Junior high school mathematics teaching plan: comparison of rational numbers I. background knowledge
The comparison of rational numbers is selected from the fifth section of the first chapter of "From Natural Numbers to Rational Numbers" in the seventh grade of mathematics (Volume I), the standard experimental textbook of compulsory education curriculum in Zhejiang Edition. The comparison of rational numbers is put forward from the familiar situation in students' life, and the comparison method of rational numbers is obtained by means of temperature and number axis. The textbook arranges a variety of teaching activities such as "doing one thing", so that students can experience the exploration process of rational numbers through observation, thinking and hands-on operation.
Second, the teaching objectives
1, let the students say the comparison rules of rational numbers.
2. I can skillfully use the law to compare the size of rational numbers with the number axis, especially the concept of absolute value to compare the size of two negative numbers. I can arrange multiple rational numbers in order with the number axis.
3, can use symbols correctly "
Third, the focus and difficulty of teaching
Key point: Compare the size of two rational numbers by using the law of number axis utilization.
Difficulty: Compare the size of two negative scores with the concept of absolute value.
Fourth, teaching preparation.
multimedia courseware
Teaching design of verb (abbreviation of verb)
(a) Exchange dialogue and explore new knowledge
1, say it
(Multimedia presentation) What information did you get from the pictures of the lowest temperature on a certain day in our five cities? (Starting with the common temperatures, to stimulate students' curiosity, some students may say that the lowest temperature in Guangzhou is higher than that in Shanghai 10℃, and some students may say that the lowest temperature in Harbin is MINUS 20℃ lower than that in Beijing. If you can't say it, the teacher will pull it out properly and let the students fill in the blanks unconsciously in cooperation and communication.
Compare the lowest temperature in the following two cities on this day (fill in "high" or "low")
Guangzhou _ _ _ _ _ Shanghai; Beijing _ _ _ _ _ _ _ Shanghai; _ _ _ _ _ _ Harbin, Beijing; Wuhan _ _ _ _ _ _ _ Harbin; Wuhan _ _ _ _ _ _ _ Guangzhou.
2. Draw a picture: (1) shows the lowest temperature in the above five cities on the axis; (2) Observe the positions of these five numbers on the number axis. What did you find?
(3) What is the position of the temperature and the corresponding number on the number axis?
Through students' hands-on operation, observation and thinking, it is found that the numbers on the left of the origin are all negative, and the numbers on the right of the origin are all positive; At the same time, it is also found that 5 is on the right of 0, and 5 is greater than 0; 10 is to the right of 5, and 10 is greater than 5. I feel that the two numbers on the right side of the origin on the number axis, the number on the right side is always greater than the number on the left. The teacher took the opportunity to ask, does the number on the left side of the origin have such a law? This stimulates students' desire to explore knowledge, and further verifies that the number on the left side of the origin also has such a law. Let students experience the fun of exploration and acquire knowledge unconsciously in the process of exploration. ) After group discussion, the teacher concluded:
Of the two numbers displayed on the axis, the number on the right is always greater than the number on the left.
Positive numbers are all greater than zero, negative numbers are all less than zero, and positive numbers are greater than negative numbers.
(B) the application of new knowledge, experience success
1, practice (teachers and students * * * cooperate to complete the example 1, students complete the exercises in class 1)
Example 1: indicates the numbers 5, 0, -4,-1 on the number axis, compares their sizes, and uses "
Analysis: How many meanings does this question have? How many steps should we take?
Summary of main points: group discussion and induction, the general steps of solving problems: ① drawing number axis ② tracing points; ③ Orderly arrangement; 4 Unequal connection.
Classroom exercise: P 19 T 1
Step 2 do this
(1) Represents the following logarithms on the number axis and compares their sizes.
①2 and 7 ②-6 and-1 ③-6 and -36 ④- and-1.5.
(2) Find the absolute value of the logarithm in the graph and compare its magnitude.
(3) What do you find from ① and ②?
(After the students discuss in groups, the delegates stand up to speak, dictate the findings of this group, explain the discovery process of this group, and gradually cultivate students' ability to observe, summarize and express mathematical laws in mathematical language. )
Summary of main points: two positive numbers are larger, and the number with larger absolute value is larger; When two negative numbers are compared in size, the number with larger absolute value is smaller.
