Tisch
Title: Complete Square Formula (1) I. Introduction.
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 learn from different people.
Seek 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 and have the ability to overcome them independently.
And have the confidence to learn math 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 students get lost
Wait, the teacher does not tell 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 mode of "problem scenario-inquiry communication-summary-intensive training"
Start teaching.
3. Teaching evaluation methods:
(1) Pay attention to students' initiative in observation, summary and training through classroom observation.
Dynamic participation and awareness of cooperation and exchange, timely encourage, strengthen, guide and correct.
(2) Give students more opportunities to relax naturally by judging and giving examples.
Revealing the thinking process and giving feedback on the mastery of knowledge and skills will help teachers diagnose the situation in time and investigate teaching.
(3) Through after-class interviews and homework analysis, timely check and fill gaps to ensure the expected results.
Teaching effect.
Verb (abbreviation of verb) Teaching media: multimedia. 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 minus 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
extreme
General description:
The complete square formula is a kind of induction and summary of the special formula in polynomial multiplication. At the same time, the derivation of the complete square formula is the beginning of the deformation of algebraic identities by using reasoning methods in junior high school mathematics. Learning the complete square formula is of great benefit to simplify the operation of some algebraic expressions and cultivate students' awareness of simplification. Moreover, the complete square formula is the necessary basis for subsequent study, which not only plays a great role in improving the speed and accuracy of students' operation. It is also an important basis for studying factorization, fractional operation, solving quadratic equations with one variable and the deformation of quadratic function identities in the future, and also has the function of cultivating students' strict logical reasoning ability. Therefore, it is of great significance to learn the complete square formula well for the subsequent algebra knowledge learning.
This section is the eighth section of the first chapter of the seventh grade mathematics book of Beijing Normal University, which takes up two class hours. This is the first class, which is mainly to let students experience the process of exploring and deducing the complete square formula, cultivate their sense of symbols and reasoning ability, and let them further understand the role of the combination of numbers and shapes in mathematics.
First, the analysis of students' learning situation
Students' skill base: Through the study of the first few lessons in this chapter, students have mastered the concept of algebra, algebra addition and subtraction, power operation, algebra multiplication and square variance formula, which laid the foundation for this lesson.
Students' experience basis: in the study of square difference formula, students have experienced the process of exploration and application, gained some experience in mathematical activities, and cultivated a certain sense of symbol and reasoning ability; At the same time, in the process of learning related knowledge, students have experienced many inquiry learning processes, and have a certain sense of independent inquiry and the ability to cooperate and communicate with peers.
Second, the teaching objectives
Knowledge and skills:
(1) Let students derive the complete square formula and apply it simply.
(2) Understand the geometric background of the complete square formula.
Mathematical ability:
(1) Students experience the process of exploring the complete square formula and further develop their sense of symbol and reasoning ability.
(2) Develop students' mathematical thinking of combining numbers with shapes.
Emotions and attitudes:
Expose and analyze the pre-concepts in students' minds to avoid forming "different concepts" in teaching.
Third, teaching focuses on difficulties.
Teaching emphasis: 1, derivation of complete square formula;
2. The application of complete square formula:
Teaching difficulties: 1. Eliminate the pre-concept in students' minds and avoid forming "different concepts";
2. Understanding and correct application of the structure of complete square formula.
Fourth, the analysis of teaching design
Eleven teaching links are designed in this lesson: students' practice, exposing problems, verifying, summarizing general situation, forming formulas, combining numbers and shapes, further expanding, summarizing formulas, applying formulas, students' feedback, students' PK, students' reflection and consolidating exercises.
The first link: students practice and expose problems.
Activity Content: Calculation: (a+2)2
Imagine that students have the following possibilities:
①(a+2)2=a2+22
②(a+2)2=a2+2a+22
③ Practice correctly;
In view of these results, a= 1 is substituted into the calculation, and it is concluded that ① and ② are both wrong, but ③ must be right? How to verify?
Activity purpose: In the minds of many students, it is considered that the complete square of the sum of two numbers is equal to the sum of the squares of two numbers, namely:
(a+2)2=a2+22, if you don't adopt this fixed thinking *, it is difficult to establish a correct concept; The purpose of this link is to make students' mistakes or other mistakes fully exposed, to make students fully realize that their original fixed thinking is wrong, and to lay the foundation for building a new thinking mode in the next step.
