The teaching goal is 1, master the concept of rational numbers, classify rational numbers according to certain standards, and cultivate the ability of classification;

2. Understand the correlation between classification standards and classification results, and preliminarily understand the meaning of "set";

3. Empirical classification is a common method to deal with problems in mathematics.

Difficulties in teaching correctly understand the classification standards and classify them according to certain standards.

Knowledge lies in correctly understanding the concept of rational numbers.

Design concept of teaching process (teacher-student activities)

In the first two periods of exploring new knowledge, we have learned many different types of numbers. Through the study of the last two lessons, we know that the current figures contain negative numbers. Now please feel free to write 3 numbers on the draft paper (and please write 3 numbers on the blackboard).

Question 1: Observe the nine numbers on the blackboard and classify them.

Classification of students' thinking, discussion and communication.

Students may only give a rough classification, such as "positive number" and "negative number" or "zero". At this time the teacher should give guidance and encouragement.

For example,

For the number 5, you can ask: Are 5 and 5. 1 the same type? Can 5 represent 5 people, 5. 1 number of representatives? (No) So they are different types of numbers. The number 5 is an integer in a positive number, so we call it a "positive integer", while 5. 1 is not an integer, so it is called a "positive fraction, ... (Since decimals can be changed into fractions, both decimals and fractions are called fractions in the future).

Through teachers' guidance, encouragement and continuous improvement, and students' own induction, we finally summed up five different numbers we have learned, namely "positive integer, zero, negative integer, positive fraction and negative fraction".

According to the concepts of integer, fraction and rational number in the book.

Read books to understand the origin of rational number names.

"Collectively" means "collectively".

Try it: according to the above classification, can you make a classification table of rational numbers Can you tell me what the above criteria for rational number classification are? Classification (divided by integers and fractions) is a common method to solve problems in mathematics. This introduction is open and students are willing to participate.

When students try to classify themselves, it may be rough. Teachers give guidance and encouragement, and the types of classification numbers should be guided by the meaning expressed in words, so that students can understand them easily.

The rational number classification table should be displayed on the blackboard or the media, and the classification standard should guide students to experience it.

Practice 1, write three rational numbers at will, and tell what kind of numbers they are, and communicate with your partners.

2, the textbook page 65438 +00 exercises.

The concept of set appears in this exercise and can be explained to students as follows.

Put some numbers together to form a set of numbers, which is called "number set" for short. A number set consisting of all rational numbers is called a rational number set. Similarly, a number set composed of all integers is called an integer set, and a number set composed of all negative numbers is called a negative number set.

The number set is generally represented by circles or braces, because the numbers in the set are infinite, and only a few numbers are given in this question, so ellipsis should be added.

Thinking: Do the four sets in the above exercise add up to the set of all rational numbers?

Teachers can also say some figures for students to judge.

The concept of set needs no further expansion.

Innovative inquiry question 2: Rational numbers can be divided into positive numbers and negative numbers, right? Why?

When teaching, ask students to sum up the numbers they have learned, encourage students to sum them up, and give appropriate guidance through exchanges and discussions, and gradually get the following classification table.

The classification of rational numbers can determine whether teaching is needed according to the level of students.

Let the students understand that the classification results are different when the classification standards are different, so the classification standards should be clear, so that every elephant involved in the classification after classification belongs to a certain category, and can only belong to this category. In teaching, teachers can give some easy-to-understand examples to illustrate, either by age or by gender and region.

Summary and homework

Class summary Up to now, all the numbers we have learned are rational numbers (except pi). Rational numbers can be classified according to different standards, and the results of different standards are different.

The homework for this lesson is 1, and the required questions are: page 65438 of the textbook +08 exercise 1.2 question 1.

2. Teachers prepare themselves.

Comments on this lesson (classroom design concept, actual teaching effect and improvement ideas)

1, after introducing negative numbers, this lesson classifies the learned numbers according to certain standards and puts forward the outline of rational numbers.

Reading classification is a common means to solve mathematical problems. Through the study of this lesson, students can understand the idea of classification and go hand in hand.

Simple classification is the embodiment of mathematical ability, and teachers should pay enough attention to it in teaching. On classification standards and scores

The relationship between class grades and the determination of classification standards can be properly infiltrated into students. The concept of set is abstract, and it takes a long time for students to really accept it. Don't expand this lesson too much.

2. This course has the characteristics of openness, which provides students with more thinking space, can promote students to actively participate in learning and experience the formation process of knowledge personally, and can avoid the boredom caused by direct classification; At the same time, it also embodies the characteristics of cooperative learning, communication and inquiry, and has a good effect on the cultivation of students' classification ability.

3, two classification methods, should be based on the first method, the second method can be carried out according to the situation of students.

