Teaching design of senior high school mathematics teaching plan 1
Monotonicity and parity of functions
Teaching objectives
1. Understand the concepts of monotonicity and parity of functions and master the basic methods of proof and judgment.
(1) Understand and distinguish the concepts of increasing function, subtraction function, monotonicity, monotone interval, odd function, even function, etc.
(2) We can understand monotonicity and parity from the perspective of numbers and shapes.
(3) The monotonicity of some functions can be judged by images, and the monotonicity of some functions can be proved by definitions; The parity of some functions can be judged by definition, and the drawing process of some function images can be simplified by parity.
2. Improve students' algebraic reasoning ability by proving monotonicity of functions; Through the formation of the concept of functional parity, students' abilities of observation, induction and abstraction are cultivated, and mathematical ideas from special to general are infiltrated at the same time.
3. Through the theoretical research on monotonicity and parity of functions, let students experience the beauty of mathematics, cultivate the spirit of seeking knowledge and form a scientific and rigorous research attitude.
Teaching suggestion
I. Knowledge structure
The concept of monotonicity of (1) function. Including the definition of increasing function and subtraction function, the method of judging monotonicity of conceptual function in monotone interval, and the relationship between monotonicity of function and function image.
(2) The concept of function parity. Including the definition of odd function and even function, the judgment method of function parity, the image of odd function and even function.
Second, analysis of key points and difficulties
(1) This section focuses on the formation and understanding of the concepts of monotonicity and parity of functions. The difficulty in teaching lies in understanding the essence of monotonicity and parity of functions and mastering the proof of monotonicity.
(2) The monotonicity of the function was known to students in junior high school, but they only observed the rise and fall of the image intuitively, but now they are required to raise it to the theoretical level and describe it with accurate mathematical language. This kind of translation from form to number, from intuition to abstraction is more difficult for senior one students. Therefore, we should focus on the formation of concepts. The proof of monotonicity is the first time that students come into contact with algebraic argument in the content of function. Students' ability of algebraic argument and reasoning is relatively weak. Many students don't even know what algebraic proof is and don't realize its importance, so monotonic proof is naturally a difficult point in teaching.
Three. Suggestions on teaching methods
When introducing the concept of monotonicity of (1) function, we can start with the images of linear function, quadratic function and inverse proportional function that students are familiar with, and from this perceptual knowledge, we can gradually approach the abstract definition by asking questions. For example, we can design such a question: how did the image get up? It can be explained from the angles of the coordinates of points, the relationship between independent variables and function values, so as to guide students to find the changing law of independent variables and function values, and then express this law in mathematical language. In this process, we can integrate the understanding and necessity of some keywords (in a certain range, both arbitrary and necessary), and combine the formation and understanding of concepts.
(2) The steps of proving monotonicity of functions are strictly stipulated. In order for students to follow the steps, it is necessary to make clear the necessity and purpose of each step, especially in the third step of transformation, so that students can make clear the transformation goal and break as much as they can. In the choice of examples, we should have different transformation goals as the standard of topic selection, which will help students sum up the law.
When introducing the concept of functional parity, courseware can be designed as follows
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For example, let the independent variables be relative and observe the changing law of the corresponding function values. First of all, from the specific values,
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At first, gradually let
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Move on the number axis, observe arbitrariness, and then ask students to write what they see in mathematical expressions. Through this process, they can get the equation.
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It is better to understand that it represents countless multiple equations, which is an identity. Regarding the symmetry of the definition domain about the origin, we can also make various changes to the function image with the help of courseware to help students discover the symmetry of the definition domain, and at the same time, we can also use images (such as
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It is proved that the symmetry of the domain about the origin is only a necessary condition for the function to have parity, but not a sufficient condition.