Step 2: Verify that (a+2) 2 = a2–4a+22.
Activity content: (a+2)2=(a+2)? (a+2)=a2+2a+2a+22
Activity purpose: On the basis of breaking the original thinking mode of students in the previous link, establish correct thinking methods for students and avoid forming "whimsy".
The third link: extend it to the general situation and form a formula.
Activity content: (a+b) 2 = (a+b) (a+b) = A2+AB+AB+B2 = A2+2AB+B2.
Activity purpose: Let students experience the process of inquiry from special to general, and experience the joy of discovery.
The fourth link: the combination of numbers and shapes
Activity content: Question: In the multiplication of polynomials, many formulas can be explained by geometric figures, so how to explain the complete square formula by geometric figures?
Play the animation and explain the geometric meaning of the complete square formula with geometric figures.
Students think: Is there any other way to interpret the complete square formula? (Thinking after class)
Purpose of the activity: Let students further realize that numbers and shapes do not exist in isolation, but can be organically combined, so as to develop students' mathematical thought of combining numbers and shapes.
The fifth link: further expansion
Activity content: Derive the complete square formula of the difference between two numbers: (a–b) 2 = A2–2ab+B2.
Method1:(a–b) 2 = (a–b) (a–b) = a2–ab–ab+B2 = a2–2ab+B2.
Method 2: (a–b) 2 = [a+(–b)] 2 = A2+2A (–b)+(–b) 2 = A2–2AB+B2.
Activity purpose: Let students experience the process of extending the complete square formula of the sum of two numbers to the complete square formula of the difference of two numbers, and realize the difference of results caused by different symbols. By quadratic derivation, they can realize the application of the complete square formula of the difference between two numbers.
The sixth link: summarize the formula and cognitive characteristics.
Activity content: Compare the similarities and differences between the two formulas: (a+b)2=a2+2ab+b2.
(a–b)2 = a2–2ab+B2
Features: ① There is a binomial square on the left, and there is only one symbol difference between them; There is a quadratic trinomial on the right, in which the first and third terms are the squares of the terms in the binomial on the left, the middle term is twice the product of the two terms in the binomial on the left, and the two terms are only one symbol apart;
② A and B in the formula can be arbitrary algebraic expressions (numbers, letters, monomials, polynomials).
Formula: the first square, the last square, twice the first and last multiplication are in the center.
Objective: To understand the characteristics of the complete square formula and summarize the formula of the complete square formula, which is convenient for students to understand and remember and avoid mistakes in the application of the formula.
The seventh link: formula application
Activity: Example: Calculation: ① (2x–3) 2; ②(4x+)2
Solution: ① (2x–3) 2 = (2x) 2–2? (2 times)? 3+32 = 4x 2– 12x+9
②(4x+)2=(4x)2+2？ (4x)()+()2= 16x2+2xy+
Activity purpose: In the first few links, students have already had a perceptual knowledge of the complete square formula. Through the explanation of this link and the practice of the next link, students will gradually experience cognition-imitation-re-cognition, thus rising to the stage of rational cognition.
The eighth link: practice in class
Activity content: calculation: ①; ②; ③(n+ 1)2–N2
Objective: Through students' feedback exercises, teachers can fully understand whether students' understanding of the complete square formula is in place and whether the complete square formula is properly applied, so that teachers can check for missing items in time.
The ninth link: student PK
Activity content: each student answers five calculation questions of complete square formula at the same table, and compares who has higher accuracy and faster speed.
Purpose of the activity: enliven the classroom atmosphere, stimulate students' competitiveness, and further consolidate students' understanding and application of the complete square formula.
The tenth link: student feedback.
Activity content: What did you gain from today's class?
Harvest 1: Understand the complete square formula and apply it simply;
Harvest 2: Understand the difference between the sum of two numbers and the complete square formula of the difference between two numbers;
Harvest 3: Feel the function of the mathematical thought of combining number and shape in mathematics.
Objective: To consolidate students' understanding of the complete square formula and experience the subtlety of mathematical thought through the induction and summary of a class.
Section 11: Homework:
Textbook P43 Exercise 1. 13
Tisso
Teaching objectives
1. Knowledge and skills: Understand the discovery and derivation process of the formula, understand the geometric background of the formula, understand the essence of the formula, and apply the formula for simple calculation.
