Chapter II Rational Numbers and Their Operations

First, the teaching objectives:

The main content of this chapter is the related concepts and operations of rational numbers. Starting from examples, the textbook introduces negative numbers from actual needs, and then leads to some concepts and classifications of rational numbers. On this basis, learn the addition, subtraction, multiplication and division of rational numbers in turn, and do complex rational number operations with a calculator.

The teaching objectives of this chapter are:

1. Understand the significance of rational numbers and their operations in specific situations; Go through the process of exploring rational number algorithm and operation law.

2. Rational numbers can be represented by points on the number axis, and the sizes of rational numbers will be compared.

3. Understand the meaning of reciprocal and absolute value with the help of number axis, and find reciprocal and absolute value of rational number.

4. Master addition, subtraction, multiplication and division of rational numbers and simple mixed operations; Understand the operation law of rational numbers, and can simplify the operation by using the operation law.

5. Be able to use rational numbers and their operations to solve simple practical problems.

6, can correctly use the calculator for more complex rational number operation.

Second, the characteristics of this chapter:

This chapter focuses on students' independent exploration. Through some familiar and concrete things, let students understand the significance of rational number in observation, thinking and exploration, explore the quantitative relationship, master the operation of rational number, and pay attention to let students acquire, understand and master this knowledge through their own activities. Compared with traditional textbooks, it reduces the requirements for calculation and deletes complicated calculations. Calculators are used for more complicated calculations.

Main features:

1, each section provides a large number of life examples, and creates problem situations by being familiar with specific things, so that students can actively think, observe, discuss, explore, discover and appreciate the meaning of rational numbers with questions, and pay attention to the formation and formation process of knowledge. Such as goal difference, temperature change and so on. Each link provides a large number of materials from real life, creates an attractive learning background, highlights the process of "mathematicization", and provides students with time and space for exploration and communication.

2. Each class provides students with rich mathematical activities, such as thinking, discussing, reading, doing, guessing, trying, summarizing, guessing, reasoning, calculating and communicating. It embodies the fundamental change of mathematical methods, thus truly realizing the student-oriented and teachers' role of guidance, cooperation and organization.

3. The textbook in this chapter embodies many mathematical ideas, such as the combination of numbers and shapes, classified discussion, transformation, transformation and operation. Pay attention to the application of mathematics and cultivate a preliminary sense of number (such as the comparison between the number axis and rational number, subtraction, division and transformation, etc.). ).

4. Pay attention to reducing the requirements for operation, pay attention to students' understanding of the meaning of operation, master the necessary basic operation skills, reduce complex written calculations, and use calculators for complex operations. (such as the integer before the fraction, focusing on the substance and dilution form. )

Third, teaching suggestions

1. When teaching the concept of rational numbers, we should introduce practical problems, choose things that students are familiar with and help them understand the concept.

2. In the teaching of rational number operation, we should guide students to understand the significance of operation in specific situations, encourage students to explore operation rules and laws, gradually form a more standardized language, advocate the diversification of algorithms, reduce complex written calculations, and do not pursue operation skills excessively. Calculators should be used for complex calculations.

3. The number axis is an important tool to understand the concept and operation of rational numbers. We should make good use of this tool in teaching, especially let students learn and understand with the help of the number axis.

4. Attention should be paid to the teaching of using rational numbers and their operations to solve practical problems, so that students can use positive and negative numbers to represent quantities in practical problems, which can reasonably explain the operation results and give practical significance.

Section 1: Why are there not enough people?

First of all, the textbook sets up a very interesting realistic scene, and introduces negative numbers in order to express quantities with opposite meanings. However, we should pay attention to make good use of the theme map in this chapter, and briefly review the development history of numbers through the theme map, so that students can understand that numbers are constantly developing with the development of society. Although primary school figures can solve many practical problems, are they enough? Then show the scene of this class, which leads to the problem of insufficient number of people. How to use new numbers to express, let students think, explore, communicate and express their views with questions, and affirm what students express with words, colors and graphics. As long as it is reasonable and points out that it has certain limitations, it is simple and clear and universal to express it with specific mathematical symbols. Finally, you can ask which team scored the highest, and the difference between the highest and the lowest is a few points, paving the way for the comparison and operation of rational numbers.

P33。 "Discussion": Let students feel the application of positive and negative numbers in real life. Let the students give several examples and try to express them with positive and negative numbers, so that they can tell the students that people often specify the temperature above zero, the rising height, traveling eastward, income, transportation, counterclockwise and so on. Is positive, and vice versa.

