teaching process
First of all, introduce the old into the new and ask questions.
1. Review the questions
Teacher: Last class, we learned how to solve a relatively simple linear equation. First, we did two exercises: (Hang a small blackboard to show the questions. )
( 1) 。 (2) 。
These two questions are based on the problems existing in students' homework. Students practice and teachers patrol. It is found that students can do (1) problems, but a few students lack the process of solving problems. (2) There are two answers to the question because of different solutions. )
Teacher: The two answers to question (2) are correct. Who will talk about the main steps and basis for solving the second problem?
Choose a student to answer each of the two solutions, and the teacher writes on the blackboard according to their answers. )
Raw armor:
Divide both sides of the shift term by 0.5.
0.5x+ 1 = 0.2 0.5x =-0.8x =-8/5
Homosolution principle 1 homosolution principle 2
Multiply both sides by two.
Health B: The first step is the same as that of health A, and the second step is x =- 1.6.
Principle of same solution II
Teacher: Please talk about the way to explain this kind of equation.
The teacher writes on the blackboard according to the students' answers. )
The simplest equation in a linear equation.
Let the coefficient of x be 1.
ask questions
Teacher: The equations encountered in practical problems are not all that simple, such as (showing a small blackboard).
(1) Solve the equation: 5x+2 = 7x–8;
(2) Solve the equation: 2 (2x-2)-3 (4x-1) = 9 (1-x);
(3) Solve the equation: (5y- 1)/6 = 7/3.
(These three questions are case 4, case 5 and case 6 of the textbook. )
Teacher: How to solve these complicated linear equations? This is what we are going to learn today. Can we also use the thinking method of "transformation" to solve it? Please have a try first.
Many methods have been introduced into this subject. Here, I introduce a new lesson by asking questions. After reviewing the topic, I immediately put forward the question: Can the complicated linear equation of one variable be solved by transformation? This question can arouse students' suspense and stimulate their thirst for knowledge. Only when students have the requirements to solve problems can they arouse the enthusiasm of their thinking. In fact, as soon as this question is raised, students' attention is directed to the correct thinking direction, thus making their thinking quickly oriented. ]
Second, observe and think and seek solutions.
Students discuss in groups to find solutions. Teachers participate in the discussion and give inspiration and guidance according to the situation. )
Teacher: To solve these complex linear equations, we can think like this:
(1) As long as these equations can be simplified to the simplest equations, we can find solutions. Think about how to simplify the equation to the goal of the simplest equation.
(2) When thinking, you can compare these equations with the simplest equations, think about where these equations are mainly "complex", and then consider what methods to simplify them.
Cultivating observation is the premise and foundation of developing thinking. We should seize every opportunity to cultivate students' observation ability in teaching. I think students should be guided to observe what and how to observe mathematical models (figures or formulas). Here, I ask students to observe the difference between these three equations and the simplest equation, lead students' unintentional observation to purposeful observation, guide students to observe and think at the same time, and find out how to turn these complicated equations into the simplest equations. ]
Students solve problems, teachers patrol and answer them one by one with students. When most of the students were about to finish the third question, the teacher asked questions. )
Teacher: How did you solve problem (3)? I asked three students to come to the blackboard to solve the problem.
Teacher: Which of the three students has the simpler solution?
Some students think that the solution of B is relatively simple, while others think that the solution of C is relatively simple. The two opinions are tit for tat. )
Teacher: Let's look at the solution of B. "Cross multiplication" is actually how many times are multiplied on both sides of the equation? (Students echo in unison: C,) Yes! C is multiplied by the least common multiple of the denominator in the equations on both sides, so that all the coefficients of the equation are converted into integers. This "naming" method is good!
The teacher drew a "√" in red before student C's answer and added the word "denominator". )
In the future, when we solve a linear equation with a denominator, we usually "remove the denominator" first.
