Design of factorization teaching plan for junior high school mathematics
I. Background of the case
According to modern education theory, teachers should fully mobilize students' learning enthusiasm, let students actively explore and learn, let students participate in classroom activities, cultivate students' observation, analysis and induction abilities through their self-feelings, and gradually improve students' self-learning, independent thinking, problem-finding and problem-solving abilities, and gradually develop a good personality.
Factorization is an important identity deformation of algebra. It is the basis of learning fractions, and it is widely used in identity deformation, algebraic operation, solving equations, solving functions and so on.
Second, the case analysis
Teaching process design
(A) "situational import"
Situation 1: How to calculate 375? 2.8+375? 4.9+375? 2.3 ? what do you think?
Question: Why 375? 2.8+375? 4.9+375? 2.3 can be written as 375? (2.4+4.9+2.3)? What is the basis?
Comments: (1), review old knowledge, deepen memory, and pave the way for later study.
(2) Students are interested in this kind of problem, and can quickly find out some different fast calculation methods, and quickly come up with the inverse deformation of multiplication and division. Setting such a situation, the formula from counting to counting is more efficient. It also creates a good mood and atmosphere for the learning of new curriculum content.
Situation 2: Analysis and comparison
Multiplication rule of polynomial multiplied by monomial
a(b+c+d)=ab+ac+ad ①
On the other hand, you will get
ab+ac+ad =a(b+c+d)②
Thinking (1) How do you know the relationship between ① and ②?
(2) Does every term of the polynomial on the left side of Equation (2) have the same factor? Can you name this factor?
Comment: (1), exploring the method of factorization is actually a re-understanding of algebraic multiplication. Therefore, in the teaching process, teachers should rely on students' existing algebraic multiplication operation, leaving them enough time and space to explore and communicate, so that they can experience the reciprocal deformation process from algebraic multiplication to factorization.
(2) This topic focuses on cultivating students' abilities of observation, analysis and induction, and permeates students with mathematical thinking methods of comparison and analogy.
(B) "Exploring factorization"
1, understand the common factor
(1), concept 1: The terms ab, ac and ad of polynomial ab+ac+ad all contain the same factor A, which is called the common factor of polynomial terms.
(2) Discuss
Do the terms of the following polynomials have common factors? If so, try to find the common factor.
① The common factor of polynomial a2b+ab2 is ab, and the common factor is letters;
② The common factor of polynomial 3x2-3y is 3, and the common factor is a numerical coefficient;
③ The common factor of polynomial 3x2-6x3 is 3x2, which is the product of mathematical coefficient and letters.
Analysis and speculation
When determining the common factor of polynomials, we should consider two aspects: sum.
① How to determine the digital coefficient of common factor?
② How to determine the letters of common factors? How to determine the index of letters?
Exercise: Write the common factors of the following polynomials.
( 1)8x- 16 (2)2a2b-ab2
(3)4x2-2x (4)6m2n-4m3n3-2mn
Comments: (1), teachers should not directly give the method and explanation of finding the common factor of polynomials, but should encourage students to explore independently, accumulate the method and experience of finding the common factor according to their own experience, and correct common mistakes in solving problems through mutual communication.
(2) Understanding the common factor is the basis of factorization, so we should pay attention to practice when solving problems, especially the common factor of power sum coefficient, and let students pay attention to it.
(3) The general steps to find the common factor can be summarized as follows: look at the coefficient 2, look at the letter 3, and look at the index.
2. Understand factorization
Concept 2: Decomposition of a polynomial factor into the product of several algebraic expressions is called polynomial factorization.
(textbook) P7 1 exercise 1 topic.
(1), the following transformations from left to right, which are factorization and which are not?
①.ab+ac+d=a(b+c)+d
②.a2- 1 =(a+ 1)(a- 1)
③.(a+ 1)(a- 1)= a2- 1
(2) What do you think is the relationship between factorization of factors and polynomial multiplication of monomials? What inspiration did you get from it?
Comments: (1), this topic is mainly to deepen students' understanding of the concept of factorization, so that students can clearly understand that the result of factorization should be in the form of algebraic expression product.
(2) The teacher's intention in assigning this topic is to guide students to analyze and discuss, encourage students to think and express their opinions diligently, and cultivate students' logical thinking ability and communication ability. Let students master that factorization is the reciprocal process of algebraic expression multiplication in active learning, and use the relationship between them to understand the idea of factorization, thus reducing the difficulty of this lesson.
(3) "case study"
Example 1: Break down the following categories.
( 1)6a3b-9a2b2c(2)-2 m3+8 m2- 12m
Solution: (1)6a3b-9a2b2c
=3a2b？ 2a-3a2b？ 3bc (Find the common factor and divide each term into the product of the common factor and the monomial)
=3a2b(2a-3bc) (extracting common factor)
(2)-2m3+8m2- 12m
=-(2m？ m2-2m？ 4m+2m？ 6) (The first symbol is a negative number, first put the polynomial in brackets with a negative sign, and pay attention to the change of the symbol in brackets. )
=-2m(m2-4m+6) (extracting common factor)
Comments: (1), the concept and significance of factorization need students' multi-level feelings. Teachers should not expect students to fully grasp it with a thorough explanation and analysis. At this time, let students have a preliminary feeling first, and then enhance their understanding of concepts through different forms of exercises.