On the basis of students' discussion, students summarize the comparison rules of rational numbers.
(1) Positive numbers are all greater than zero, negative numbers are all less than zero, and positive numbers are greater than negative numbers.
(2) Comparing the sizes of two positive numbers, the number with larger absolute value is larger.
(3) When two negative numbers are compared in size, the number with larger absolute value is smaller.
3. Teachers and students * * * After completing Example 2, students complete classroom exercises 2, 3 and 4.
Example 2 Compare the following logarithms and explain the reasons: (Both teachers and students complete * * *)
(1) 1 and-10, (2)-0.00 1 and 0, (3)-8 and+2; (4)- and-; (5)-(+) and -0.8 |
Analysis: Questions (4) and (5) are more difficult, so questions (4) are divided first, and questions (5) are simplified first and then compared. At the same time, pay attention to the format when explaining.
Note: when comparing absolute values, the denominator is the same, and the larger the numerator, the larger the number; If the numerator is the same, the number with a large denominator will be smaller; When the numerator and denominator are different, we should divide before comparing, or compare after the numerator is the same.
When two negative numbers are relatively large, the general steps are as follows: ① Find the absolute value; ② compare absolute values; ③ Compare the size of negative numbers.
Thinking: Is there any other way? (Discuss in groups and think positively)
4. Think about it: How many ways do we judge the size of rational numbers? What do you think are their characteristics?
After students' discussion, two methods to compare the size of rational number * * * * are summarized, one is the rule, and the other is to use the number axis. When comparing two numbers, generally choose the first one. When multiple rational numbers are relatively large, generally choose the second one.
Exercise: p19t2,3,4
5. Test you: Please answer the following questions:
(1) Is there a maximum rational number and a minimum rational number? Why?
(2) Is there a rational number with the smallest absolute value? If so, please write it down?
(3) There are _ _ _ _ _-1.5 integers less than 4.2, which are _ _ _ _.
(4) If a>0, b<0 and a & lt|b|, can you compare the sizes of the four numbers A, B, -a and -b? (This topic is entitled Promotion, and it is not required for all students to master it.)
Novel questions will stimulate students' curiosity, and cultivate students' thinking habits and mathematical language expression ability through cooperative communication and independent inquiry. )
6. Have a discussion and talk about what you have gained in this class.
The summary of this lesson is completed by both teachers and students. ) In this lesson, we mainly learned two methods to compare rational numbers, one is to compare them according to the law, and the other is to use the number axis. When using this method, we should first express the numbers to be compared on the number axis, and then use ""). This method is very simple when comparing the sizes of multiple rational numbers.
Task of intransitive verbs: group P 19 A and group B.
Do group a and group b at the same time if you have a good foundation.
Students with poor foundation choose group a.
Junior high school mathematics teaching plan: judgment of parallel lines I. teaching objectives
1. Understand the format of reasoning and proof, and understand the proof method of judgment theorem.
2. Master the second judgment theorem of parallel lines and use judgment axioms and theorems for simple reasoning.
3. Cultivate students' analytical reasoning ability through the deduction of the second judgment theorem.
4. Let students understand that knowledge comes from practice and serves practice. Only by learning cultural knowledge well can they solve practical problems and educate students for the purpose of learning.
Second, the guidance of learning methods
1. Teacher's teaching method: heuristic guided discovery method.
2. Students learn the law: actively participate, take the initiative to discover and develop their thinking.
Three. Key points, difficulties and solutions
(1) key points
Deduction of judgment theorem and solution of examples.
(2) Difficulties
Use symbolic language for reasoning.
(3) Solutions
1. Through the correct guidance of teachers, students can actively think, discover theorems and solve key points.
2. Under the guidance of the teacher, students complete the reasoning process by themselves and solve the difficulties and doubts.
Fourth, the class schedule
1 class hour
Verb (abbreviation for verb) Prepare teaching AIDS and learning tools.
Triangle board, projector, homemade film.
Sixth, the design of teacher-student interaction activities.