[Problems (1) and (2) can be solved by students themselves, but they can also be touched simply. The emphasis is on the evaluation of several different solutions to problem (3). I encourage students to express different opinions and discuss, and follow the thinking of most students, so that they can quickly believe and accept the method of "naming", which is better than injecting students with "naming". In the latter, students' intellectual potential and non-intellectual factors are in a static state, and they have not invested in the teaching process to play a role; The former pays full attention to students' main role, and through the encouragement of teachers, students' intellectual factors and non-intellectual factors change from static to dynamic, which is conducive to stimulating the internal driving force of learning, thus fully mobilizing students' learning enthusiasm. ]
Third, summarize and organize knowledge.
Teacher: What method did we use to solve the above equation? What are the bases of these methods?
According to the students' answers, the teacher fills in the correct conclusions in the table drawn in advance below. Three lines (1), (3) and (5) in the table are written in red chalk. )
Teacher: Who will tell us what the purpose of the above steps is?
(Students answer, the teacher repeats, and uncover the purpose column in the table below. )
Methods according to the purpose
(1) denominator sharing principle 2 converts all coefficients into integers.
(2) Non-load-bearing and non-load-bearing regular multiplication and distribution laws are beneficial to shifting terms and merging similar terms.
(3) The principle of moving the term to the same solution 1 makes the unknown term and the known term concentrate separately.
(4) The rule of merging similar items reduces the items containing unknowns.
(5) The coefficient of dividing two sides by the unknown is the same as that of solving principle 2, and the coefficient of the unknown is converted into 1.
Teacher: Here, we can clearly see that no matter how complicated a linear equation with one variable is, it can be transformed into a "simplest equation" by using the same solution principle of the equation [pointing out the (1) column in the table], and then divide the two sides of the equation by unknown coefficients to find the solution, which is the general step to solve the linear equation with one variable. (blackboard writing)
( 1)-(4)
The simplest equation of "more complex one-dimensional linear equation"
(5)
The processing method of this part of the textbook usually focuses on teaching students to master the steps of solving, but I firmly grasp the principle of the same solution, so that students can deform the equation accordingly and achieve the purpose of gradually forming the equation. In this way, the solution steps become the natural induction of this process. This lesson focuses on guiding students to explore the ideas of solution, which is conducive to guiding students to use their hands and brains to sort out their own ideas. Students can not only master the basic knowledge comprehensively, but also cultivate the organization of students' thinking through logical analysis, synthesis, classification and induction. ]
Fourth, practice oral English and two-way feedback.
1. Consolidate exercises and comments
Exercise 1 Solve the following equation:
[Question (2) is Example 7 of the textbook. When students remove the denominator, it is easy to ignore the constant 1 and multiply it by the common denominator. So the question (1) was compiled. ]
Student internship, teacher patrol, individual counseling. When most students begin to do problem (2), ask a classmate to act out problem (2). )
Student: If you remove the denominator, you will get.
……
Teacher: (pointing to the question) Here 1 should be multiplied by 12, (pointing to the students' blackboard solution) Oh! He didn't forget to take the bus. Ok, let's see if he got it right.
Health: No, remove the denominator and put the numerator in brackets.
Teacher: (pointing to the students performing on the blackboard, smiling) Did you omit the steps wrong? X X is right (in front of everyone). We should pay attention to prevent such mistakes, and don't omit the steps when we are beginners. Here, if you add the step "",it is not easy to make mistakes. Pay attention to the general steps (referring to the table) of solving a linear equation with one variable.
2. Thinking exercises, discussions and comments
Exercise 2 Solve the following equation:
( 1)2(x- 1)+5(x- 1)= 1-8(x- 1)；
(2)5(y- 1)-3( 1-y)+7( 1-y)= 100
(There are generally two different solutions to solve this kind of equation: the software center of Wangma Computer Company solves it according to "general steps"; The other is to solve an algebraic expression, such as the one in the problem, as the unknowns y and x respectively. The teacher consciously chooses two students with different solutions to solve the problem on the blackboard (2). )
Teacher: (pointing to two students) Did they get it right? (Students echo in unison: Yes! Why is his process so simple? (Pointing at the students) What do you think?