Subject: 1.2.2 number axis

The teaching goal is 1, master the concept of number axis, and understand the corresponding relationship between points on number axis and rational numbers;

2, the number axis will be drawn correctly, the given rational number will be represented by points on the number axis, and the rational number will be read according to the points on the number axis;

3. Feeling number and shape can be transformed into each other under certain conditions, and mathematics can be experienced in life.

The concept of number axis and the representation of rational numbers on the number axis are difficult points in teaching.

Knowledge focus

Design concept of teaching process (teacher-student activities)

Set the situation

Through examples and courseware demonstrations, this paper introduces how the project teacher obtains thermometer readings.

Question 1: Thermometer is an important tool for measuring temperature in our daily life. Can you read a thermometer? Would you please try to read the temperature displayed by the three thermometers in the picture?

(The multimedia shows three pictures, which are above zero, below zero and below zero.)

Question 2: On an east-west road, there is a bus stop. There are a willow tree and a poplar tree at 3m and 7.5m east of the bus stop, and a locust tree and a telephone pole at 3m and 4.8m west of the bus stop. Try to draw a picture to illustrate this situation.

(Group discussion, communication and cooperation, hands-on operation) Create problem situations to stimulate students' enthusiasm for learning and discover mathematics in life.

Points represent the perceptual knowledge of logarithm.

Cooperation and communication

Exploring new teachers: What can we learn from the above two questions? Can you use points on a straight line to represent rational numbers?

Let the students operate on the basis of discussion and summarize on the basis of operation: What conditions must a straight line that can represent rational numbers meet?

Thus, the three elements of the number axis are obtained: the origin, the positive direction and the idea of combining numbers and shapes per unit length; Only the characteristics of the number axis are described, and the requirements of the number axis three are not particularly emphasized.

Learn mathematics from the game and play the game: the teacher prepares a rope to let eight students come up, adjust the position to equal distance, and stipulate that the fourth student is the origin and the positive direction is from west to east. Every student has an integer. Please remember, now please ask the students in the first row to issue the password in turn. When the password is a number, the student corresponding to the number should answer "to"; When the password is a classmate's name, the classmate should report his corresponding "number". If the third student is designated as the origin, can the game still be played? Students' game experience and understanding of the concept of number axis

Looking for patterns

Conclusion question 3:

1, can you give some practical examples of numbers represented by straight lines in real life?

2. If you are given some numbers, can you find their exact positions on the number axis accordingly? If you are given some points on the axis, can you read the numbers it represents?

3. Which numbers are on the left of the origin and which numbers are on the right of the origin, what rules will you find?

4. What is the distance from each count to the origin? What rules will you find from it?

(Group discussion, communication and induction)

Summarize the general conclusion, textbook number 12. These questions are the skills that need to be learned in this course. Teaching should focus on students' inquiry learning, and teachers can give students appropriate guidance in combination with textbooks.

Consolidation exercise

Textbook exercises 12 pages.

Summary and homework

Class summary Let students summarize:

1, three elements of number axis;

2. The work of number axis and the transformation method between number and point.

The assignment for this lesson is 1, and the required question is: Exercise 2 on page 18 of the textbook 1.2.

2, choose to do the problem: the teacher arranges it himself.

Comments on this lesson (classroom design concept, actual teaching effect and improvement ideas)

1, the number axis is an important medium for number-shape conversion and combination. The prototype of situational design comes from real life and is easy for students to experience and accept. Through observation, thinking and hands-on operation, students can experience and appreciate the formation process of number axis, which can deepen their understanding of the concept of number axis and cultivate their ability of abstract generalization, which also reflects the cognitive law from perceptual knowledge to rational knowledge to abstract generalization.

2. The teaching process highlights the main line from emotion to abstraction to generalization, and the teaching method embodies the mathematical thinking method of combining numbers and shapes from special to general.

3. Pay attention to students' knowledge and experience, give full play to students' subjective consciousness, let students actively participate in learning activities, guide students to feel the generation, development and change of knowledge in class, and cultivate students' independent exploration of learning methods.

Subject: 1.2.3 Countdown

The teaching goal is 1, master the concept of inverse number, and further understand the corresponding relationship between points and numbers on the number axis;

2. Cultivate inductive ability by summarizing the characteristics of the points represented by the reciprocal on the number axis;

3. Experience the idea of combining numbers with shapes.

Teaching difficulty: summarize the characteristics of the points represented by the opposite numbers on the number axis.

The concept of the opposite number of knowledge focus

Design concept of teaching process (teacher-student activities)

Set the situation

Introduce the topic 1: Please divide the following four numbers into two categories and explain why they are so classified.

4,-2,-5,+2

It is difficult to encourage students to have differences, as long as they can tell the truth, but teachers should guide them appropriately and gradually come to the conclusion that 5 and -5, +2 and -2 are distinctive points.

