Teaching objectives?
1. Knowledge and skills?
Understand the significance of factorization and its relationship with algebraic expression multiplication.
2. Process and method?
Through the analogy process from factorization to factorization, master the concept of factorization and feel the role of factorization in solving problems.
3. Emotions, attitudes and values?
In the activity of exploring factorization method, we should cultivate students' orderly thinking, expression and communication ability, cultivate positive enterprising consciousness, and realize the intrinsic meaning and value of mathematical knowledge.
Key points, difficulties and key points?
1. key: understand the significance of factorization and feel its role.
2. Difficulties: the relationship between algebraic expression multiplication and factorization.
3. Key points: Introduce factorization factors into factorization factors, and deepen understanding by analogy.
Teaching methods?
Adopt the teaching method of "stimulating interest and guiding learning".
Teaching process?
First, create a situation to stimulate interest?
Problem traction?
Ask students to explore the following two questions:
Question 1: 720 is divisible by which numbers? Talk about your thoughts.
Question 2: When a= 102 and b=98, find the value of A2-B2.
Second, enrich associations and show thinking?
Exploration: Can you fill in the blanks below? ?
1 . ma+m b+MC =()()； ?
2 . x2-4 =()()； ?
3.x2-2xy+y2=()2。 ?
Teachers and students all know that turning a polynomial into the product of several algebraic expressions is called decomposing this polynomial, which is also called decomposition.
Third, group activities, * * * with exploration?
Problem traction?
(1) Are the following left and right deformations factorized?
①(x+ 1)(x- 1)= x2- 1； ?
②a2- 1+B2 =(a+ 1)(a- 1)+B2； ?
③7x-7=7(x- 1)。 ?
(2) Fill in the appropriate items in the brackets below to make the equation hold.
①9 x2(_ _ _ _ _ _ _)+y2 =(3x+y)(_ _ _ _ _ _ _)； ?
②x2-4xy+(_______)=(x-_______)2。 ?
Fourth, consolidate and deepen the class exercises?
Textbook exercises.
Space-time calculation of exploration and research: Can 993-99 be divisible by 100? ?
5. Class summary and development potential?
Summarized by the students themselves, the teacher put forward the following outline:
1. What is factoring? ?
2. What is the difference between factorization and algebraic expression operation? ?
Sixth, homework, special breakthrough?
Select a supplementary job.
Blackboard design?
15.4.2 How to improve the common factor?
Teaching objectives?
1. Knowledge and skills?
Can determine the common factor of polynomial and decompose polynomial by improving the common factor.
2. Process and method?
Make students go through the process of exploring the common factor of polynomial and decompose it according to the thinking method of mathematical reduction.
3. Emotions, attitudes and values?
Cultivate students' thoughts of analysis, analogy and induction, enhance students' awareness of cooperation and exchange, actively accumulate preliminary experience in determining common factors, and realize their application value.
Key points, difficulties and key points?
1. key point: master the factorization of polynomials by proposing common factors.
2. Difficulties: Correctly determine the common factor of polynomials.
3. Key: The key of common factor method is how to find common factor. The method is: look at the coefficient and look at the letters. The coefficients of the common factor take the common divisor of each coefficient; Letters take the same letters, and the index of each letter takes the lowest power.
Teaching methods?
Adopt "heuristic" teaching method.
Teaching process?
First, review exchanges and introduce new knowledge?
Review and communicate?
Is the following transformation from left to right factorized? Why? ?
( 1)2 x2+4 = 2(x2+2)； (2)2t2-3t+ 1 =(2t3-3t2+t)； ?
(3)x2+4xy-y2 = x(x+4y)-y2； (4)m(x+y)= MX+my； ?
(5)x2-2xy+y2=(x-y)2。 ?
Question:?
Do the terms in 1. polynomial mn+mb contain the same factor? ?
2. What about polynomials 4x2-x and xy2-yz-y? ?
Please write the above polynomial as the product of two factors and explain the reasons.
The common factor of each term in a polynomial summed up by the teacher is called the common factor of this polynomial. For example, the common factor in mn+mb is m, the common factor in 4x2-X is x, and the common factor in xy2-yz-Y is y?
Concept: If every term of a polynomial contains a common factor, then this common factor can be put forward, so that the polynomial can be transformed into the product form of two factors. This method of decomposing factors is called extracting common factors.
Second, work in groups and explore ways?
What is the common factor of polynomial 4x2-8x6, 16a3b2-4a3b2-8ab4? ?
Teachers and students first determine the common factor of each item, then divide it by polynomial to get another factor, look for the common factor to see the coefficients and letters, and take the common factor of each coefficient as the coefficient of the common factor; Letters take the same letters, and the index of each letter takes the lowest power.
Third, learn by example and apply what you have learned?