1. Review basic knowledge through design exercises, create situations and introduce new courses.
2. Under the guidance of teachers, students explore new knowledge, practice consolidation and complete new teaching.
3. Finish the summary by the students themselves.
Seven, teaching steps
Clear goal
Master the reasoning of the second theorem of parallel lines and use it for simple proof to cultivate students' logical thinking ability.
(B) the overall perception
Create situations, design suspense, lead to topics, guide students' thinking, discover new knowledge and consolidate new knowledge with variant training.
(3) Teaching process
Create a situation and review the introduction.
Teacher: Last class, we learned axioms and a method to judge parallel lines. Look at the following questions according to what we have learned.
Student activities: Students answer questions 1 and 2.
Teacher: Can you tell me under what conditions two straight lines can be judged to be parallel?
Student activities: from question l and question 2, students think and analyze that as long as the congruent angle or the internal angle are equal, they can judge that two straight lines are parallel.
The teacher drew a picture of question 3 on the blackboard.
Student activities: students answer the reasons orally, and the same angle and complementary angle are equal.
Teacher: Let the students write the process of symbolic reasoning and write it on the blackboard.
The teaching method shows that this class is a continuation of the last class and is based on the last class. Therefore, through the questions 1 and 2, review the two methods of judging parallel lines learned in the last lesson, so that students can make it clear that two straight lines can be judged as parallel as long as their included angles or internal angles are the same. The third problem is to pave the way for deducing the theorem of this lesson, that is, if the internal angles of the same side are complementary, it can be deduced that the same angle is equal.
Teacher: The fourth question is a practical one. What are the two known angles in the question?
Student activity: share the inner corner equally.
Teacher: What's the relationship between them?
Student activities: Complementarity.
Teacher: The question is to know that the internal angles of the bisector are complementary, so are the two straight lines parallel? This is what we are going to learn in this class.
Junior high school mathematics teaching plan: one-dimensional linear inequality group 1. One-dimensional linear inequality group: several one-dimensional linear inequalities about the same unknown quantity are combined to form a one-dimensional linear inequality group. The concept of one-dimensional linear inequalities can be understood from the following aspects:
(1) The inequalities that make up the inequality group must be one-dimensional linear inequalities;
(2) In quantity, the number of inequalities must be two or more;
(3) The position of each inequality in the inequality group is not fixed, they are parallel.
2. Solution set and solution set of one-dimensional linear inequality group: In one-dimensional linear inequality group, the common part of the solution set of each inequality is called the solution set of this one-dimensional linear inequality group. The process of finding the solution set of this inequality group is called solving inequality group. Steps to solve a set of unary linear inequalities:
(1) First, the solution set of each inequality in the inequality group is found separately;
(2) Using the number axis or formula, find the common part of these solution sets, that is, get the solution sets of inequality groups.
3. The number axis representation of inequality solution set (group):
Knowledge points of one-dimensional linear inequality system
1. When using the number axis to represent the solution set of inequality, remember the following rules: draw more to the right and less to the left, draw a solid origin with an equal sign, and draw a hollow circle without an equal sign;
2. First, we can draw the solution set of inequality group with the solution set of each inequality on the number axis, and find out that the common part is the solution set of inequality. The common part is the overlapping part of the solution set of each inequality on the number axis;
3. According to the linear inequality group, we simplify it to the simplest inequality group and then classify it. Usually we can divide linear inequalities into the above four categories.
Note: When the inequality group contains "≤" or "≥", we can ignore this equal sign when solving problems, so this kind of inequality can be classified into one of the above four basic inequality groups. But in the process of solving problems, this equal sign should be connected with that equal sign and cannot be separated.
4. Find some special solutions: find special solutions such as positive integer solutions and integer solutions of inequality (group) (these special solutions are often limited), and the steps to solve such problems: first find the solution set of this inequality, and then find the required special solution with the help of the number axis.
Test point analysis of one-dimensional linear inequality system
(1) Investigate the concept of inequality group;
(2) Investigate the solution set of linear inequalities in one variable and its representation on the number axis;
(3) Test the special solution of the inequality group;
(4) Determine the value of letters.