Teaching design of high school mathematics teaching plan II
The first volume of senior high school mathematics (1) 1. 1 set (1) teaching case teaching goal: 1, understand the concept and elements of set; 2. Understand the three characteristics of set elements; 3. Representation of commonly used number sets; 4, will judge the relationship between elements and sets,
Teaching case set (1)
. Teaching emphasis: 1, the concept of set; 2. Three characteristics of set elements: 1, three characteristics of set elements; 2. The relationship between several episodes: preparation before class: 1, preparation of teaching AIDS: introduction of multimedia production mathematician Cantor, including head portrait, life and contribution to the development of mathematics; Examples, figures, etc. This class needs. 2. Arrange students to preview the compilation of 1. 1. Instructional design: 1. [Creating a situation] Multimedia display to stimulate interest: a man crazy about science-Cantor (1845-1918), Russia. Cantor was born in St. Petersburg, Russia, his parents were Dan _, and his father was born in Copenhagen _, Denmark. He is a wealthy businessman. His mother Mary is an artist. When his parents were young, they moved to St. Petersburg, where Cantor was born. Cantor, the eldest son of his family, moved to Frankfurt in 1856. Because Cantor has changed its nationality many times, many countries think that they have cultivated Cantor's achievements. Cantor was interested in mathematics since he was a child. He received his doctorate at the age of 23 and has been engaged in mathematics teaching and research ever since. The set theory he founded has been recognized as the basis of all mathematics. The concept of infinity put forward by Cantor in 1874 shocked the intelligentsia. With the help of the infinite thought in ancient and medieval philosophical works, Cantor derived a new thinking mode about the nature of numbers, established the basic skills of dealing with infinity in mathematics, and greatly promoted the development of analysis and logic. He studied number theory, expressed functions with trigonometric functions, and found amazing results: he proved that rational numbers are countable, but all real numbers are uncountable. Because the study of infinity often leads to some logical but absurd results (called "paradox"), many great mathematicians are afraid of falling into it and adopt an evasive attitude. During 1874- 1876, Cantor, who was less than 30 years old, declared war on the mysterious infinity. With hard sweat, he successfully proved that points on a straight line can correspond to points on a plane one by one, and can also correspond to points in space one by one. In this way, it seems that there are "as many" points on the 1 cm long line segment as there are points in the Pacific Ocean and the whole earth. In the following years, Cantor published a series of articles about this kind of "infinite set" and drew many amazing conclusions through strict proof. Cantor's creative work has a sharp conflict with the traditional mathematical concept, and some people oppose, attack and even abuse it. Some people say that Cantor's set theory is a kind of "disease", Cantor's concept is "fog in fog", and even Cantor is a "madman". Great mental pressure from mathematics finally destroyed Cantor, making him exhausted, insane and sent to a mental hospital. Many of his outstanding achievements in set theory were obtained during the period of mental illness. At the first international congress of mathematicians held in 1897, his achievements were recognized. Russell, a great philosopher and mathematician, praised Cantor's work as "probably the greatest work that this generation can boast about." But Cantor is still in a trance, unable to get comfort and joy from people's reverence. 1918 65438+1October 6th, Cantor died in a mental hospital. Today, we learn the first chapter of senior high school mathematics, set 1. 1 set simple logic (1). Let's review the knowledge related to collection in junior high school. Second, [Review old knowledge] Review questions: 1. What sets did we learn in junior high school? Real number set, solution set of binary linear equation, solution set of inequality (group), point set, etc. 2. In junior high school, what did we describe with set? Angle bisector, perpendicular bisector, circle, inside circle, outside circle, etc.
Real rational number irrational number integer fraction positive irrational number negative irrational number positive fraction negative integer natural number positive integer zero 3. Classification of real numbers 3. Classification of real numbers:
Real number positive real number negative real number zero
4. The following is completed by students: (1). Fill in the following numbers in the corresponding circles.
0、、2.5、、、-6、、8%、 19
Integer set, fraction set, irrational number set
(2) Fill in the following numbers in the corresponding braces: 1,-10, -2, 3.6, -0. 1 8, negative rational number set: {}
