This lesson is selected from the first content of the fourth quarter of Chapter 15 in the first volume of the eighth grade mathematics of People's Education Press (P 165- 167). Factorization is one of the important means to transform algebraic identities, and it has important applications in future algebraic learning, such as simple operation of polynomial division, operation of fractions, solving equations (groups) and identity transformation of quadratic functions. Therefore, learning factorization well is of great significance to the subsequent study of algebraic knowledge.

This section is factorized into 1 section, which takes up one class hour. Mainly to let students experience the process from factorization to factorization, experience the mathematical thought-analogy thought, understand the reciprocal relationship between factorization and multiplication of algebraic expressions, and feel the role of factorization in solving related problems.

Analysis of learning situation

Based on the experience that students have been exposed to factorization in primary school, but they are completely unfamiliar with the concept of factorization, we should consciously cultivate students' mathematical ability of knowledge transfer, such as analogical thinking and reverse operation ability.

Basic analysis of students' skills: students have been familiar with the distribution law of multiplication and its inverse operation, and have learned the multiplication operation of algebraic expressions. Therefore, students will not be unfamiliar with the introduction of factorization, which laid a good foundation for learning factorization today.

According to the analysis of students' experience, the method of factorization by multiplication of algebraic expressions is a process of reverse thinking, but reverse thinking is still unfamiliar to eighth-grade students and difficult to accept. Moreover, this section does not involve the specific method of factorization, so it is difficult for students to seek the method of factorization.

Teaching objectives

1. Knowledge and skills: (1) Make students understand the meaning and concept of factorization.

(2) Understand the reciprocal relationship between factorization and algebraic expression multiplication, and use this relationship to find the method of factorization.

2. Process and method: (1) Students explore ways to solve problems independently. In this process, the relationship between factorization and factorization is found by observation and analogy, so as to cultivate students' observation ability and further develop their analogical thinking.

(2) Transition from the inverse operation of algebraic expression multiplication to factorization to develop students' reverse thinking ability.

(3) Through the observation and comparison of factorization and algebraic multiplication, cultivate students' ability of analyzing problems and comprehensive application.

Third, emotional attitude and values: let students initially feel the dialectical view of unity of opposites and the scientific attitude of seeking truth from facts.

Teaching emphases and difficulties

Teaching emphasis: the concept of factorization and the method of putting forward common factor.

Teaching difficulties: correctly find out the common factor of polynomial terms, the difference and connection between factorization and algebraic expression multiplication.

teaching process

Teaching link

Teachers' activities

The presupposition of students' behavior

Design intent

Activity 1:

Introduction to comments

See who can calculate quickly: calculate in a simple way;

( 1)7/9 × 13-7/9 ×6+7/9 ×2= ; (2)-2.67× 132+25×2.67+7×2.67= ;

(3)992– 1= 。

Students are divided into two categories in calculation: one is to correctly apply the method of factor decomposition to simple calculation; Second, I don't know how to correctly apply the factorization method for simple calculation, but take the real calculation method for calculation.

If students are quite unfamiliar with factorization, I believe they should be quite familiar with simple calculation methods. The purpose of introducing this step is to review the special algorithm of simple calculation method-factorization, so that students can naturally transition to correctly understand the concept of factorization through analogy and clear the way for mastering factorization. The calculated value of 992–1in this link design is to reduce the difficulty of the next link.

Note: Students are familiar with the method of (1) (2) using the distribution law of multiplication in reverse, but it is difficult to use the square difference formula in reverse (3). Therefore, it is necessary to guide students to review the square difference formula in algebraic expression multiplication that they learned in grade seven, and help them to use the square difference formula in reverse smoothly.

Activity 2:

Import theme

The exploration of 1 P 165 (omitted);

2. See who thinks fast: What numbers can divide 993–99? How did you get it?

Student thinking: Do you know the key to solving these problems from the solution of the above problems?

