The Concept of Quadratic Radical in Junior Middle School Mathematics
The application of secondary roots is mainly reflected in two aspects:
1. Use important thinking methods from special to general and from general to special to solve some regular exploratory problems;
2. Use the quadratic root formula to solve the calculation problem of length and height, calculate some lengths or heights according to the known quantity, or design a material-saving scheme, as well as the splicing and segmentation of graphics. This process needs to calculate the quadratic root, which is actually a simplified evaluation.
Common inspection methods
(1) design some rules to explore problems and improve students' imagination and creativity; (2) Design some project inquiry questions in combination with real life.
Misunderstanding reminder
(1) It is impossible to find out the law of * * through observation, induction and conjecture, and use this law to solve problems;
(2) Unable to apply mathematical knowledge to solve problems in real life.
A typical example: Xiaoli wants to use a square paper with an area of 400cm2 to cut out a rectangular paper with an area of 300cm2 along the edge direction, so that its length-width ratio is 3: 2. I don't know if I can cut it out. I'm worried. Can you help him solve it?
The operation of quadratic root is mainly to learn multiplication, division, addition and subtraction of quadratic root.
Addition and subtraction of (1) quadratic radical;
It is necessary to simplify the quadratic roots first, and then add and subtract the coefficients of quadratic roots with the same number of roots (that is, similar quadratic roots), and the number of roots remains unchanged.
Note: For the addition and subtraction of quadratic roots, the key is to merge similar quadratic roots. Usually, the simplest quadratic root is found first, and then similar quadratic roots are merged. However, when simplifying the quadratic root, the square root of the quadratic root should contain neither denominator nor factors that can be completely opened.
(2) Multiplication of quadratic roots:
(3) Division of quadratic roots:
Note: Multiplication and division should be used flexibly. In practice, it often changes from the right of the equation to the left of the equation. At the same time, the range of letters should be considered, and finally the operation result should be transformed into the simplest quadratic root.
(4) Mixed operation of quadratic roots:
First multiply (or root), then multiply and divide, and finally add and subtract. If there are brackets, count them first; If you can use arithmetic or multiplication formula, you can change the operation order appropriately to perform simple operations.
Note: When performing radical operation, we should correctly use the operation rules and multiplication formulas, analyze the characteristics of the topic, master the methods and skills, and simplify the operation process. The result of quadratic radical operation should be simplified as much as possible. In addition, the score of radical operation must be written as false score or true score, not as score.
Lecture notes on the second-level radicals of junior high school mathematics
First of all, talk about textbooks.
This lesson is selected from the first section of the second part of chapter 2 1 in the first volume of ninth grade mathematics of People's Education Press. ? Quadratic radical? Is it the Curriculum Standard? Numbers and algebra? The important content of. In this chapter, the real number (13.1square root; 13.2 cube root; 13.3 real number), the concept, properties and operation of quadratic roots are further studied. What is the content of this chapter and what have you learned? Algebraic Pythagorean Theorem of Real Numbers? Close contact, but also for future study? Acute trigonometric function? 、? Quadratic equation with one variable? And then what? Quadratic function? And other content to lay an important foundation.
Second, talk about learning.
Students have learned the square root (arithmetic square root) and other related knowledge, and have a certain knowledge base and cognitive ability. The knowledge learning in this class and beyond needs students' rigorous thinking, classified discussion and analogical mathematical thinking. If students can't understand and recognize correctly here, it will have a great influence on the subsequent study. Therefore, students are required to actively explore and think, train and consolidate in time and overcome learning difficulties. Study? .
Third, talk about teaching objectives.
According to the requirements of the syllabus and the content analysis of the textbook structure, combined with the actual level of ninth-grade students and taking into account the psychological characteristics of students' existing cognitive structure, the following teaching objectives can be determined in this lesson:
1. Knowledge and skills: master the concept of quadratic root, the value range of quadratic root and the value range of root.
2. Process and method: the ability to deal with problems according to conditions and the ability to discuss problems by classification.
3. Emotional attitude and values: rigorous scientific spirit
Fourth, talk about the key points and difficulties in teaching.
Teaching emphasis: the range of square root number of quadratic form
Teaching difficulty: the range of square root
Verb (abbreviation for verb) Speaking and teaching methods
The essence of teaching activities is a kind of cooperation and communication. Students are the masters of mathematics learning, and teachers are the organizers, guides and collaborators of mathematics learning. According to the age characteristics of students and the existing knowledge base, this course focuses on strengthening the vertical connection between knowledge, expanding the space for students to explore and embodying the cognitive process from concrete to abstract. In order to lay a solid foundation for further study, for example, in? Acute trigonometric function? In this chapter, we will encounter many practical problems, such as the constraints on quadratic roots in the process of solving practical problems. This class will properly strengthen the exercises, so that students can form the habit of learning mathematics from the perspective of connection and development.
