Mathematics knowledge points in Grade Two
Chapter 12 congruent triangles
I. Knowledge framework:
Second, the concept of knowledge:
1. Basic definition:
(1) Coincidence: two figures can completely coincide, which is called congruence.
Congruent triangles: Two triangles that can completely coincide are called congruent triangles.
⑶ Corresponding vertices: The mutually coincident vertices in congruent triangles are called corresponding vertices.
⑷ Corresponding edges: The overlapping edges in congruent triangles are called corresponding edges.
5] Correspondence angle: The mutually coincident angles in congruent triangles are called correspondence angles.
2. Basic nature:
(1) Stability of a triangle: Once the lengths of three sides of a triangle are determined, the shape and size of the triangle are determined, which is called stability of a triangle.
⑵ The nature of congruent triangles: the corresponding edges of congruent triangles are equal, and the corresponding angles are equal.
3. congruent triangles's judgment theorem;
(1) SSS: three sides correspond to the congruence of two triangles.
⑵ SAS: Two triangles with equal included angles are congruent.
⑶ Angle and Angle (ASA): Two triangles with two angles, with equal sides.
⑷ Angular edge (AAS): The opposite side of two angles and one of them corresponds to the congruence of two triangles.
(5) hypotenuse and right-angled edge (HL): hypotenuse and right-angled edge correspond to the congruence of two right-angled triangles.
4. Angle bisector:
(1) Painting:
⑵ Property Theorem: The distance between a point on the bisector of an angle and both sides of the angle is equal.
(3) The inverse theorem of the property theorem: the point with equal distance from the inside of the angle to both sides of the angle is on the bisector of the angle.
5. The basic method of proof:
(1) Make clear what is known and verified in the proposition (including implied conditions, such as the angle relationship implied by the edge, angle, bisector, midline, height and isosceles triangle).
⑵ Draw a picture according to the meaning of the question, and use digital symbols to indicate the known and verified.
(3) After analysis, find out the method of proof from the known and write the proof process.
Chapter 13 Axisymmetric
I. Knowledge framework:
Second, the concept of knowledge:
1. Basic concepts:
(1) Axisymmetric graph: If a graph is folded along a straight line, the parts on both sides of the straight line can overlap each other, and this graph is called an axisymmetric graph.
⑵ Two graphs are symmetrical: one graph is folded along a straight line, and if it can overlap with another graph, the two graphs are said to be symmetrical about this straight line.
⑶ Midline of the line segment: The line passing through the midpoint of the line segment and perpendicular to the line segment is called the midline of the line segment.
⑷ isosceles triangle: A triangle with two equal sides is called an isosceles triangle. Two equal sides are called waist and the other side is called bottom. The angle between the two waists is called the top angle, and the angle between the buttocks and the waist is called the bottom angle.
5. equilateral triangle: A triangle with three equilateral sides is called an equilateral triangle.
2. Basic nature:
The essence of (1) symmetry;
(1) whether an axisymmetric figure or two figures are symmetrical about a straight line, and the symmetry axis is the middle perpendicular of the line segment connected by any pair of corresponding points.
② Symmetric figures are congruent.
(2) The nature of the vertical line in the line segment:
① The distance between the point on the vertical line of a line segment and the two endpoints of the line segment is equal.
② The point with the same distance from the two endpoints of a line segment is on the middle vertical line of this line segment.
(3) Coordinate properties of axisymmetrical points.
① The coordinate of the point P(x, y) about the axis symmetry of X is P'(X, y).
(2) The coordinate of the point P(x, y) which is symmetrical about the Y axis is p "(x, y).
(4) the nature of isosceles triangle:
The waist of an isosceles triangle is equal.
② The two base angles of an isosceles triangle are equal (equilateral and equiangular).
③ The bisector of the top corner of the isosceles triangle, the median line on the bottom edge and the height on the bottom edge coincide. ④ The isosceles triangle is an axisymmetric figure, and the symmetry axis is the combination of three lines (1).