Misunderstanding of knowledge points of one-dimensional linear inequality group
(1) Misunderstanding, confusing inequality and equality;
(2) The common part of the solution set of inequality group cannot be determined correctly;
(3) When the solution set of inequality group is represented on the number axis, the representation method of boundary points is confused;
(4) thoughtlessness and omission of implied conditions;
(5) When there are multiple constraints, the exploration of inequality relations is not comprehensive, which leads to the expansion of the unknown range;
(6) For inequalities with letters, there is no classification to discuss the values of letters.
Mathematics teaching design in junior high school: complete square formula I. Brief introduction of content
Topic of this lesson: Through a series of inquiry activities, guide students to sum up two complete square formulas from the calculation results.
Key information:
1, based on the teaching materials and according to the mathematics curriculum standards, to guide students to experience and participate in the scientific inquiry process. Firstly, the relationship between the two multiplication polynomials on the left side of the equal sign and the three terms on the right side of the equal sign is proposed. Students discover problems independently, make assumptions and guesses about possible answers, and draw correct conclusions through repeated tests. Students acquire knowledge, skills, methods, attitudes, especially innovative spirit and practical ability through activities such as collecting and processing information, expressing and communicating.
2. Draw conclusions with standard mathematical language, so that students can feel the rigor of science and inspire their learning attitudes and methods.
Second, the learner analysis:
1, the basic knowledge and skills that should be possessed before learning this course:
(1) Definition of similar projects.
② Rules for merging similar projects
③ Polynomial multiplication polynomial rule.
2. Learners' level of what they will learn:
Before learning the complete square formula, students have been able to sort out the correct form of the formula. The purpose of this lesson is to let students summarize the application methods of formulas from the relationship between the left and right forms of equal signs.
Three. Teaching/learning objectives and corresponding curriculum standards;
Teaching objectives:
1, by exploring the process of complete square formula, the sense of symbol and thrust ability are further developed.
2. A complete square formula can be derived, and simple calculation can be made by using the formula.
(b) Knowledge and skills: Understanding is reasonable through the process of abstracting symbols from specific situations.
Numbers, real numbers, algebraic expressions, defensive cities, inequalities, functions; Master the necessary calculation (including estimation) skills; Explore the quantitative relations and changing laws in specific problems, and describe them with algebraic expressions, guarding cities, inequalities, functions, etc.
(4) Problem solving: being able to find and put forward mathematical problems in combination with specific situations; Try to find solutions to problems from different angles, and effectively solve problems, and try to evaluate the differences between different methods; Through the reflection on the process of solving problems, we can gain experience in solving problems.
(5) Emotion and attitude: dare to face the difficulties in mathematics activities, have successful experience in overcoming difficulties independently and using knowledge to solve problems, and have confidence in learning mathematics well; And respect and understand the opinions of others; Can benefit from communication.
Fourth, educational ideas and teaching methods:
1. Teachers are the organizers, promoters and collaborators of students' learning: students are the masters of learning, learning actively and individually under the guidance of teachers, experiencing with their own bodies and feeling with their own hearts.
Teaching is a process of communication, positive interaction and common development between teachers and students. When a student gets lost, the teacher does not tell him the direction easily, but instructs him how to distinguish the direction; When a student is afraid of climbing, the teacher does not drag him away, but arouses his inner spiritual motivation and encourages him to keep climbing.
2. Adopt the teaching mode of "problem scenario-inquiry and communication-induction and summary-intensive training".
3. Teaching evaluation methods:
(1) Through classroom observation, pay attention to students' active participation and awareness of cooperation and exchange in activities such as observation, summary and training, and encourage, strengthen, guide and correct them in time.
(2) By judging and giving examples, give students more opportunities to reveal the thinking process, and feedback the mastery of knowledge and skills in a natural relaxed state, so that teachers can diagnose the situation in time and investigate teaching.
(3) Through after-class interviews and homework analysis, timely check and fill the gaps to ensure the expected teaching effect.
Verb (abbreviation of verb) Teaching media: multimedia
Six, teaching and activity process:
The teaching process is designed as follows:
< 1 >, ask questions.
[Introduction] Students, we have learned the rule of polynomial multiplication and the rule of merging similar items. By operating the following four small questions, can you sum up the relationship between the result and the two monomials in the polynomial?
(2m+3n)2 = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _，(-2m-3n)2=____________，
(2m-3n)2 = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _，(-2m+3n)2=____________ .
