The teaching goal of factorization of teaching plan in grade two;
1. knowledge and skills: master factorization by putting forward common factors and formulas, and cultivate students' ability to solve problems by factorization.
2. Process and method: By exploring the process of factorization method, students are trained to discuss problems, and the factorization method is obtained through guessing, reasoning, verification and induction.
3. Emotion, attitude and values: Through factorization, students can realize the beauty of mathematics, the self-confidence of success and the spirit of unity and cooperation, and the overall mathematical thinking and transformed mathematical thinking.
Emphasis and difficulty in teaching: decompose factors by putting forward common factors and formulas.
Teaching aid preparation: multimedia courseware (small blackboard)
Teaching method: activity inquiry method
Teaching process:
Introduction: In the transformation of algebraic expressions, it is sometimes necessary to write a polynomial as the product of several algebraic expressions. This deformation is factorization. What is factorization?
Detailed explanation of knowledge
Definition of knowledge point 1 factorization
Transforming a polynomial into the product of several algebraic expressions is called decomposing this polynomial, which is also called decomposing this polynomial.
It shows that (1) factorization and algebraic expression multiplication are deformations in opposite directions.
For example:
(2) Factorization is the deformation of identity, so it can be tested by algebraic expression multiplication.
How to factorize a polynomial?
Knowledge point 2 refers to common factor method
Every term in the polynomial ma+mb+mc has a common factor M, which we call the common factor of this polynomial. Ma+mb+mc=m(a+b+c) is to decompose ma+mb+mc into the product of two factors, one of which is the common factor M of each item, and the other is the common factor (A+B+). This method of decomposing factors is called the common factor extraction method. For example, x2-x = x (x- 1), 8a2b-4ab+2a = 2a (4ab-2b+ 1).
Inquiry communication
Have the following deformations been decomposed? Why?
( 1)3x2y-xy+y = y(3 x2-x)； (2)x2-2x+3 =(x- 1)2+2；
(3)x2 y2+2xy- 1 =(xy+ 1)(xy- 1)； (4)xn(x2-x+ 1)= xn+2-xn+ 1+xn。
Analysis of typical examples of teacher-student interaction
Example 1 decompose the following factors by extracting common factors.
( 1)-x3z+x4y； (2)3x(a-b)+2y(b-a)；
Analysis: (1) extracting the common factor can directly decompose the problem. (2) The problem must be properly deformed first, then b-a is transformed into -(a-b), and then the common factor is extracted.
Summarize the following problems when decomposing factors by extracting common factors:
(1) factorization results If there are similar items in each bracket, it cannot be decomposed again.
(2) If the minor events like (2) need to be unified, they should be unified first, and the fewer the unified items, the better. Then note that (a-b)n=(b-a)n(n is an even number).
(3) Factorization should be written as a power if it finally has the same base power.
Students do the following factorization.
( 1)(2a+b)(2a-3b)+(2a+5b)(2a+b)； (2) 4p( 1-q)3+2(q- 1)2
Knowledge point 3 formula method
(1) square difference formula: a2-b2=(a+b)(a-b). That is, the square difference of two numbers is equal to the product of the sum of these two numbers and the difference of this number. For example, 4x2-9=(2x)2-32=(2x+3)(2x-3).
(2) Complete square formula: a2? 2ab+b2=(a？ B)2。 Among them, a2? 2ab+b2 is a completely flat way. That is to say, the sum of squares of two numbers plus (or minus) twice the product of these two numbers is equal to the square of the sum (or difference) of these two numbers. For example, 4x2- 12xy+9y2=(2x)2-2? 2x？ 3y+(3y)2=(2x-3y)2。
Inquiry communication
Are the following deformations correct? Why?
( 1)x2-3 y2 =(x+3y)(x-3y)； (2)4x 2-6xy+9 y2 =(2x-3y)2； (3)x2-2x- 1=(x- 1)2。
Example 2 breaks down the following categories.
( 1)(a+b)2-4a 2； (2) 1- 10x+25 x2； (3)(m+n)2-6(m+n)+9。
Analysis: The purpose of this question is to investigate the factorization of the complete square formula.
Students do the following factorization.
( 1)(x2+4)2-2(x2+4)+ 1； (2)(x+y)2-4(x+y- 1)。
Comprehensive application
Example 3 Factorization.
( 1)x3-2 x2+x； (2)x2(x-y)+y2(y-x)；
Analysis: The purpose of this question is to investigate the comprehensive application of common factor method and formula method to decompose factors.
When solving the factorization problem, first consider whether there is a common factor, and if there is, first mention the common factor; If no common factor is binomial, consider whether it can be decomposed by square difference formula. If it is a trinomial, consider it completely flat, and finally, until each factor can no longer be decomposed.
Exploration and innovation issues
Example 4 If 9x2+kxy+36y2 is completely flat, then k=.
Analysis: Completely flat way shape image: a2? 2ab+b2 is the sum (or difference) of the sum of squares of two numbers and twice the product of these two numbers.
If x2+(k+3)x+9 is completely flat, then k=.
Course summary
Factorization is used to solve the calculation problem when factoring and formula are used to decompose factors.
Every term has a "public", and the first term is often negative. One item is "1", which is divided into "bottom" in brackets.
Self-assessment knowledge consolidation
1. If x2+2(m-3)x+ 16 is completely flat, the value of m is equal to ().
A. 3 B- 5 c. 7d. 7 or-1
2. If (2x) n-81= (4x2+9) (2x+3) (2x-3), then the value of n is ().
A.2 B.4 C.6 D.8
3. Decomposition factor: 4x2-9y2=.
4. Given X-Y = 1, XY = 2, find the value of x3y-2x2y2+xy3.
5. Factorization polynomial 1-x2+2xy-y2.
Thinking problem decomposition factor (x4+x2-4)(x4+x2+3)+ 10.
Attachment: blackboard design
factoring
Discussion on the Definition of Factorization, Communication and Innovation
A summary of classroom analysis of common factor method
Comprehensive application of self-evaluation by formula method
Reflective factorization in mathematics factorization teaching in Grade Two is the difficulty in Chapter Nine. When students learn factorization for the first time, they are often confused with multiplication. The main reason is unclear concept.
In teaching, the difference between factorization and multiplication is intuitively obtained by changing the positions of formulas on both sides of the equal sign. For factorization, students can experience it through a series of exercises. So there is no need to pave the way at the beginning and waste a certain amount of time.
Among several methods of factorization, the most basic method of extracting common factor mage is easy for students to master. However, in some comprehensive application problems, students always forget to observe whether there is a common factor first, and directly think about decomposition by formula. This directly leads to errors in the decomposition of some topics and incomplete decomposition of some topics. Therefore, in the factorization step, this piece should continue to be strengthened. Formula method is actually the decomposition of factors. Students will confuse square difference with perfect balance method. This is because we don't understand the formula thoroughly and don't really grasp the difference between the two. Generally, it can be distinguished from the following aspects. If it is the square difference of two terms, the square difference formula is given priority after extracting the common factor. If it is three terms, it is best to factorize it in a completely flat way.