Quadratic roots belong to the field of numbers and algebra. They are carried out on the basis of students' learning square roots and cubic roots, and they are an extension and supplement to the contents of "real numbers" and "algebraic expressions" in the first volume of grade seven. The operation of quadratic root is based on algebraic expression, and the algorithm used in quadratic root operation is similar to algebraic expression and fraction. When adding and subtracting secondary roots, the method adopted is similar to merging similar items; In the multiplication and division of quadratic roots, the rules and formulas used are similar to the multiplication algorithms and formulas of algebraic expressions, which explain the internal relationship between the former knowledge and the latter knowledge.

The main contents of this chapter are quadratic root, its properties and its operation.

First, teaching materials and teaching objectives

Teaching requirements of this chapter.

(1) Understand the concept of quadratic root and the letter range of simple quadratic root;

(2) Understand the nature of quadratic radical;

(3) Understand the addition, subtraction, multiplication and division of quadratic roots;

(4) We will use the properties of quadratic root and operation rules to perform four simple operations on real numbers (the denominator is not required to be rational).

This chapter is about teaching material analysis.

On the basis of reviewing the arithmetic square root, the textbook introduces the concept of quadratic square root through three questions of "cooperative learning" and explains that the arithmetic square root of previously learned numbers is also called quadratic square root. In the arrangement of examples and exercises, three requirements are emphasized: one is to find the range of letters in the quadratic root; The second is to find the value of quadratic root; The third is to express related problems with quadratic roots.

For the properties of quadratic roots, the textbook uses the figure 1-2 on page 4, which means that if the area of a square is, then the side length of the square is; On the other hand, if the side length of a square is 0, then the area of the square is 0, so there is. Thus, the first property of quadratic radical is obtained. As for the second nature, it can be found by students' calculation, so the textbook arranges a "cooperative learning" for students to discover and summarize themselves. The first lesson of this section focuses on the understanding and application of these two properties. Examples and exercises are designed around these two properties. The second lesson is to learn two other properties of quadratic roots. The textbook arranges two groups of exercises, which are intended to let students discover these two properties through their own attempts and cooperation with their classmates. Through two examples and a set of exercises, students can know that using the properties of quadratic roots can simplify the operation of real numbers and get the formula of quadratic roots. "Inquiry Activities" on Page 9 of the Textbook

1.3, the operation of quadratic root, including four operations of addition, subtraction, multiplication and division of quadratic root and simple application. The textbook is arranged in three classes, step by step and comprehensively used. The first category focuses on the multiplication and division of two quadratic roots (equivalent to two monomials), and its law is derived from the nature of quadratic roots, which is more natural. Example 65438. Example 2 is a practical application, including Pythagorean theorem and triangle area calculation. The second kind is the mixed operation of addition, subtraction, multiplication and division of quadratic root, which is similar to polynomial single multiplication, polynomial multiplication (including multiplication formula and power) and polynomial single division. There is no concept of "similar quadratic root" in textbooks. Only students who are "similar to merging similar items" and "same quadratic root" can understand and operate similar algebraic expressions. The third category is the application of quadratic radical operation. The figure in Example 6 looks very complicated, and its purpose is to apply quadratic radical operation. Example 7 comprehensively applies the knowledge of right triangle, division of figure and calculation of area. Its solution process is long, and it is also a comprehensive application of quadratic root knowledge.

Second, the writing characteristics of this chapter

Pay attention to cultivating students' abilities of observation, analysis, induction and inquiry.

In the way of presenting knowledge in this chapter, the textbook highlights the narrative mode of "problem situation-mathematical activities-generalization-consolidation, application and expansion", which is mostly completed through "cooperative learning". "Cooperative learning" creates opportunities for students to engage in mathematical activities such as observation, guessing, verification and communication. For example, on page 5, ask students to calculate the specific values of three groups of sums, and then discuss the relationship between sums. Then, we get the property of the quadratic radical "=". Several other properties of quadratic radical are also adopted in the textbook. After learning the related properties of quadratic roots, the textbook designed an "inquiry activity". By simplifying the related quadratic roots, students can discover, express, verify and communicate with their peers. These are the directions of teaching materials compilation to stimulate teaching and learning.

Pay attention to the connection between mathematical knowledge and real life.