P34。 The benchmark in the example 1 is abstract for beginners. Here, it is suggested to briefly discuss the representable meaning of zero after number expansion (if zero can indicate no, it can also indicate serial number, which is often used as a benchmark), and some benchmarks are not zero.

P34。 Do One Thing classifies rational numbers. When students try to classify, their thinking is quite active. However, most students are disorganized, and repetition or omission are common, such as odd number, even number, composite number, even number and even number. Teachers should never rush into things. They should affirm its reasonable part and point out its unreasonable part. We can make up an example to guide students to review how numbers learned in primary school are classified, and then supplement them according to the symbolic characteristics, encourage students to express them in their own languages, and gradually form a system. Pay attention to time control.

Part 2: Number Axis

The starting point of the combination of numbers and shapes plays an important role in the comparison of opposites, absolute values and rational numbers and the formation of addition algorithms.

The first thermometer to be used in this class is the best model of the number axis, which is intuitive and sharp in contrast and easy for students to accept. By comparing thermometers, the three elements of the number axis and the drawing method of the number axis are explained, but students are not required to memorize the drawing method of the number axis.

P37。 After learning Example1-Example 2, let the students practice and experience the mutual transformation process of numbers and shapes. Add the example 1 to the example and tell example 2.

P38, "Think about it", let students get the concept of reciprocal by observing special cases, and let students express it on the number axis, and understand reciprocal from two aspects: number and shape.

P38。 Example 3: To compare the sizes of two rational numbers, we should advocate the diversification of comparison methods. You can apply the comparison method and the number axis comparison method, and you can also quote the comparison of three numbers appropriately, and pay attention to the inequality of directions.

P39. In-class exercises 2. Thinking is the reverse thinking process of P38. We can compare the mistakes made by students through the number axis or "thinking", so that students can find the reasons for their mistakes and prepare for the study of absolute value.

Section 3: Absolute Value

The concept of absolute value is the most abstract concept in this chapter, and it is also the most difficult concept for students to understand. Need a preliminary understanding, and the requirements should not be too high. There is no need to go deep, and there is no need to introduce what is not mentioned in the textbook. In teaching, we can first review the concept of reciprocal and the position relationship of reciprocal on the number axis, and give the concept of absolute value intuitively by using the number axis, so that students can feel that absolute value is a numerical representation, regardless of direction significance.

P4 1, "think about it" and "discuss it" should be organically combined, and several discussion points should be appropriately added, such as what number is equal to 2 in absolute value? What is the relationship between two numbers with equal absolute values? What is the absolute value of a number equal to itself? What is the absolute value of a number equal to its reciprocal? What is the absolute value of a number and so on. Let the students draw their own conclusions through repeated comparisons.

P42, "Do one thing", let students complete it step by step, communicate fully, show the results of each group discussion, compare and summarize, and get the law of comparing two negative numbers.

P42, example 2, advocates that the methods of comparing two negative numbers are diversified, which can be based on law or number axis. However, it is necessary to use the absolute value rule to compare the sizes of two negative numbers. However, quite a few students are confused about the writing format. For students who have difficulties, try to express them orally. The writing format should not be uniform, but can be compared.

P43, Exercise 3, use absolute value to explain which ball has better quality. Some students have difficulties, so we should guide them to compare which football is closest to the standard quality, that is, the error is the smallest, and then explain it with absolute value, so that students can understand the meaning and function of absolute value.

Completes P43 "Try" completed in class, permeates letters to indicate the meaning of numbers, and prepares for algebra study. Not all students are required to master it. For students with difficulties, we should try our best to reduce the difficulty and concretize abstract problems. For example, 1 question, suppose a=4, 0, -4, discuss it first, and then let the students try to find a few numbers. Finally,

The teaching of rational number concept should be slow rather than fast. Editing textbooks takes 6 class hours, while editing new textbooks only takes 3 class hours, which greatly increases the classroom capacity and makes the whole process feel too fast. It is suggested to add a class hour to sort out the knowledge learned, so as to make it more organized and systematic and make full preparations for the follow-up study.