Six, said the learning method
The new curriculum standard points out that students are the main body of learning. In order to make students become real masters, teachers should guide students to think independently, explore cooperatively and sum up together in the process of mathematics teaching, thus reflecting students' dominant position in learning. This course mainly adopts the methods of autonomous learning, cooperative inquiry, guidance and promotion, heuristic, teaching and practice. First, ask questions, let students discuss and analyze problems, and teachers and students jointly summarize the concepts; Then the connotation of the concept is analyzed, and several important conclusions are drawn, and these important conclusions are used to calculate and simplify the quadratic root. Through the study of this lesson, we can inspire students' divergent thinking, train students' ability to observe, analyze and find problems, and cultivate students' dialectical materialistic views.
Suggestions on learning junior high school mathematics well
First, master the preview learning method and cultivate the ability of self-study in mathematics.
Preview is a learning method to learn new knowledge in textbooks before class. To learn junior high school mathematics well, we must first learn to preview new knowledge of mathematics, because preview is the premise of listening to a good lesson and mastering classroom knowledge, and it is an essential link in mathematics learning. Is the preview available? One move, two batches, three trials and four points? Preview method of. A row? Is to circle the main points of knowledge and basic concepts. Two batches? It is to annotate the experiences, opinions and contents that you can't understand temporarily in the preview in the blank space of the book; ? Three tests? Just try to do some simple exercises to test the effect of your preview. Four points? It is to list the main points of this section of knowledge that you have previewed, and to distinguish which knowledge you have mastered through previewing and which you don't understand through previewing, which needs further study in classroom learning.
Second, master the classroom learning methods to improve the classroom learning effect
Classroom learning is the most basic and important link in the learning process, which should be adhered to? Arrive at five o'clock? Namely, ears, eyes, mouth, heart and hands;
Handwriting: it is to write down the main points and thinking methods of the lecture simply and clearly, so as to review, digest and rethink, but the lecture should be the main part, supplemented by records;
Listening: Listen to the teacher attentively, how to analyze and summarize. In addition, listen to the students' answers to see if they are enlightening, especially the questions that they didn't understand beforehand;
Mouth-to-mouth: actively cooperate with teachers and classmates to explore, dare to ask questions and express opinions, and not follow suit;
Eye-catching: look at the teacher's expression, the meaning expressed by gestures, the teacher's demonstration experiment and the content on the blackboard, look at the textbook content that the teacher asks to read, and connect the knowledge in the book with the knowledge that the teacher said in class;
Heart orientation: that is, we should think carefully in class, pay attention to understanding new knowledge in class, and think positively in class. The key is to understand and be able to integrate and apply flexibly. It is necessary to grasp the key words and understand the new concept spoken by the teacher from another angle.
Third, master the practice methods and improve the ability of solving mathematical problems.
The ability to solve mathematical problems is mainly improved through practical exercises. Mathematics exercises should pay attention to the following points:
1. Correct attitude and fully understand the importance of mathematical practice. Practice can not only improve the answering speed and master the answering skills, but also often lead to many new problems in practice.
2. Have confidence and willpower. Mathematical exercises often involve complicated calculations and profound proofs. You should have enough confidence, tenacious will and patient and meticulous habits.
3. Develop the good habit of thinking first, then answering, and then checking. Don't practice blindly when you encounter problems, the calculation is invalid. We must first understand the meaning of the question deeply, think carefully, grasp the key points, and then answer. I'll check it when I'm finished.
4. Observe carefully, use flexibly, find the rules and become a skill.
Fourth, master the review methods and improve the comprehensive ability of mathematics.
Review is the mother of memory. We should constantly review what we have learned. Review and consolidation should pay attention to the following methods.
1. Arrange the review time reasonably. The homework on the day when the iron is hot must be reviewed on the same day. No matter how difficult the homework is, it must be consolidated.
2. Adopt the comprehensive review method, that is, by finding out the left-right relationship of knowledge and the internal relationship between vertical and horizontal, improve the whole. How to divide the comprehensive review? Three steps? First, look at the overall situation, browse all the contents, and initially form the impression of the knowledge system by evoking memories; Second, deepen understanding and comprehensively analyze what you have learned; Finally, consolidate and form a complete knowledge system.
3. Review methods to break through weak links. We should work hard on the weak links and strengthen the consolidation of textbook knowledge. Only by breaking through the weak links can we improve the overall comprehensive ability of mathematics.
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