Integer set: {}
Positive real number set: {}
Irrational number set: {}
3. Solve the inequality group (1) 2x-3 < 5.
4. The integer whose absolute value is less than 3 is ————————————— [Learning Interaction] 1. Observe the following objects (1) 2,4,6,8, 10, 12. (2) All right triangles; (3) Points with equal distance on both sides of the angle; (4) satisfy x-3 >; All real numbers of 2; (5) All boys in this class; (6) four great inventions of ancient china; (7) subjects of the 2007 college entrance examination; (8) Ball events in the 2008 Olympic Games,
In the teaching case of set (1), after the students observed the above objects, the teacher asked: [Concept of set] (1) What is set? When some specified objects are put together, they become a set, which is called a set for short. (2) What are the elements of a set? Each object in a collection is called an element of the collection. (3) How to express a set and its elements? General sets are represented by braces, usually in capital letters; Elements in the collection are represented by lowercase letters. (4) The relationship between the elements in the set and the set A is the element of the set A, which is called A ∈ A; A is not an element of the set A, so A does not belong to A, and is recorded as aA. 2. Discuss whether the following question (1){ 1, 2,2,3} is a set containing 1 1, 2 2, 1 3. (2) Can scientists form a set? (3) Do 3){ a, b, c, d} and {b, c, d, a} represent the same set? Through the discussion between teachers and students, the following conclusions can be drawn: The nature of elements in a set is certain: the elements in a set must be certain. The characteristics of the elements in the set are different from each other: the elements in the set must be different from each other. Disorder: The elements in the set are out of order. The elements that make up a set can be numbers, figures, people, things, etc. [Representation of common number set] (1) natural number set: n means (2) positive integer set: n or N+ means (3) integer set: z means (4) rational number set: q means (5) real number set: r means (positive real number set is R_R+). The root example 2 of the number (D) equation x2-3x+2=0, which is very close to 2004, is filled with symbols (1) 3.14q (2) π q (3) 0n+(4) 0n.
32(5)(-2)0N_6)Q
3232(7)Z(8)—R
Verb (abbreviation of verb) [stratified exercise] 1, multiple-choice question (1) The following () A, all triangles B, all questions C and integer D in "Mathematics for Senior One" are greater than π, so irrational number 2, true or false (1) {x2, 3x+.
The set of common numbers belongs to a∈AN, N_ or N+), z, q, R. The relationship between the conceptual elements of the set and the set; The nature of the elements in the set is deterministic, anisotropic and disorderly, and does not belong to aA.
The purpose of this lesson design is to stimulate students' interest in learning, prepare before class and cultivate students' autonomous learning ability by creating situations; Multimedia-assisted teaching improves classroom efficiency and enriches teaching presentation. Explore the integration of modern teaching methods and high school mathematics teaching.
Teaching Design of Senior High School Mathematics Teaching Plan III
The concept of set
Teaching purpose:
(1) enables students to have a preliminary understanding of the concept of set, and to know the concept and notation of common number set.
(2) Let students understand the meaning of "belonging" relationship.
(3) Make students understand the meaning of finite set, infinite set and empty set.
Teaching emphasis: the basic concept and expression method of set
Difficulties in teaching: Use two commonly used representation methods of sets-enumeration method and description method to correctly represent them.
Some simple sets
Teaching type: new teaching
Class schedule: 1 class hour
Teaching AIDS: multimedia, physical projector.
Content analysis:
1. set is an important basic concept in middle school mathematics. In primary school mathematics, the initial concept of set is permeated. Junior high school further expresses some problems in set language, such as number set and solution set used in algebra. As for logic, it can be said that from the beginning of learning mathematics, it is inseparable from the mastery and application of logic knowledge. Basic knowledge of logic is also an indispensable tool for understanding and studying problems in daily life, study and work. These can help students understand the significance of learning this chapter, and they are also the basis of learning this chapter.
The reason why the set of preparatory knowledge and simple logic knowledge are arranged at the beginning of high school mathematics is because in high school mathematics, these knowledge are closely related to other contents and are the basis for learning, mastering and using mathematical language. For example, the concepts and properties of functions in the next chapter are inseparable from sets and logic.
This section introduces the concepts of set and the elements of set from the examples involved in algebra and geometry in junior high school, and illustrates the concept of set with examples. Then it introduces the common representation methods of sets, including enumeration method and description method, and gives an example of representing sets by drawing.
This lesson mainly studies the introduction of the whole chapter and the basic concepts of set. The introduction is to arouse students' interest in learning and let them know the significance of learning this chapter. The teaching focus of this lesson is the basic concept of set.
Set is an original and undefined concept in set theory. When we came into contact with the concept of set, we got a preliminary understanding of the concept mainly through examples. The sentence "In general, some specified objects will become a set together, also called a set" given in the textbook is only a descriptive explanation of the concept of set.