Section 4 Addition of Rational Numbers

Rational number addition is introduced, and the textbook adopts symbolic method and exploration algorithm, combined with number axis for verification. The teacher of symbolic method is controversial, and students are advised to teach themselves or delete them. If 1 means+1 and 1 means-1, then or means 0. When exploring the result of (-2)+(-3), in the textbook, one group of two and the other group of three all got five together, so (-2)+ (. It is equivalent to (-1)+(-1)+(-1)+(-1) =-5 in the law of addition, which is the goal of this lesson. It is suggested to create another teaching situation and introduce teaching. When comparing and summarizing the addition rules, it is better to add another formula, such as (+2)+(+3)=+5, which reflects two possible situations when the numbers are the same. In teaching practice, it is difficult for students to describe the rules in more accurate language at first. The key is not to step on the idea of absolute value. At this time, you can review the concept of absolute value first, such as observing +4 and -4, -3 and. Encourage students to express themselves in their own language. Take the exercise as the dividing line and the algorithm as the next teaching time.

It is not difficult for students to get the operation law of addition through "discussion" and "thinking", but don't ignore the process of trying several numbers after drawing a conclusion, so that students can understand that verification is an important link in reasoning and reflection.

P49, Example 2, several similar exercises can be appropriately added to make students realize that using the algorithm can simplify the operation.

P49, Example 3, students have many different algorithms besides those in books, such as the combination of multiplication and addition, and the benchmark number is 450. Let the students show their own practices, talk about their own thinking process, communicate with each other and compare with each other.

P5 1, Exercise 4, Question 2: How many meters did Xiaoming run? I didn't expect many students to be at a loss, but I still thought how far Xiao Ming was from A as how many meters Xiao Ming ran. The main thing is to confuse the position with the itinerary. Position is a directional quantity; The journey is an absolute value, which has nothing to do with the direction, but this explanation is not helpful. Then the problem can be simplified. For example, if an object starts from A and moves eastward 10 meter to B, it moves westward from B 10 meter. Where is the object at this moment and how many meters has it moved back and forth? Let the students understand for themselves.

Section 5: subtraction of rational numbers

First, make good use of teaching materials for example, so that students can work out the method of 3-(-3) by themselves. Both inverse operation and thermometer should be encouraged to guide students to observe the two formulas and their results. Lay the foundation for the law of subtraction. The law of rational number subtraction is not difficult for students to get by comparing two sets of formulas. The key is to cultivate students' abstract generalization ability and oral expression ability in this process. The plus sign of a positive number is omitted in the operation (the concept of algebraic sum of plus signs is not omitted)

P55, Exercise 5, for students with difficulties, guide them to compare the known data in the table to be filled with the corresponding positions in the sample table to find out the rules.

P55, the trial filling method is not unique, but 5 should be in the middle, so that students can exchange solutions.

Section 6: Mixed operation of rational number addition and subtraction (2 class hours)

1. By citing two algorithms, namely, the conversion from subtraction to addition and practical algorithm, we can further understand the law of subtraction of rational numbers and prepare for algebraic sum.

2. After unifying subtraction into addition in Yi Yi, let students read the formula of plus sign in the algebra and method before operation, reflecting algebra and meaning, and don't mention algebra and name.

3. Fractions and decimals first appeared in the formula. You can add some exercises appropriately, but don't increase the difficulty. The key is to be clear.

4. In the process of operation, advocate the diversification of algorithms, and pay attention to guiding students to simplify the operation by using the law of addition operation.

5, P59, "hands-on", the teacher should arrange for students to make cards before class, judge the winners of each group in group activities, and stimulate students' enthusiasm for learning.

6.P59。 Exercise 2 is more open. As for the overall change of water level in a week, the students' answers are varied. Some answers are ups and downs, some answers are first down and then up, some answers are which day the water level is the highest, which day the water level is the lowest, or how much the difference between the highest water level and the lowest water level is. As long as it is reasonable, students should be encouraged to estimate first and then calculate accurately.

Section 7: Water Level Change

The teaching goal is to comprehensively apply the knowledge of rational numbers and their addition and subtraction operations, solve simple practical problems and experience the connection between mathematics and real life. This section is closely related to statistics and functions.

The example in this section is a long question for junior one students, with many questions. Students are unfamiliar with technical terms and have high comprehensive application ability of knowledge. It is suggested that students should be organized to complete classroom exercises first. After introducing the example of water level, ask the students to estimate the overall change of water level, draw a picture before answering the question 1, and add a water record to the table before drawing. The change of water level is cumulative, which is noticed when drawing. After students have accumulated some experience, it is suggested that teachers and students complete the solution of examples and communicate with each other.