(2) When explaining examples, teachers should encourage students to find common factors themselves, so that students can practice with their hands and brains. Teachers can collect mistakes and comment on them below to deepen their understanding of factorization methods.
(3) In teaching, teachers should not simply ask students to memorize arithmetic rules, but should pay more attention to students' understanding of arithmetic, let students try to tell the truth of each step of operation, and consciously and methodically cultivate students' thinking ability and language expression ability.
The error-prone point of this question:
(1), missing items: the number of items in brackets after the common factor is put forward should be the same as that of the original polynomial, so as to check whether there are missing items.
(2) Symbol: Since the law of brackets was not involved in last semester, it is necessary to emphasize the law of brackets here: What is before brackets? +? No, all items in brackets remain unchanged; What's before the brackets? -? Numbers and everything in brackets should be changed.
"consolidation exercise"
Exercise: Distinguish the right and wrong of the following factorization.
( 1)8a3b 2- 12ab 4+4ab = 4ab(2a2b-3 B3)
(2)4x2- 12x3=2x2(2-6x)
(3)a3-a2=a2(a- 1)= a3-a2
After solving the (1) error and decomposing the factor, the number of terms of the polynomial in brackets is one less.
(2) Error. After factorization, the polynomials in brackets still have common factors.
(3) Error, after factorization, returns to the multiplication of algebraic expressions.
Comments: (1), most of them are mistakes that students are easy to make. The purpose of this topic is to let students use the results of example 1 to accurately identify common mistakes in factorization and have a clearer understanding of factorization. In this example, group discussion and communication are still used to let students participate in classroom activities.
(2) When a polynomial term happens to be a common factor, it should be regarded as the product of it and 1, and the remainder after extracting the common factor should be 1. 1 As a term, the coefficient can usually be omitted, but if it is a single term, it cannot be omitted in factorization.
(3) When using polynomials to decompose factors, each factor must be decomposed until it cannot be decomposed.
(4) Teachers arrange this process so that students can complete it independently, fully expose students' thinking process, show students' lively and active knowledge and rich personality, make students truly become the main body of learning, strengthen the connection between factorization and algebraic expression multiplication, and disperse the difficulties of this lesson.
(5) "think about it":
How to factorize the polynomial 3a(x+y)-2b(x+y)?
Solution: 3a(x+y)-2b(x+y)= (x+y)(3a-2b)
Comment: The common factor formula (x+y) is a polynomial, which requires higher requirements. When there are the same integers (polynomials) in polynomials, don't disassemble them. When extracting the common factor, it is put forward as a whole, sometimes with appropriate deformation, such as: (2-a)=-(a-2). In teaching, we can initially infiltrate the idea of substitution and introduce factors.
Concept 3 transforms a polynomial into the product of a common factor and another polynomial. This method of decomposing factors is called extracting common factors.
Reflections on factorization teaching in junior high school mathematics
1. According to the students' knowledge structure, the teaching process adopted in this class is: Ask questions? Actual operation? Induction? Classroom exercises? Class summary? Assign six parts of homework to reflect the process of knowledge occurrence, formation and development, so that students can further develop their ability of observation, induction, analogy, generalization and reverse thinking, and develop their ability of organizational thinking and language expression;
2. Factorization is a kind of deformation, and the result of deformation should be the product of algebraic expressions. Factorial decomposition and algebraic multiplication are reciprocal, that is, factorization is regarded as a deformed process, and then algebraic multiplication is the inverse process of factorization. On the one hand, this reciprocal relationship reflects their close relationship, on the other hand, it also illustrates their fundamental differences. Exploring the method of factorization is actually a re-understanding of algebraic expression multiplication. Therefore, in the teaching process, teachers should provide students with rich and interesting problem situations with the help of students' existing algebraic multiplication operations, leave enough time and space for students to explore and communicate, and let them experience the reciprocal deformation process from algebraic multiplication to factorization.
3. In the aspect of putting forward common factors, students' lack of understanding of common factors and unclear requirements for putting forward common factors have caused the following mistakes in factorization: (1) common factors are wrong; (2) Incomplete finding of common factor (such as missing the coefficient of common factor (or the coefficient is not the greatest common divisor of each coefficient), missing the coefficient or letter factor when the common factor contains polynomial), resulting in incomplete factorization;
4. Because the parenthesis rule is not involved in the textbook of the first volume of the seventh grade, there are many symbol errors when students decompose polynomials with negative first coefficient;
Factorization is a key and difficult point, and the above problems need to be further strengthened in future teaching.