Guide students to decompose this formula into the product of several numbers, continue to strengthen students' understanding of factorization, and provide necessary ideological preparation for students to analogize factorization.

Activity 3: Explore new knowledge

See who's right:

Calculate the following formula:

( 1)3x(x- 1)=；

(2)m(a+b+c)=；

(3)(m+4)(m-4)=；

(4)(y-3)2 =；

(5)a(a+ 1)(a- 1)=；

Fill in the blanks according to the above formula:

( 1)ma+m b+ MC =；

(2)3 x2-3x =；

(3)m2- 16 =；

(4)a3-a =；

(5)y2-6y+9= .

Students get factorization (improved common factor method) from the reverse calculation of algebraic expression multiplication.

In the calculation of multiplication of the first set of algebraic expressions, students can get the results of the second set by observing the first set. Then, by comparing the results of the two groups, students can have a preliminary understanding of factorization, and gradually transition from the inverse operation of algebraic expression multiplication to factorization, thus developing students' reverse thinking ability.

Activity 4:

Induce and acquire new knowledge

Compare the connections and differences between the following two operations:

( 1)a(a+ 1)(a- 1)= a3-a

(2) a3-a= a(a+ 1)(a- 1)

Are there any other similar examples of the operation in the third link? Besides, can you find similar examples?

Conclusion: Transforming a polynomial into the product of several algebraic expressions is called factorization of this polynomial. Among them, the common factor of each term in the polynomial is extracted as one factor of the product, and the rest of the polynomial is taken as another factor of the product. This factorization method is called common factor method.

Distinguish: Are the following deformations decomposed? Why?

( 1)a+b=b+a

(2)4x2y–8xy 2+ 1 = 4xy(x–y)+ 1

(3)a(a–b)= a2–ab

(4)a2–2ab+B2 =(a–b)2

Students discuss and speak about the knowledge, understanding and viewpoints of factorization, especially the common factor method, and summarize the definitions of factorization and common factor method.

Through students' discussion, make students more aware of the following facts:

(1) factorization factor and algebraic expression multiplication are reciprocal relations;

(2) The result of factorization should be expressed in the form of product;

(3) Each factor must be an algebraic expression, and the degree of each factor must be lower than that of the original polynomial;

(4) It must be decomposed until every polynomial can no longer be decomposed.

Activity 5: Applying new knowledge

Example learning:

P 166 cases 1 case 2 (omitted)

Under the guidance of teachers, students use common factor method to complete examples.

By putting forward the common factor method, students can further understand the factorization.

Activity 6: Classroom Practice

1.P 167 exercise;

2. See who can connect correctly

x2-y2 (x+ 1)2

9-25 x 2 y(x -y)

x 2+2x+ 1 (3-5 x)(3+5 x)

xy-y2 (x+y)(x-y)

3. Which of the following deformations is factorization, and why?

( 1)(a+3)(a -3)= a 2-9

(2)a ^ 2-4 =(a+2)(a-2)

(3)a2-B2+ 1 =(a+b)(a-b)+ 1

(4)2πR+2πr=2π(R+r)

Students finish the exercises by themselves.

Through students' feedback exercises, teachers can fully understand whether students' understanding of the meaning of factorization is in place, so that teachers can check for missing items in time.

Activity 7: Class Summary

What did you learn from today's class? What methods have you mastered? you do not get it , do you?

Students speak.

Through students' review and reflection, we can strengthen students' understanding of the significance of factorization, further understand the reciprocal relationship between factorization and algebraic expression multiplication, and deepen their understanding of analogical mathematics.

Activity 8: Homework after class

Textbook P 170 exercise no. 1 and 4.

Students do it independently.

Through the consolidation of homework, the understanding and application of factorization, especially the common factor method.

Blackboard book design (the motherboard book that needs to be left on the blackboard all the time)

15.4. 1 common factor method example

Definition of 1. factorization

2. Common factor method