"In-class exercise" can directly show the average height and table first, let students guess who is the tallest and who is the shortest, and encourage comparison by height or the difference with the average height.

Exercise 2, guide students to pay attention to the meaning of the data in the table, and also ask students to estimate which day the systolic blood pressure is the highest and which day is the lowest, and whether the systolic blood pressure is rising or falling this Friday compared with last weekend. And guide students to take 160 as the benchmark when making statistical charts.

In the teaching process, students should be consciously guided to compare the data in the two exercises. What are the differences in their handling methods? Their benchmark is 160. The data in the first few tables are relatively independent, and the data in each table is compared with 160. The following table data and the previous data are interdependent and compared with each other, which is beneficial to the ability of collecting, sorting and processing data in the future statistical study.

Section 8: rational number multiplication

Rational number multiplication is the continuation of primary school multiplication. The symbol processing method used in this textbook is easy for students to learn and master, which reduces the difficulty of understanding.

1, the derivation of the multiplication rule should pay attention to guiding students to carefully observe the changing law of the factor and product of the formula "111", find out the law, then guess the result of the formula, and finally classify, summarize and communicate the two formulas to get the multiplication formula.

2. The determination of the product sign of multiple rational numbers and the exploration of the operation law. It is not difficult for students to find out through examples that this process is from special to general. After drawing a conclusion, teachers should ask students to find more examples to verify and cultivate their abilities of observation, induction, guessing and verification.

3. The concept of reciprocal is obtained under special circumstances, and there is no need to deduce the method of reciprocal.

4. The laws of symbols can be obtained through "discussion", which can be consolidated through practice.

Section 9: Division of Rational Numbers

As the inverse operation of multiplication, the division of rational numbers has the same meaning as the division of positive numbers in primary school. Textbooks embody the principle of continuity of knowledge system, which is helpful to quickly integrate new knowledge into the structure of old knowledge. There are two key points:

1, meaning division.

2. Division becomes multiplication.

Pay attention to the reciprocal of negative numbers. 0 cannot be divided in division.

Section 10: Power of Rational Numbers

Compared with the old textbooks, the requirements have been greatly improved, and many situations have been added to explore the law and develop the sense of numbers.

1, citing examples to observe the scene of cell division, let students "draw a picture" to explore the law, list the formulas, and pay attention to students' understanding of the meaning of power.

2. Pay attention to the writing format of fractions, negative numbers and powers, and don't omit brackets. Notes -2 and (-2),

The difference between the two, what is their basis, what is the reduction formula, and the different reading methods. Pay attention to understanding the meaning of power.

3. Example 2 can also supplement the situation when the cardinality is 0. 1, and explore the law as broadly as possible. In addition to the power sign law, it can also prepare for the future scientific notation.

4. Cell division and origami in this section have a lot in common with P74, Exercise 3, P76, Exercise 2 and "Reading", and the operation is simple, so that students can explore the law on the basis of operations such as "drawing a picture" and "folding a picture". Suggestion: Show the number of cell division in a list, so that students can understand the general expression. If we can use multimedia, the effect will be better. It is necessary to organize students to compare and reflect, and pay attention to guiding students to feel that when the radix is greater than 1, the power result will increase rapidly, and when the radix is less than 1, the power result will decrease rapidly.

"Reading" and learning on the chessboard are interesting mathematical allusions. In addition to direct reasoning, you can also compare the number of cells in cell division with the total number of scratches in origami questions, and list the chessboard as 1- 10, which makes students feel that the number is very large, 1+2+8+65438.

Section 1 1: Mixed Operation of Rational Numbers

It is suggested that "doing one thing" be listed as two classes and one class.

The operation of rational numbers is the sum of the operations in the whole chapter. Students are prone to confusion and mistakes, so it takes time to integrate. Add some exercises appropriately, but the topics should not be too complicated and difficult. Based on the difficulty of the textbook (mainly divided into three steps). Attention should be paid to cultivating students' orderly thinking in the process of operation, rationally using the algorithm to simplify the operation and form basic skills. Such as -24+32 ÷ (-6) and so on.

"Hands-on" is a good way to train your thinking by playing 2 1 point. It is very demanding for students to try the corresponding calculation methods and programs according to the extracted digital features. Students' oral expression is often good, but there are many mistakes in writing format, so it is impossible to list the correct formula. The fundamental reason is that it is difficult to determine the order of operation with brackets, because this problem basically does not exist in the operation of this chapter, and it is necessary to strengthen guidance.

Section 12: Calculator

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