Health: I regard it as an unknown number, which is represented by X, so it is-X.
Teacher: His solution is good and a little "creative"! Usually he is willing to use his head, so a good idea comes out. This method of treating Y and X as unknowns is called "method of substitution", and we will study it further in the future. As can be seen from the above, due to the different forms of the equation, we don't have to follow the "general steps" or use every step when solving the equation. Observe carefully, be good at grasping the characteristics of the equation and choose a reasonable solution.
3. Summary
Teacher: What knowledge and methods did we mainly learn in this class? Who will sum it up?
Student A: In this class, we learn the general steps to understand the linear equation of one yuan.
Student B: I also learned the basics of each step.
Student C: I also learned the substitution method.
Teacher: The students summed it up very well. This is what we should do. Learn to sum up and sort out. In this lesson, we will explore and study the solutions of complex linear equations with one variable. The basic idea is to "transform" (the word "blackboard writing", the same below, refers to the original blackboard writing. The goal of transformation is "the simplest equation", and the basis of transformation is mainly two "same solution principles" of angular equation. The general steps of transformation have been summarized in the textbook. Ask the students to open their textbooks and read the relevant bold words silently. However, this is not absolute. We should be good at observation, think seriously, "adapt to the topic", pay attention to the "art" of transformation, try our best to use reasonable methods to be correct and rapid.
It is very important to form "two-way feedback" between teachers and students and between classmates in practice, evaluation and summary. For example, the mistakes found in students' exercises can be remedied in time to achieve the goal of "taking the old with the new, checking for leaks and filling gaps". In this way, the teacher's guidance has obvious pertinence. For another example, students only list the knowledge and methods they have learned in the summary, so I will supplement the summary from the general way of thinking, so that students can be influenced by the way of thinking. ]
Fifth, teach students in accordance with their aptitude and develop their personality.
1. Design
Choose three questions in the textbook, choose one question, and then add the following two questions.
Solve the equation:
( 1)7(2x+ 1)-3(4x-2)-5(x+ 1/2)= 1；
(2) 1/3[2(9x+2)- ]= 1。
Step 2 think about the problem
Find twenty-five consecutive integers; Make the sum of the first fifteen integers equal to the sum of the last ten integers.
Leave five minutes for spare students to think and answer the above questions, while the rest do their homework. In the teaching of connecting primary and secondary schools, students have been exposed to the problem of finding the sum of continuous natural numbers, so there are always some students in the software center of Wangma Computer Company who use the following methods to find the answer, and only publish the answer before class, so that students who are interested but have not found the answer can do it after class. )
Let the smallest integer be x.
X+(x+ 1)+(x+2)+……。 +(x+ 14)
=(x+ 15)+(x+ 16)+……。 (x+24)，
15x+( 1+2+3+……+ 14)= 10x+( 15+ 16+……+24)。
15x+ 105 = 10 x+ 195，
The solution is x= 18.
A: These 25 consecutive integers are 18, 19, 20, ... 42.
