design
Analysis of learning situation:
Students learned algebraic expressions, fractions, quadratic roots, linear equations with one variable, linear equations with two variables and fractional equations in the seventh and eighth grades. On this basis, this lesson will start with practical problems and abstract the concept of quadratic equation of one variable and the general form of quadratic equation of one variable.
Teaching objectives
Knowledge and skills:
1, understand the concept of quadratic equation in one variable.
2. Master the general form of quadratic equation with one variable and correctly understand quadratic coefficient, linear coefficient and constant term.
Mathematical thinking:
1, through the introduction of a quadratic equation, cultivate students' modeling thinking, induction, analysis and problem-solving ability.
2. By learning the concept of quadratic equation in one variable, we can cultivate students' completeness and profundity in understanding the concept.
3. From the knowledge from reality, establish the idea of reduction, and infiltrate the idea of equation into students from setting unknowns and listing equations, thus further improving students' ability to analyze and solve problems.
Solve the problem:
In the process of analyzing and revealing the quantitative relationship of practical problems and transforming practical problems into a mathematical model of quadratic equation, students feel that the equation is a tool to describe the quantitative relationship in the real world, which increases their perceptual knowledge of quadratic equations.
Emotional attitude:
1. Cultivate students' awareness of autonomous learning, knowledge exploration and cooperation and exchange.
2. Stimulate students' interest in learning mathematics, experience the happiness of learning mathematics, and cultivate the consciousness of using mathematics.
Teaching focus:
The concept and general form of quadratic equation with one variable.
Teaching difficulties:
1, the transformation process from practical problems to mathematical problems.
2. Correctly identify the terms and coefficients in the general form of quadratic equation with one variable.
Teaching interactive design:
First, feel the new knowledge of autonomous learning
Question 1 There is a rectangular green space with an area of 900 square meters, the length is more than the width 10 meter. What is the length and width of the green space?
The width of rectangular green space is assumed to be x meters, and according to the meaning equation, it is xx+10 = 900;
Finishing: x2+ 10x-900=0 ①.
Question 2: There were 50,000 books in the school library at the end of last year, and it is expected to increase to 72,000 by the end of next year, so as to seek the average annual growth rate in these two years.
The analysis assumes that the average annual growth rate in these two years is X, and the equation obtained according to the topic is: 51+x2 = 7.2;
Finishing: 5 x2+ 10x-2.2=0 ②
Question 2: The school should organize volleyball invitational tournament, and there should be a match between every two teams. According to the venue, time and other conditions, the schedule is arranged for 7 days, and 4 games are arranged every day. How many teams should the organizer invite to participate?
Analysis of all games ***4×7=28 games, assuming that X teams are invited to participate, then each team will play 1 game with other x- 1 teams, and the whole game will be * * * *, and the equation is obtained according to the meaning of the question:
Finishing: x2-x-56=0 ③
Design intention: Find and ask simple questions in real life, attract students' attention, and stimulate students' interest and enthusiasm in autonomous learning. At the same time, by solving practical problems, the concept of quadratic equation with one variable is introduced, and at the same time, the ability of students to solve practical problems by using equation thinking is improved.
Second, independent exchanges and exploration of new knowledge
Explore the left and right sides of the above three equations 1 are algebraic "algebraic expression" and "fraction" with unknown number;
2 equation contains an unknown number after sorting;
According to polynomials in algebraic expressions, their highest degree is quadratic.
give rise to
1, the definition of quadratic equation in one variable
There are algebraic expressions on both sides of the equal sign, which only contains the knowledge of a unary number and the knowledge of the highest degree is quadratic, so it is called a unary quadratic equation.
2. The general form of quadratic equation with one variable
Generally speaking, any univariate quadratic equation about x can be transformed into the following form after finishing:
ax2+bx+c=0a≠0
This form is called the general form of quadratic equation with one variable. Where ax2 is a quadratic term, A is a quadratic term coefficient, bx is a linear term, B is a linear term coefficient, and C is a constant term.
It is emphasized that the equation ax2+bx+c=0 is a quadratic equation with one yuan only when a≠0, and a linear equation with one yuan only when a=0 and b≠0. Therefore, in the general form, the condition of a≠0 must be included.
Design intention: Because students have mastered the concepts of algebra, fraction, linear equation, etc., it is in line with students' cognitive basis to ask questions from the number and maximum number of unknowns and guide students to sum up the same points. Students' independent observation, comparison and induction are effective guarantees for activities, and students should be allowed to fully explore and communicate in teaching. At the same time, in concept teaching, analogy is an effective way to help students understand concepts correctly.
According to the exercises, judge the following equations, which are quadratic equations with one variable? Which ones are not? Why?