Teaching process:
First, review the introduction:
1. Introduce the development of number sets and review common divisors and least common multiples, prime numbers and sums;
2. Introduction of chapters in the textbook;
3. The founder of set theory-Cantor (German mathematician) (see appendix);
4. "Birds of a feather flock together" and "Birds of a feather flock together";
5. Examples in textbooks (P4)
Second, explain the new lesson:
Read the first part of the textbook. These questions are as follows:
What are the concepts of (1)? How is it defined?
(2) What symbols are there? How is it expressed?
(3) What are the characteristics of the elements in the set?
(a) the related concepts of set:
It is composed of some numbers, some points, some figures, some algebraic expressions, some objects and some people. We say that all the objects in each group form a set, or that some specified objects together form a set, which is also called a set for short. Every object in a collection is called an element of this collection.
Definition: Generally, some specified objects are brought together to form a set.
1, the concept of set
(1) Set: Set some specified objects together to form a set.
(2) Element: Each object in a set is called an element of this set.
2, commonly used digital sets and symbols
(1) non-negative integer set (natural number set): the set of all non-negative integers is denoted as n,
(2) Positive integer set: the set without 0 in the non-negative integer set is recorded as N_N+
(3) Integer set: the set of all integers is denoted as z,
(4) rational number set: the set of all rational numbers is marked as q,
(5) Real number set: the set of all real numbers is recorded as r.
Note: (1) natural number set is the same as non-negative integer set, that is, natural number set includes.
Count 0
(2) The set without 0 in the non-negative integer set is denoted as N_N+Q, z, r, etc.
A set that excludes 0 from a number set is also expressed in this way, such as excluding 0 from an integer set.
A collection of, denoted as z.
_
3. The connection between elements and sets
(1) belongs to: If A is an element of the set A, it is said that A belongs to A and marked as A ∈ A.
(2) Does not belong to: If A is not an element of the set A, it is said that A does not belong to A, and it is recorded as
4. Characteristics of elements in the set
(1) Determinism: Given an element or in this set according to clear criteria,
Or not, not ambiguous.
(2) Reciprocity: the elements in the set are not repeated.
(3) Disorder: The elements in the set have no certain order (usually written in normal order).
5.( 1) sets are usually represented by capitalized Latin letters, such as A, B, C, P and Q. ...
Elements are usually represented by lowercase Latin letters, such as A, B, C, P, Q. ...
(2) The opening direction of "∈" should not be written as A ∈ A in reverse.
Third, exercise questions:
1, textbook P5 exercise 1, 2
2. Can the following groups of objects determine a set?
(1) All very large real numbers (uncertain)
(2) Good-hearted people (uncertain)
(3) 1, 2, 2, 3, 4, 5. (copy)
3. Let A and B be non-zero real numbers, then the possible values of the set are _-2, 0, 2__.
4. The set consisting of real numbers x, -x, |x| contains at most (a).
(A)2 elements (B)3 elements (C)4 elements (D)5 elements
5. Let all the elements in the set G be numbers in the form of a+b(a∈Z, b∈Z), and prove that:
(1) when x∈N, x∈G;
(2) if x∈G and y∈G, then x+y∈G does not necessarily belong to the set g.
Proof (1): in a+b(a∈Z, b∈Z), let a=x∈N, b=0,
Then x=x+0_a+b∈G, that is, x ∈ g.
Proof (2):∫x∈G, y∈G,
∴x=a+b(a∈Z,b∈Z),y=c+d(c∈Z,d∈Z)
∴x+y=(a+b)+(c+d)=(a+c)+(b+d)
∫a∈Z，b∈Z，c∈Z，d∈Z
∴(a+c)∈Z,(b+d)∈Z
∴x+y=(a+c)+(b+d)∈G，
Again =
And are not necessarily all integer,
∴ = does not necessarily belong to the set g.
Fourth, summary: This lesson has learned the following points:
1. Related concepts of set: (set, element, attribution, non-attribution)
2. The essence of set elements: certainty, mutual difference and disorder.
3. Definitions and symbols of commonly used number sets
V. Homework:
Six, blackboard design (omitted)
Seven, after class: