The main method to judge whether a quadratic root is the simplest quadratic root is according to the definition of the simplest quadratic root, or intuitively observing that the index of each factor (or factor) of the root number is less than the root index 2, and the root number does not contain the denominator. When the root number is polynomial, factorization should be carried out first, and then observation should be carried out. The following is the teaching plan of quadratic multiplication and division that I compiled. I hope everyone will read it carefully!
1 quadratic radical multiplication and division teaching plan I. Content and content analysis
1. Content
Division rule of quadratic root and its inverse application, the concept of simplest quadratic root.
2. Content analysis
The exploration of the division rule of the arithmetic square root of quadratic root and quotient, and the proposal of the simplest quadratic root point out the direction for the operation of quadratic root. After learning the division rule, I have rich operation rules and formula foundation. Turning a quadratic root into the simplest quadratic root is the basis of addition and subtraction.
Based on the above analysis, the teaching focus of this course is determined: the division rule of quadratic root, the nature of the arithmetic square root of quotient, and the simplest quadratic root.
Second, the goal and goal analysis
1. Teaching objectives
(1) The division rule of square root and the properties of arithmetic square root of quotient are obtained by inductive analogy;
(2) simple secondary division;
(3) Understand the concept of the simplest quadratic root.
2. Target analysis
(1) Students can find and describe the division rule of quadratic roots by analogy with the multiplication rule of quadratic roots.
(2) Students can understand the meaning of the reverse use of the division rule, and combine the concept, properties and multiplication and division rules of the quadratic root to operate the simple quadratic root.
(3) By observing the operation result of quadratic form, we can understand the characteristics of the simplest quadratic form and change the operation result of quadratic form into the simplest quadratic form.
Third, the diagnosis and analysis of teaching problems
The content of this section is mainly that students may have difficulties or make mistakes in the way that the denominator contains the root sign when doing the division operation of the quadratic root. In the division operation, we can use the nature of the arithmetic square root of the quotient after calculation, or we can use the nature of the fraction to remove the root sign in the denominator first, and then combine the multiplication rule with the nature of the arithmetic square root of the product. The division of square root is similar to the operation of fraction. If the numerator and denominator contain the same factor, they can be directly subtracted to simplify the operation. In teaching, we should not just list the questions, but take exercises at all levels as the carrier to guide students to master the operation process, predict the operation results and make clear the operation direction.
The teaching difficulties of this lesson are: the relationship between the division rule of quadratic square root and the nature of the arithmetic square root of quotient and its application.
Fourth, the teaching process design
1.
Question: What is the multiplication rule of 1 quadratic root? How about the general steps of simplifying quadratic roots?
Teacher-student activities Students answer.
The design aims to remind students to explore the process of multiplication law. By analogy, students can explore the law of division.
Five, the target detection design
2 quadratic radical multiplication and division teaching plan 1. Teaching objectives
(1) has experienced the multiplication rule of square root and the formation process of the arithmetic square root property of product; Can perform simple quadratic multiplication;
(2) The quadratic root can be simplified by formula.
2. Target analysis
(1) Students can find the law through calculation and popularize it to get the content of the multiplication law;
(2) Students can simplify the quadratic root by using the multiplication rule of the quadratic root and the nature of the arithmetic square root of the product.
Diagnosis and analysis of teaching problems
In this lesson, after learning the nature of the multiplication rule and the arithmetic square root of the product, students find it difficult to choose which formula to simplify the operation. The cultivation of operation habits is closely related to the cultivation of symbol consciousness and the formation of operation ability, because the content is more related to the real number content they have learned before. For example, the multiplication formula in algebraic expressions is also valid in the operation of quadratic roots. In teaching, we should make more efforts from contact to cultivate students' good operating habits.
In teaching, there are generally two situations in which a quadratic root can be transformed into the simplest quadratic root through example operation: (1) If the root number is a fraction or a fraction (including a decimal), it can be simplified by directly using the properties of the fraction and combining the properties of the quadratic root (for example, the solution 1 see Example 6 in the textbook), or it can be written as the quotient of the arithmetic square root first. (2) If the root sign does not contain a denominator, it can be decomposed into factors or factors first, and then the root sign or factor can be derived, thus simplifying the formula.
The teaching difficulties of this lesson are: the nature of quadratic root, the correct application of multiplication rule and the simplification of quadratic root.
Teaching process design
1. Review and introduce to explore new knowledge.
We have learned the concept and properties of square root. At the beginning of this class, we will learn the multiplication and division of quadratic roots. This lesson will first learn the multiplication of quadratic roots.
Question 1 What is a quadratic radical? What are the properties of quadratic roots?
Teacher-student activities Students answer.
The design intent of quadratic roots Multiplication and simplification need to make use of the properties of quadratic roots.
Question 2: What is the calculation result of the "Inquiry" column on page 6 of the textbook? What are the rules?