(Guide students to observe the distance from the origin)

Thinking conclusion: Thinking textbook page 13.

Try two other similar numbers.

Conclusion: Summary of page 13 of the textbook. Create situations in an open way, discuss with students and cultivate classification ability.

Cultivate students' ability of observation and induction, and infiltrate mathematical thinking.

Deepen the definition of theme refinement and give the definition of reciprocal.

Question 2: How to understand the meanings of the words "only different symbols" and "interaction" in the definition of opposites? What is the reciprocal of zero? Why?

Students think, discuss and communicate, and the teacher summarizes.

Law: Generally speaking, the reciprocal of the number A can be expressed as-A..

Thinking: What is the relationship between the two points representing the opposite number on the number axis and the origin?

Exercise: The first exercise on page 14 of the textbook is to experience the characteristics of symmetrical figures and prepare for the characteristics of opposites on the number axis.

Deepen the concept of reciprocal; "The inverse of zero is zero" is part of the definition of inverse.

Strengthen the geometric meaning of points represented by mutually opposite numbers on the number axis.

Give the law

Question 3: What do-(+5) and -(-5) mean respectively? Can you simplify them?

Student exchange.

The antonyms of +5 and -5 are -5 and +5, respectively.

Exercise: the second exercise on page 14 of the textbook, using the concept of reciprocal, obtains the method of finding the reciprocal of a number.

Summary and homework

Definition of class summary 1, reciprocal

2. Characteristics of points represented by mutually opposite numbers on the number axis.

3. How to find the inverse of a number? How to express the reciprocal of a number?

The assignment for this lesson is 1, and the required question is the third question on page 18 of the textbook.

2, choose to be the teacher's own arrangement.

Comments on this lesson (classroom design concept, actual teaching effect and improvement ideas)

1, the concept of inverse number is convenient to express the arithmetic rules of rational numbers, and also reveals the characteristics of two special numbers. These two special numbers have the same absolute value in quantity, and their sum is zero. When expressed on the number axis, the distance from the origin is in phase, and so on. So this teaching design is based on the idea of combining numbers with shapes.

2. Teaching attracts people with open questions and cultivates students' ability of classification and divergent thinking; Representing numbers on the number axis and observing their characteristics, while reviewing the knowledge of the number axis, infiltrating the mathematical method of combining numbers and shapes, and transforming numbers and shapes can also deepen the understanding of the concept of reciprocal; Question 2 can help students master the concept of reciprocal accurately; Question 3 actually gives a method to find the reciprocal of a number.

3. This teaching design embodies the teaching concept of the new curriculum. Under the guidance of teachers, students learn independently, explore independently, observe and summarize, attach importance to students' thinking process, and leave room for students to play.

Subject: 1.2.4 Absolute value

Teaching goal 1, master the concept of absolute value and the comparison rule of rational numbers.

2. Learn to calculate absolute values and compare the sizes of two or more rational numbers.

3. The concepts and rules of empirical mathematics come from real life and are permeated with the idea of combination and classification of numbers and shapes.

Comparison of two negative numbers in teaching difficulties

The concept of absolute value in knowledge set

Design concept of teaching process (teacher-student activities)

Set the situation

On Sunday, Mr. Huang started from school and drove to play. She first went 20 kilometers east to Zhujiajian Island Island, and then 30 kilometers west in the afternoon, and returned home (school, Zhujiajian Island Island and home are on the same line). If the rule is Dongzheng, ① use rational number to represent the distance between Miss Huang's two trips; (2) If the car consumes 0. 15 liter per kilometer, how many liters does the car consume on this day?

After the students thought, the teacher explained as follows:

Some problems in real life only focus on the specific value of quantity, but the opposite is true.

Meaning is irrelevant, that is, positive and negative are irrelevant. For example, we only care about the distance traveled by cars and the price of gasoline, but have nothing to do with the direction of travel;

Observe and think: draw a number axis, and the origin represents the school. Draw points on the axis representing Zhujiajian Island Island and Miss Huang's home. Look at the picture and tell the distance from Miss Huang's home to Zhujiajian Island Island School.

After the students answered, the teacher explained as follows:

The distance between a point representing a number on the number axis and the origin is only related to the length of the point from the origin, and has nothing to do with the positive or negative of the number it represents;

Generally speaking, the distance between the point representing the number A on the number axis and the origin is called the absolute value of the number A, and it is recorded as |a|.

For example, the above question |20|=20, |- 10 | = 10. Obviously, in the example of |0|=0, the first question is a quantity with opposite meaning, with positive and negative numbers.

Numbers indicate that the answer to the latter question has nothing to do with symbols, which shows that there are some problems in real life. People only need to know their specific values without paying attention to their meanings, so as to prepare for introducing the concept of absolute value and make students feel better.