Because there are some differences in students' thinking quality, in order to make students with poor foundation "eat enough" and students with good foundation "eat enough", the principle of "teaching students in accordance with their aptitude" should be implemented in teaching, so I leave room for arranging teaching, including homework, to make teaching flexible and easy to choose. For students with poor foundation, pay attention to the basic training of thinking, pay attention to the cultivation of study habits, and appropriately reduce the difficulty to improve the confidence of learning mathematics well; For students with a good foundation, it is necessary to increase the difficulty appropriately, and sometimes broaden their knowledge appropriately, so that students have room for thinking and improve their interest in learning mathematics. For the above topics, I usually make up several topics for students who have spare capacity and interest in learning when preparing lessons. Over time, it has a positive effect on training and cultivating their thinking quality and developing students' personality. ]
self-criticism
I think the process of students is actually the process of their constant thinking, and teachers should be "directors" rather than "actors" in this thinking process. Therefore, the design of this teaching plan aims at cultivating and training students' thinking quality, and adopts "teachers first, then talk; Students want to be in front and listen to the back. Its main features are:
(1) "Design the whole chapter and dredge it by sections", that is, pay attention to the overall structure of knowledge, design the best learning process for students according to their understanding level, and give necessary help when students are confused about the implementation by sections. On the basis of the first stage (five lessons), let the students understand and master the principle of simultaneous solution of equations and the linear equation of one yuan (textbook example 1- example 3). This lesson focuses on guiding the students to explore the basic ideas of solving the linear equation of one yuan by using the principle of simultaneous solution of equations, and then arrange three lessons for consolidation and digestion to help them form certain problem-solving skills. This arrangement, on the one hand, makes the whole unit coherent, compact and focused, on the other hand, it conforms to the students' cognitive laws, with a strong sense of rhythm and dialectical unity of "fast" and "slow". It is precisely because of the slowness of the first five classes (it takes more time to teach the same solution principle and solution method of a linear equation synchronously, so that students can get into contact with the thinking method of "transformation" initially), and the "quickness" of this class has a foundation. After students have mastered the basic idea of understanding the linear equation of one yuan, it is a stage of "exquisite carving". In two or three classes, students can "step by step" through active thinking (different from the customary "step by step").
(2) "Seeing, thinking and remembering = asking, discussing, practicing, checking and concluding" is the main activity form of students' positive thinking in the teaching process. The so-called "optimal combination" means that teachers choose several forms of activities according to the teaching objectives of a class and their position in unit learning. For example, this class focuses on "discussion", "practice" and "summary", aiming at the weak links of students' knowledge, grasping their different views on a certain issue and organizing them to discuss. For example, the introduction of "denominator" in this course will sometimes make some students hit a nail and suffer a little, and then organize them to exchange "hit a nail" experience. Through this kind of information exchange between students, we can learn from each other's strengths and make each student's eyes, ears, mouth, hands and brain move together. When students get information and ideas from other students or teachers, they can make their thinking smooth or stimulate their enthusiasm for further thinking. Doing so is conducive to mobilizing the thinking initiative of students at all levels in the class to the maximum extent and effectively ensuring students' dominant position in bilateral teaching activities.
(3) "Guiding, dredging, teaching and motivating" is the main way for teachers to give full play to their leading role in the teaching process. For example, at the beginning of this class, the teacher asks questions to guide students to observe the difference between a given equation and the "simplest equation" and explore the basic idea of solving it; When students make mistakes in "naming" and it is difficult to summarize, the teacher helps solve the problem and dredge it; When a small number of students are found not to follow the "general steps", the teacher will affirm them in time and give some guidance on the way of thinking, so as to broaden the students' thinking and stimulate their enthusiasm for seeking reasonable solutions. In short, in the bilateral activities of teaching, as long as teachers judge the quality of students' feedback accurately and deal with it promptly and decisively, the teaching effect will be better. I hope to ignite the spark of students' thinking through the teacher's "introduction"; Through the teacher's "sparse", students' thinking is smooth; Through the teacher's "point", students' thinking will step into a new height; Through the "stimulation" of teachers, the enthusiasm of students' thinking is mobilized, so that students' thinking is in the best state, and teachers really become the "directors" of students' thinking process. At the same time, teachers should also be "agitators" of students' positive thinking. Teachers should grasp the characteristics of students' psychological changes, and "introduction" is the key, "sparseness" is the need, "point" is the key, and "excitement" is the heart. In the dialogue and communication between teachers and students in classroom teaching, we should not only exchange thinking methods, but also attach importance to the exchange of thoughts and feelings. For example, in this class, a student skillfully solved the problem, but he couldn't explain it clearly, so I praised him and made him feel encouraged. It can be said that the success of a class, in a sense, is the success of teacher-student communication and the result of the simultaneous development of students' "intellectual factors" and "non-intellectual factors".