The textbook strives to overcome the complexity of learning quadratic roots in traditional concepts, avoid the simplification or calculation of a large number of pure formulas, properly insert practical applications or give formulas some practical significance. Whether learning the concept of secondary roots or learning the properties and operations of secondary roots, we should try our best to link what we have learned with real life and attach importance to the cultivation of the ability to solve practical problems by using what we have learned. For example, in learning the concept of secondary roots, three practical problems are introduced into textbooks. Its purpose is to pay attention to the actual background and formation process of concepts and overcome the learning mode of mechanical memory concepts. For example, on page 3 of the textbook, the distance traveled by ships is represented by the square root, the area of road signs is calculated at 1 1 page, and the planting area of flowers and plants is discussed at 2 1 page. Especially in the operation of quadratic roots, a special section is arranged to learn the application of quadratic roots.

Make full use of graphics to organically combine algebra and geometry.

For the content of number and algebra, the textbook attaches importance to the geometric background of the content, and it is a writing feature of the textbook to help students understand and solve algebraic problems with geometric intuition, which is also a guide to teaching. In this chapter, if the quadratic root is closely related to the calculation of each side of the right triangle, the textbook selects a certain number of problems in this respect, which not only enriches the application of Pythagorean theorem, but also learns the calculation of quadratic root. Another example is the introduction of secondary roots. When learning the properties of secondary roots is used in textbooks, the textbook helps students understand its meaning from both positive and negative aspects by reading the graph 1-2, and obtains the properties of secondary roots. Design some figures in homework or textbook exercises and calculate the length of line segments; Draw a triangle by square and rectangular coordinate system to determine the position of the point, etc. When arranging the application of quadratic radical operation in daily life and production practice, the selected questions are also to reflect the connection between students' knowledge, feel the integrity of knowledge, constantly enrich students' problem-solving strategies and improve their problem-solving ability.

Third, teaching suggestions

Pay attention to the use of pre-holiday words.

There are not many words before this chapter, but a specific question is put forward in close connection with this section. They can be used to create problem situations and introduce topics in teaching. For example, paragraph 1. 1, "The height of volleyball net is 2.43 meters, and CB is meters. Can you use algebra to represent the length of AC? " A few short sentences are not only a familiar problem situation for students, but also a seemingly familiar but challenging problem, which is related to mathematics learning. Teachers can ask a question related to this lesson. This role should not be ignored in teaching.

Pay attention to the difficulty of teaching.

Compared with the previous textbooks, the quadratic root formula reduces the requirements. For example, if the quadratic radical is simplified in nature, it only needs to be simple, and it is clear that the radical does not contain letters. The four operations of quadratic radical are limited to simplicity, and the radical sign does not contain letters, so there is no need to supplement the problems beyond the requirements of textbook topics in teaching. Of course, it should be flexible for students of different levels.

Make full use of analogy.

The operation of quadratic roots is based on algebraic expressions, and its rules and formulas are similar to algebraic expressions, especially the addition and subtraction of quadratic roots. The textbook does not put forward the concept of similar quadratic root, and completely refers to the method of merging similar items; The multiplication, division and multiplication operations of quadratic roots are similar to algebraic expressions. Therefore, analogy should be fully used in the teaching of the four operations of quadratic roots, so that students can understand their arithmetic and algorithms and improve their operational ability.

Chapter II Quadratic Equation in One Variable

First, the content of teaching materials and course learning objectives

(A) the contents of textbooks

This chapter includes three parts:

2. 1 unary quadratic equation;

2.2 solution of quadratic equation in one variable;

2.3 the application of a quadratic equation.

Section 2. 1 is the basic part of the whole chapter, section 2.2 is the key content of the whole chapter, and section 2.3 is the content of knowledge application and extension. In addition, the reading material introduces the development of quadratic equation in one variable, so that students can understand the development history of mathematics.

(C) curriculum objectives

(1) Understand the concept of unary quadratic equation and solve the equation in the form of (b≥0) by direct Kaiping method;

(2) Understand the collocation method, and use it to solve the unary quadratic equation of digital coefficient; Master the derivation of the root formula of quadratic equation in one variable, and use the root formula to solve quadratic equation in one variable; Factorization can be used to solve the quadratic equation of one variable, so that students can flexibly use various solutions of the quadratic equation of one variable to find the root of the equation according to the characteristics of the equation.

(3) Experience the process of estimating the solution of the equation by observation, drawing or calculator.

(4) According to the quantitative relationship in specific problems, list the univariate quadratic equations to solve practical problems, find and put forward practical problems that can be solved by univariate quadratic equations in daily life, production or other disciplines, and correctly express the problems and solving processes in language. Empirical equation is an effective mathematical model to describe the real world.

(5) Further cultivate students' logical thinking ability in combination with the teaching content, and educate students on dialectical materialism. Through the teaching of quadratic equation with one variable, students can further gain the understanding that things can be transformed. Quadratic root, unary quadratic equation, judgment and proof of proposition, quadrilateral