Teacher-student activities Students calculate, think and try to summarize, and guide students to describe the content of multiplication formula in their own language.
This design aims to enable students to discover the law in the process of independent inquiry, and draw the multiplication law of quadratic roots from special to general by analogy and incomplete induction. Students are required to describe the rules in mathematical language and words respectively, so as to cultivate students' symbol consciousness.
2. Observe and compare, and understand the law
Question 3: Simple radical operation.
Teacher-student activities: students operate by hand and teachers check.
Question 4. What are the conditions for quadratic square root multiplication and division? What are the values of the equation in turn?
Teacher-student activities Students answer, and after giving the correct answer, the teacher gives the nature of the arithmetic square root of the product.
The purpose of the design is to let students use this law to multiply simple quadratic roots, so as to test their mastery of this law. The law of multiplication, in turn, is the nature of the arithmetic square root of the product, which serves the operation. The nature of arithmetic square root of product decomposes the arithmetic square root of product into several factors or products of several factors. Algebraic expressions and multiplication formulas can simplify quadratic roots and cultivate students' operational ability.
3. Example demonstration, apply what you have learned.
Example 1 simplification: (1) multiplication and division of quadratic roots; (2) Multiplication and division of quadratic root.
Teacher-student activities: How to understand the example (1)?
If the student's answer is not perfect, ask again: in this question, can you directly calculate the result as the multiplication and division of the quadratic root? How do you think this problem can be simplified?
Teachers and students cooperate to answer the above questions. For the final result of radical operation, there are factors in the radical number or factors that can be thoroughly explored, which should be removed from the radical number according to the multiplication and division of the quadratic root.
Ask again: Can you imitate the answer to the question (1) and solve (2) by yourself?
The design aims at cultivating students' computing ability and clarifying the direction of quadratic root simplification through operation. The nature of arithmetic square root of product can be simplified by quadratic root.
Example 2 calculation: (1) multiplication and division of quadratic root; (2) multiplication and division of quadratic root; (3) Multiplication and division of quadratic root
Teacher-student activities Student calculation, teacher test.
(1) When multiplying roots, you can consider factorization or factorization, and you can directly get the multiplication and division of quadratic roots without writing the multiplication and division of quadratic roots before decomposition.
(2) The multiplication operation of quadratic root is similar to algebraic expression, and both exchange method and combination method are applicable. For roots with coefficients other than the root sign, the coefficient can be multiplied by the coefficient as the product first, and then the root operation can be performed;
(3) The operation in Example (3) is an elective course. Let spare students learn the operation of "quadratic square root with letters under the root sign" In this problem, the multiplication and division of quadratic root are obtained by using the property of arithmetic square root of product, and then the multiplication rule of quadratic root is used to become the multiplication and division of quadratic root. Because the multiplication and division of the quadratic root can judge the multiplication and division of the quadratic root, X is directly moved out of the root sign.
The design aims to guide students to sum up in time, emphasize the operation by using the algorithm and simplify the operation by using the multiplication formula, so that students can realize that the quadratic root is a special real number, so it satisfies the algorithm of real number, and the formula and method of algebraic expression operation are also applicable.
Although it is pointed out in the textbook that all letters in this chapter represent positive numbers unless otherwise specified, it should be emphasized that you should pay attention to its symbols when you see the root sign. We can judge the symbol of letters according to the concept of quadratic root, and correctly handle the symbol problem when removing the root sign.
4. Consolidate concepts and apply what you have learned
Exercise: Exercise 1 on page 7 of the textbook. Page 10, question 16.2.
Design intention consolidation exercises to test the mastery of multiplication rules.
5. Summary, reflection and improvement
Teachers and students review what they have learned in this lesson and ask students to answer the following questions:
(1) Can you explain how the multiplication rule of quadratic root is obtained?
(2) Can you explain the significance of the reversal of multiplication law?
(3) What are the basic steps to simplify quadratic roots? What are the general requirements for the final result?
6. Homework: Page 7, Questions 2 and 3. Exercise 16.2, questions 1 and 6.
Five, the target detection design
1. Of the following categories, () is definitely true.
A. multiplication and division of quadratic root B. multiplication and division of quadratic root
C. multiplication and division of quadratic root D. multiplication and division of quadratic root
This design aims to examine the concept and properties of quadratic root, which is the basis of quadratic root multiplication.
2. Simplify the multiplication and division method of quadratic roots _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
The quadratic root of design intention is a special real number, and the related algorithm of real number is also suitable for quadratic root.
3. Given the multiplication and division method of quadratic root, the result of simplifying the multiplication and division method of quadratic root is ()
A. multiplication and division of quadratic root B. multiplication and division of quadratic root C. multiplication and division of quadratic root D. multiplication and division of quadratic root
The design intention is to use the properties of the arithmetic square root of the product to consolidate the properties of the quadratic root and simplify the quadratic root correctly.
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