Example 1 factorization -4x2yz- 12 XYZ+4xyz.
Solution: -4x2yz- 12x2z+4xyz?
=-(4x2yz+ 12 XYZ-4xyz)？
=-4xyz(x+3y- 1)？
Example 2 Factorization, 3A2 (x-y) 3-4B2 (y-x) 2?
The common factor (y-x) 2 or (x-y) 2 can be found by observing the given polynomial, so there are two variants, (x-y) 3 =-(y-x) 3 and (x-y) 2 = (y-x) 2, and the following two decomposition methods can be obtained.
Solution 1: 3A2 (x-y) 3-4B2 (y-x) 2?
=-3a2(y-x)3-4b2(y-x)2？
=-[(y-x)2？ 3a2(y-x)+4b2(y-x)2]？
=-(y-x)2[3a2(y-x)+4b2]？
=-(y-x)2(3a2y-3a2x+4b2)？
Option 2: 3A2 (x-y) 3-4B2 (y-x) 2?
=(x-y)2？ 3a2(x-y)-4b2(x-y)2？
=(x-y)2[3a2(x-y)-4b2]？
=(x-y)2(3a2x-3a2y-4b2)？
Example 3 is calculated by a simple method: 0.84×12+12× 0.6-0.44×12.
Teachers' activities guide students to observe and analyze how to calculate more conveniently.
Solution: 0.84×12+12× 0.6-0.44×12?
= 12×(0.84+0.6-0.44)?
= 12× 1= 12.？
Teacher's activity points out that Example 3 is the application of factorization in calculation after students finish Example 3, and puts forward the differences between Example 1, Example 2 and Example 3. ?
Fourth, consolidate and deepen the class exercises?
Textbook P 167 exercises 1, 2, 3.
Exploring time and space?
Calculate by raising the common factor:?
0.582×8.69+ 1.236×8.69+2.478×8.69+5.704×8.69?
5. Class summary and development potential?
1. The key to factorization is to find the common factor. When seeking the common factor, we should pay attention to: (1) coefficient requires the common factor; (2) Find all the letters; (3) Find the lowest power of the index.
2. Factorization should be thoroughly decomposed, that is, until it can no longer be decomposed.
Sixth, homework, special breakthrough?
Textbook P 170 exercises 15.4 questions 1, 4( 1) and 6.
Blackboard design?
15.4.3 formula method (1)?
Teaching objectives?
1. Knowledge and skills?
Will apply the square difference formula to factorize and develop students' reasoning ability.
2. Process and method?
By exploring the process of factorization using square difference formula, students can develop reverse thinking and feel the integrity of mathematical knowledge.
3. Emotions, attitudes and values?
Cultivate students' good habit of interactive communication and realize the application value of mathematics in practical problems.
Key points, difficulties and key points?
1. key: decompose the factor with the square difference formula.
2. Difficulties: Understand the problem-solving steps of factorization and the thoroughness of factorization.
3. Key: Derive the square difference formula by applying the direction of reverse thinking. In the application of the formula, we should first pay attention to its characteristics, and then do a good job of transforming the formula to turn the problem into an aspect where the formula can be applied.
Teaching methods?
The teaching method of "problem solving" is adopted, so that students can promote their thinking under the traction of problems.
Teaching process?
First, observe and discuss new knowledge?
Problem traction?
Please calculate the following categories.
( 1)(a+5)(a-5)； (2)(4m+3n)(4m-3n)。 ?
Students solve the above two problems and perform on the stage.
( 1)(a+5)(a-5)= a2-52 = a2-25； ?
(2)(4m+3n)(4m-3n)=(4m)2-(3n)2 = 16 m2-9 N2。 ?
Teachers' activities guide students to complete the following two questions, and use the idea of "reciprocal" in mathematics to find the law of factorization.
1. Decomposition factor: A2-25; 2. Factorization factor 16m2-9N.
Students' activities start with reverse thinking and quickly get the following answers:?
( 1)a2-25=a2-52=(a+5)(a-5)。 ?
(2) 16 m2-9 N2 =(4m)2-(3n)2 =(4m+3n)(4m-3n)。 ?
Teachers' activities guide students to complete A2-B2 = (A+B) (A-B), and at the same time, the topic is derived: factorization with square difference formula.
Square difference formula: A2-B2 = (a+b) (a-b).
Comments: The letters A and B in the square difference formula should be emphasized in teaching, and letters can be used to represent numbers and algebraic expressions (monomials and polynomials).
Second, learn by example and apply what you have learned?
Example 1 decompose the following factors: (projection display or blackboard writing)?
( 1)x2-9 y2； (2) 16x 4-y4； ?
(3) 12a2x 2-27b2y 2； (4)(x+2y)2-(x-3y)2； ?