2. Process and method: By making students go through the process of exploring the complete square formula, we can cultivate students' exploration and innovation abilities such as observation, discovery, induction, generalization and conjecture, develop their reasoning ability and organizational expression ability, and cultivate their ability of combining numbers with shapes.
3. Emotion, attitude and values: Experiencing mathematics activities is full of exploration and creativity, and you can gain successful experience and happiness in mathematics activities and build up your self-confidence in learning.
Emphasis and difficulty in teaching
Teaching focus:
Understand the formula, including its derivation process, structural characteristics, language expression (students' own language) and geometric explanation.
2. Simple calculations can be made with formulas.
Teaching difficulties:
Derivation of 1 and complete square formula and its geometric explanation.
2. The structural characteristics and application of the complete square formula.
teaching tool
courseware
teaching process
First, review old knowledge and introduce new knowledge.
Question 1: Please state the square difference formula and its structural characteristics.
Question 2: How is the square difference formula derived?
Question 3: What problem can the square difference formula solve? For example.
Question 4: Think, do and say the following results.
( 1)(a+b)2 2(a-b)2
(At this time, teachers can ask students to explain the reasons separately, and continue to stimulate students' interest in learning, without directly giving correct evaluation. )
Second, create problem situations and explore new knowledge.
A square experimental field with a side length of one meter needs to be increased by b meters to form four experimental fields and plant different new varieties (as shown in the figure).
(1) The areas of the four blocks are:,,,;
(2) Two forms represent the total area of the experimental field:
① On the whole: a big square with a side length of s, s =；;
② Part: the sum of four areas, S=.
Summary: What have you found through the above exploration?
Question 1: Through the above exploration and study, students should know what the correct result of question 4 is.
Question 2: If some students don't agree with this result, let's look at the following questions and continue our discussion. What does (a+b)2 mean? Please use the multiplication rule of polynomial to verify.
(In the teaching process, teachers should consciously mention that guesses and feelings are not necessarily correct, and real knowledge can only be obtained through verification, but students should be encouraged to make bold guesses and express their opinions, but they should be verified. )
Question 3: Can you say (a+b)2=a2+2ab+b2?
What are the structural characteristics of this equation? Speak in your own language.
Structural characteristics: the square of binomial (sum of two numbers) is on the right, and there are three terms on the right, which is twice the sum of the squares of two numbers plus the product of these two numbers.
Question 4: According to the structural characteristics of the above formula, can you tell what (a-b)2 equals? Please use the multiplication rule of polynomial to verify it again.
Summary: We call (a+b) 2 = A2+2ab+B2 (a–b) 2 = A2–2ab+B2 a complete square formula.
Question: ① What are the similarities and differences between these two formulas? Can you describe these two formulas in your own language?
Language description: the square of the sum (or difference) of two numbers is equal to the sum of the squares of these two numbers plus (or minus) twice the product of these two numbers.
Strengthening memory: head side, tail side, the first and second time in the center, and the sum is positive or negative.
Third, give examples to consolidate new knowledge.
Example 1: Calculate using the complete square formula.
( 1)(2x-3)2(2)(4x+5y)2(3)(Mn-a)2
Solution: (2x-3) 2 = (2x) 2-2O (2x) O3+32
=4x2- 12x+9
(4x+5y)2 =(4x)2+2o(4x)o(5y)+(5y)2
= 16x2+40xy+25y2
(mn-a)2=(mn)2-2o(mn)oa+a2
=m2n2-2mna+a2
Ac summary: the general steps of calculation by using complete square formula
(1) Determine the head and tail, and square them respectively;
(2) Determine the intermediate coefficient and sign, and get the result.
Fourth, practice consolidation.
Exercise 1: Calculate with the complete square formula.
Exercise 2: Calculate with complete square formula
Exercise 3:
Exercise can take many forms. Students can perform on the blackboard and teachers and students can evaluate each other. Students can also correct each other after completing independently, so that students can master the formula comprehensively. If students have problems, students and teachers should help them in time. )
Five, variant exercises
Sixth, talk about the harvest and summarize.
1. In this lesson, we learned the complete square formula of multiplication.
2, when we use the formula, we should pay attention to the following points:
The letters A and B in formula (1) can be arbitrary algebraic expressions;
(2) There are three items in the formula, so don't miss them and write wrong symbols;
(3) Errors such as ① and ② may occur. Don't confuse them with the square variance formula.
Seven, homework settings