< 2 >, analyze the problem
1, [student answers] discuss in groups
(2m+3n)2= 4m2+ 12mn+9n2，(-2m-3n)2= 4m2+ 12mn+9n2，
(2m-3n)2= 4m2- 12mn+9n2，(-2m+3n)2= 4m2- 12mn+9n2 .
(1) The characteristics of the original formula.
(2) The item number characteristics of the results.
(3) The characteristics of trinomial coefficients (especially the characteristics of symbols).
(4) The relationship between three terms and two monomials in the original polynomial.
2. [Student answers] Summarize the language description of the complete square formula:
The square of the sum of two numbers is equal to the sum of their squares, plus twice their product;
The square of the difference between two numbers is equal to the sum of their squares.
Subtract twice their product.
3. [Student's solution] Mathematical expression of complete square formula:
(a+b)2 = a2+2ab+B2；
(a-b)2=a2-2ab+b2。
(3) Using formulas to solve problems
1, oral answer: (the form of rushing to answer, active classroom atmosphere, stimulate students' enthusiasm for learning)
(m+n)2 = _ _ _ _ _ _ _ _ _ _ _ _ _，(m-n)2 = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _，
(-m+n)2 = _ _ _ _ _ _ _ _ _ _ _ _ _ _，(-m-n)2=______________，
(a+3)2=______________，(-c+5)2=______________，
(-7-a)2=______________，(0.5-a)2=______________。
2. Judges:
()① (a-2b)2= a2-2ab+b2
()② (2m+n)2= 2m2+4mn+n2
()③ (-n-3m)2= n2-6mn+9m2
()④ (5a+0.2b)2= 25a2+5ab+0.4b2
()⑤ (5a-0.2b)2= 5a2-5ab+0.04b2
()⑥ (-a-2b)2=(a+2b)2
()⑦ (2a-4b)2=(4a-2b)2
()⑧ (-5m+n)2=(-n+5m)2
3, small test knife
①(x+y)2 = _ _ _ _ _ _ _ _ _ _ _ _ _ _； ②(-y-x)2 = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _；
③(2x+3)2 = _ _ _ _ _ _ _ _ _ _ _ _ _； ④(3a-2)2 = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _；
⑤(2x+3y)2 = _ _ _ _ _ _ _ _ _ _ _ _； ⑥(4x-5y)2 = _ _ _ _ _ _ _ _ _ _ _ _ _ _；
⑦(0.5m+n)2 = _ _ _ _ _ _ _ _ _ _ _ _； ⑧ (a-0.6b)2 =_____________。
< 4 > student summary
What problems do you think should be paid attention to in the application of complete square formula?
(1) Formula * * * has three terms on the right.
(2) The sign of two square terms is always positive.
(3) The symbol of the middle item is determined by whether the two symbols on the left side of the equal sign are the same.
(4) The middle term is twice the product of the two terms on the left of the equal sign.
(5) Adventure Island:
( 1)(-3a+2b)2 = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
(2)(-7-2m)2 = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
(3)(-0.5m+2n)2 = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
(4)(3/5a- 1/2b)2 = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
(5)(Mn+3)2 = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
(6)(a2 B- 0.2)2 = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
(7)(2xy 2-3x2y)2 = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
(8)(2n 3-3 m3)2 = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
(6) Students' self-evaluation
[Summary] What have you gained and learned from this lesson?
In this lesson, we summed up the complete square formula by calculating and analyzing the results ourselves. In the process of knowledge exploration, students actively think, boldly explore, unite and cooperate and make progress together.
【 Homework 】 P34 Classroom exercise P36
Seven, after-school reflection
Although this lesson is not a difficult point in the textbook, it is the focus of the chapter on algebraic expressions. This is a simple operation in a special form of polynomial multiplication. Students need to be familiar with the use of two forms of formulas to improve the operation speed. In the teaching process, we should pay attention to let students summarize the characteristics of the equal sign on both sides of the formula, let students express the content of the formula in language, and let students explain the problems that are easy to appear in the process of using the formula and the details that pay special attention to. Then, through the in-depth practice step by step, the application of two forms of complete square formula is consolidated.