(5)m2( 16x-y)+N2(y- 16x)。 ?
It is found in observation that 1 ~ 5 questions all conform to the characteristics of the square difference formula, which can be used for factorization.
Teachers' activities inspired students to factorize from the perspective of square difference formula, and invited five students to perform on stage.
Students' activities are divided into four groups for cooperative inquiry.
Solution: (1) x2-9y2 = (x+3y) (x-3y); ?
(2) 16x 4-y4 =(4x 2+y2)(4x 2-y2)=(4x 2+y2)(2x+y)(2x-y)； ?
(3) 12a2x 2-27b2y 2 = 3(4a2x 2-9b2y 2)= 3(2ax+3by)(2ax-3by)； ?
(4)(x+2y)2-(x-3y)2 =[(x+2y)+(x-3y)][(x+2y)-(x-3y)]= 5y(2x-y)； ?
(5)m2( 16x-y)+N2(y- 16x)？
=( 16x-y)(m2-N2)=( 16x-y)(m+n)(m-n)。 ?
15.4.3 formula method (2)?
Teaching objectives?
1. Knowledge and skills?
Understand the method of factorization by using complete square formula and develop reasoning ability.
2. Process and method?
After exploring the process of factorization with complete square formula, I feel the significance of reverse thinking and master the basic steps of factorization.
3. Emotions, attitudes and values?
Cultivate good reasoning ability, understand "transformation" and "substitution" thinking methods, and form flexible application ability.
Key points, difficulties and key points?
1. key: understand the factorization of the complete square formula and learn to apply it.
2. Difficulties: Flexible application of factorization formula method.
3. The key is to apply the thinking methods of "transformation" and "substitution" to transform the problem into a form, so as to achieve the purpose of decomposing factors by formula.
Teaching methods?
Adopt the teaching method of "independent inquiry" and complete the content of this lesson under the appropriate guidance of teachers.
Teaching process?
First, review exchanges and introduce new knowledge?
Problem traction?
1. Decomposition factor:?
( 1)-9 x2+4 y2； (2)(x+3y)2-(x-3y)2； ?
(3)x2-0.0 1y2。 ?
Knowledge transfer?
2. Calculate the following categories:?
( 1)(m-4n)2； (2)(m+4n)2； ?
(3)(a+b)2； (4)(a-b)2。 ?
Teachers' activities guide students to complete the following two questions, and use the idea of "reciprocal" in mathematics to find the law of factorization.
3. Factorization:?
( 1)m2-8mn+ 16 N2(2)m2+8mn+ 16 N2； ?
(3)a2+2ab+B2； (4)a2-2ab+b2。 ?
Student activities from the perspective of reverse thinking, quickly get the following answers:?
Solution: (1) m2-8mn+16n2 = (m-4n) 2; (2)m2+8mn+ 16 N2 =(m+4n)2； ?
(3)a2+2ab+B2 =(a+b)2； (4)a2-2ab+b2=(a-b)2。 ?
The inductive formula completely squares the formula A2A2 AB+B2 = (AB) 2.
Second, learn by example and apply what you have learned?
Example 1 decompose the following factors:?
( 1)-4a2b+ 12ab 2-9 B3； (2)8a-4a 2-4； ?
(3)(x+y)2- 14(x+y)+49； (4)+n4。 ?
Example 2 if x2+axy+ 16y2 is a complete square, what is the value of a?
According to the definition of completely flat mode, the solution of this problem should be divided into two cases, that is, the square of the sum of two numbers or the square of the difference between two numbers, from which the value of A can be calculated accordingly, and then A3 can be calculated.
Third, consolidate and deepen the class exercises?
Textbooks P 170 exercises 1 and 2.
Exploring time and space?
1. Given x+y=7 and xy= 10, find the following values.
( 1)x2+y2； (2)(x-y)2？
2. Given x+=-3, find the value of x4+.
Fourth, class summary, development potential?
Because polynomial factorization is just the opposite of algebraic expression multiplication, the formula of polynomial factorization can be obtained by writing algebraic expression multiplication formula backwards, and there are three main types:?
a2-B2 =(a+b)(a-b)； ?
a2 ab+b2=(a b)2。 ?
When using formula factorization, we should pay attention to:
(1) The form and characteristics of each formula are determined by the overall analysis of the number and degree of the polynomial, and whether it can be decomposed by the formula and which formula to use. Usually, when the polynomial is binomial, the square difference formula is considered for decomposition; When polynomial is trinomial, it should be decomposed by complete square formula. (2) In some cases, polynomials may not be directly formulated, and they need to be properly combined, deformed and replaced before they can be decomposed by formula method; (3) When the polynomial term has a common factor, we should first consider putting forward the common factor and then decompose it with a formula.
5. Homework, special breakthrough?