Test the connection between mathematics knowledge and real life.

Because the geometric meaning of the concept of absolute value is a typical transformation from number to shape.

It is difficult for students to accept the model for the first time. Configure this observation and thinking to prepare for establishing the concept of absolute value.

Cooperation and communication

Explore the example of law 1 and find the absolute value of the following number, and find the absolute value of rational number a.

What are the rules? 、

-3,5,0,+58,0.6

Group discussion and cooperative learning are required.

Teachers guide students to use the meaning of absolute value to find the answer first, then observe the characteristics of the original number and its absolute value, and combine the meaning of the inverse number, and finally summarize the law of finding the absolute value (see textbook 15).

Consolidation exercise: textbook 15 page exercise.

Among them, the answer to the question 1 is written directly according to the law, which is the basic training for finding the absolute value; The second problem is to distinguish the concept of reciprocal and absolute value, which requires students' analytical judgment ability. Pay attention to the thoroughness of thinking and let students understand the difference between different statements. The law of finding the absolute value of a number can be regarded as an absolute value summary.

Look at an application, so arrange this example.

Students do what they can, and teachers are only organizers in the teaching process. Based on this concept, this discussion is designed.

Guide the students to look at the pictures on page 16 of the textbook and answer the related questions:

Arrange from low to high 14 temperature;

The number 14 is represented by points on the number axis;

Observe and think: observe the positions of these points on the number axis and think about their relationship with temperature. Do you think two rational numbers can be compared?

How should I compare the sizes of two numbers?

After the students exchanged ideas, the teacher concluded:

14 The order of numbers from left to right is the order of temperature from low to high:

Rational numbers are represented on the number axis, and the order from left to right is from small to large, that is, the number on the left is smaller than the number on the right.

In the above 14 number, select two numbers to compare, and then select two numbers to try. By comparison, we can sum up the comparison rules of rational numbers.

Imagination exercise: imagine that there is a number axis in your mind, and there are two points on the axis, which represent the numbers-100 and -90 respectively. Realize the distance between these two points and the origin (that is, their absolute values) and the relationship between the sizes of these two numbers.

Students are required to have clear graphics in their minds, so that students can realize that all the laws of mathematics come from life and each law has its rationality.

The number in the second point of the size comparison method is difficult for students to master. It is necessary to combine the meaning of absolute value with the number on the number axis, configure imagination exercises, and strengthen the imagination of logarithm and shape.

Classroom exercise example 2, compare the following figures (textbook page 65438 +07)

The process of comparing sizes should be carried out in strict accordance with the rules and pay attention to the writing format.

Exercise:/kloc-exercise on page 0/8

Summary and homework

How to find the absolute value of a number and how to compare the sizes of rational numbers?

The assignment for this lesson is 1, and the required questions are: teaching production book 19 page exercise 1, 2, 4, 5, 6, 10.

2, choose to do the problem: the teacher arranges it himself.

Comments on this lesson (classroom design concept, actual teaching effect and improvement ideas)

1, the reasons for scenario creation are as follows: ① It reflects the close connection between mathematical knowledge and real life, so that students can learn.

Obtaining mathematical experience in these familiar daily life situations not only deepens the understanding of absolute value, but also feels learning.

The necessity of learning the concept of absolute value and stimulating learning interest. ② The concept of absolute value of numbers in textbooks is based on geometric meaning.

Meaning to define (its essence is to interpret numbers as forms, which is a difficult point), and then get it through practice and seek rationality.

The law of absolute value of numbers, if the concept of absolute value is given directly, the taste of instilling knowledge is very strong and too abstract.

It is not easy for students to accept.

2. The law of absolute value of a number is actually a direct application of the concept of absolute value, and it also embodies the mathematical idea of classification, so it is very concise and is the focus of teaching directly through examples 1; From the perspective of knowledge development and students' ability training, teachers should pay more attention to the process of students' autonomous learning and inquiry, pay attention to students' thinking, do a good job in teaching organization and guidance, and leave enough space for students.

3. The comparison rule of rational number size is a direct induction of the law of size, among which item (2) is difficult for students to understand and teach.

The meaning and stipulation of absolute value should be combined: "Rational numbers are expressed on the number axis, and the order from left to right is from small to large."

"Grand order" helps students to build a model combining numbers and shapes, that is, "the farther the point to the left on the number axis is from the origin, the smaller the number represented". So I set up imagination exercises.

4. The content of this lesson includes the concept of absolute value, the solution of absolute value of numbers, the comparison law of rational numbers, and teaching.

There is a lot of learning content, which may be difficult for students to accept. It is suggested that the comparison of rational numbers be moved to the next class.