon content P is 14.
(4) When the organism dies 10000 years later, the value of carbon content P in the organism is
2. Review the operational properties of exponential powers of integers.
Operational properties of exponential powers of integers;
3. Thinking: Do these operational properties apply to fractional exponential powers?
This is what we are going to learn today, the operation of exponent and exponent power.
Exponential and Exponential Power Operation of 2. 1. 1 on the blackboard
2. Radical concept:
Let's look at a few simple examples. The concepts of oral square root and cubic root guide students to summarize the concept of n square root. ..
Write the symbols of square root, cube root and n square root on the blackboard, and give some simple square root operations for students to observe and summarize.
Now, let the students sum up the concept of n square root. ..
The concept of 1. radical
Blackboard writing concept
That is to say, if the n power of a number is equal to a (n >; 1, and n∈N_, then this number is called the n-th root of a.
Through the example just given, it is not difficult to see that the odd-even number of n and the positive and negative of a will affect the n-th root of a. Let's finish such a table together.
Blackboard form
From this table, we know that negative numbers have no even roots. What is the nth root of 0?
The nth root of student 0 is 0.
Now let's explain this symbol.
Example 1. Find the following values.
Note that this question is relatively simple, students can answer it orally, and the process is omitted here.
Three. Properties of n-degree roots
Note: For 1, ask the students about the value range of A, and let them think and draw a conclusion.
2 points for attention, give a few examples less, let students observe, get up and say their own conclusions.
Properties of 1.n power root
Four. Power of fractional exponent
These two roots can be written in the form of fractional exponential power, because the index of the root can be divisible by the index of the root sign, so please think about the following questions.
Thinking: Can the root exponent be written as the power of fractional exponent if it is not divisible by the exponent of the square root?
If the teacher is established, what is its significance? We have such a rule.
(A) the significance of fractional exponential power:
1. We specify that the positive fractional exponential power of positive numbers means:
2. We stipulate that the significance of the negative fractional exponential power of positive numbers is:
A positive fractional exponent power of 3.0 is equal to 0, and a negative fractional exponent power of 0 is meaningless.
(2) Generalization of the nature of exponential power operation:
Examples of verbs (short for verb)
Example 2. assess
Note here that Example 2 asks the students to do it on the blackboard, Example 3 asks the teacher to perform on the blackboard after the students finish it, and Example 4 asks the students to do it on the blackboard, and then corrects the mistakes.
Summary of intransitive verbs course
The definition of 1. radical;
2. The nature of n-th root;
3. Fractional exponential power.
Seven. Homework after class
P59 Exercise 2.1Group A 1.2.4.
Eight. Reflection after class
1. In the first class, the important content was not written on the blackboard, and the conditions of A, R and S in the operational properties were not given. In addition, there is an error in the courseware. The mistakes in the first class were corrected in the second class.
There are many questions that should be answered by students, not by themselves. Radical thinking is not clear, so students should be given more time to answer and think about questions and less interaction with them.
3. There are still many details that have not been handled well during the lecture, and the lecture sound is small and there is no fluctuation.
4. The knowledge structure of the chapters before class is very good, the introduction is simple and in place, and the highlight is the table after the concept.
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