1x 3-2 x2+5 = 0； 2 x2 = 1；
35 x2-2x-= x2-2x+； 42x+ 12 = 3x+ 1；
5x 2-2x = x2+ 1； 6ax2+bx+c=0
Design intention: This question takes the form of answering first and then answering, so as to improve students' interest and enthusiasm in learning mathematics. Its purpose is to consolidate the concept of a quadratic equation in time, and at the same time let students know that to judge whether an equation is a quadratic equation in one variable, we must first organize it into a general form, and then judge it according to the definition.
Third, apply independently and consolidate new knowledge.
Example 1 known equation a-3x|a- 1|-2x+5=0. When a=- 1, this equation is quadratic, and when a=0, 2 or 3, this equation is linear.
Design intention: Through the study of the example 1, firstly, let students further consolidate the concept of quadratic equation with one variable, and pay attention to its most basic conditions: the maximum number of unknowns is 2, and the coefficient of quadratic term is not 0; The second is to let students understand the connection and difference between quadratic equation and linear equation. When filling in the first blank, students should pay attention to choosing a value, and when filling in the second blank, they should pay attention to guiding students to discuss in different categories.
In Example 2, the equation 3xx- 1=5x+2 is transformed into the general form of quadratic equation, and the quadratic coefficient, the linear coefficient and the constant term are written.
The general form of analyzing a quadratic equation with one variable is ax2+bx+c=0a≠0. Therefore, the equation 3xx- 1=5x+2 must be sorted by algebraic expression operation, including removing brackets and shifting terms.
Solution: Remove the brackets and get:
3x2-3x=5x+ 10
To transfer projects and merge similar projects, you must:
3x2-8x- 10=0
The quadratic term coefficient is 3, the linear term coefficient is -8, and the constant term is-10.
Design intention: Through the study of Example 2, firstly, students can further master the general form of quadratic equation with one variable, and pay attention to emphasizing that quadratic term, quadratic term coefficient, linear term, linear term coefficient and constant term all contain the previous symbols; The second is to make students further understand the deformation process of the equation.
Fourth, independently summarize and expand new knowledge.
What did you learn in this class? What did you get from it?
1 and a≠0 are the necessary conditions for ax2+bx+c=0 to become a quadratic equation, otherwise the equation ax2+bx+c=0 becomes bx+c=0, which is not a quadratic equation.
2. To find the quadratic term coefficient, the linear term coefficient and the constant term in the univariate quadratic equation, we must first turn the equation into a general form.
Design intention: guide students to review the learning content of this lesson and strengthen the formation of knowledge.
Five, independent test feedback new knowledge
1, the following equation is a quadratic equation, which is 14⑤.
①3x2+x=20，②2x2-3xy+4=0，③，④ x2=0，⑤
A school is going to build a rectangular garden with an area of 200 square meters. Its length is more than its width 10 meter. If the width of the garden is x meters, the equation can be listed as xx+ 10=200, which can be transformed into the general form of x2+ 10x-200=0.
3. If the equation m-2x|m|+3mx+ 1=0 is a quadratic equation about x, then m= -2.
4. Transform the equation X+ 12+X-2x+2 = 1 into a quadratic equation with the general form 2x2+2x-4=0, in which the quadratic term is 2x2, the quadratic term coefficient is 2, the linear term coefficient is 2x, the linear term coefficient is 2 and the constant term is -4.
Design intention: test students' mastery of new knowledge in class, know the feedback in time, and adjust the follow-up teaching content and teaching methods.
Sixth, homework after class
Page 28 of the textbook 1 2 5 6 7
Teaching idea and reflection of quadratic equation in junior middle school
This section is the first lesson of the second chapter of ninth grade mathematics. Through the study of this section, students will master the concept and general form of quadratic equation ax2+bx+c=0a≠0 as well as quadratic term, quadratic term coefficient, linear term, linear term coefficient and constant term, which is a typical concept teaching class.
Concept teaching always follows this law: introducing concepts, forming concepts, consolidating concepts, applying concepts and deepening concepts, and also follows this law in design teaching. The teaching task is completed through five links: learning, communication, application, summary and detection. Firstly, let students establish a quadratic equation of one variable and introduce the new lesson smoothly through three questions; Then, through communication and exploration, the concept of quadratic equation in one variable is summarized, so that students can realize the necessity of learning quadratic equation in one variable, explore the general form of quadratic equation in one variable and related concepts, learn to use equations to solve practical problems, and gain new knowledge in this course; Thirdly, through two examples, the concept can be consolidated and applied. Finally, by summing up and testing, we can deepen what students have learned and apply it to practical problems, so that students can master what they have learned skillfully.
In the teaching process, we emphasize autonomous learning and cooperation, so that students can communicate and cooperate with each other in the process of inquiry, so that they can understand and master the concept and general form of a quadratic equation in the process of autonomous inquiry, gain experience in mathematical activities, and improve their inquiry ability, discovery ability and innovation ability.