5] the properties of equilateral triangle:
All three sides of an equilateral triangle are equal.
② All three internal angles of an equilateral triangle are equal and equal to 60?
③ There are three lines on each side of an equilateral triangle.
An equilateral triangle is an axisymmetric figure, and the symmetry axis is the combination of three lines (3 lines).
3. Basic judgment:
Determination of (1) isosceles triangle;
A triangle with equal sides is an isosceles triangle.
If the two angles of a triangle are equal, then the opposite sides of the two angles are equal.
(2) Determination of equilateral triangle:
A triangle with three equilateral sides is an equilateral triangle.
A triangle with three equal angles is an equilateral triangle.
③ There is an angle of 60? An isosceles triangle is an equilateral triangle.
4. Basic methods:
(1) perpendicular to a known straight line:
(2) The midline of the known line segment:
(3) Symmetry axis: connect two corresponding points and make the middle perpendicular of the connecting line segment.
(4) Make a symmetrical figure of a known figure about a straight line:
5. Make a point on the straight line so that the sum of the distances from it to two known points on the same side of the straight line is the shortest.
Chapter 14 multiplication, division and factorization of algebraic expressions.
I. Knowledge framework:
Second, the concept of knowledge:
1. Basic operation:
(1) Multiplication with Base Number
(2) the power of power
(3) the power of the product
2. Calculation formula:
(1) variance formula
⑵ Complete square formula
3. Factorization: It is called factorization of this formula to convert a polynomial into the product of several algebraic expressions.
4. Factorization method:
(1) Improving the common factor method: find the greatest common factor.
(2) Formula method:
① Variance formula
Math learning methods in grade two
(1) Explore concepts and formulas carefully.
Many students pay insufficient attention to concepts and formulas. This problem is reflected in three aspects: first, the understanding of the concept only stays on the surface of the text, and the special situation of the concept is not paid enough attention. For example, the concept of algebraic expression (an expression expressed by letters or numbers is algebraic expression) has been neglected by many students? Is a single letter or number algebraic? . Second, concepts and formulas are blindly memorized and have nothing to do with practical topics. The knowledge learned in this way can't be well connected with solving problems. Third, some students do not pay attention to the memory of mathematical formulas. Memory is the basis of understanding. If you can't memorize the formula, how can you skillfully use it in the topic?
Our suggestions are: be more careful (observe special cases), go deeper (know the common test sites in the topic), and be more skilled (we can use it freely no matter what it looks like).
(2) Summarize similar topics.
This work is not only for teachers, but also for our classmates. When you can summarize the topics, classify the topics you have done, and know what types of problems you can solve, what common problem-solving methods you have mastered, and what types of problems you can't do, you will really master the tricks of this subject and really do it. Let it change, I will never move? . If this problem is not solved well, after entering the second and third grades, students will find that some students do problems every day, but their grades will fall instead of rising. The reason is that they do repetitive work every day, and many similar problems are repeated, but they can't concentrate on solving the problems that need to be solved. Over time, the problems that can't be solved have not been solved, and the problems that can be solved have also been messed up because of the lack of overall grasp of mathematics.
Our suggestion is: summarize? Asking fewer and fewer questions is the best way.
(3) Collect your typical mistakes and solve the problems that you can't solve.
The most difficult thing for students is their own mistakes and difficulties. But this is precisely the problem that needs to be solved most. There are two important purposes for students to do problems: First, to practice the knowledge and skills they have learned in practical problems. The other is to find out your own shortcomings and make up for them. This deficiency also includes two aspects, mistakes that are easy to make and contents that are completely unknown. However, the reality is that students only pursue the number of questions and deal with their homework hastily, rather than solving problems, let alone collecting mistakes. We suggest that you collect your typical mistakes and problems that you can't do, because once you do, you will find that you thought you had many small problems before, but now you find this one is recurring; You thought you didn't understand many problems before, but now you find that these key points have not been solved.
Our suggestion is: doing problems is like digging gold mines. Every wrong question is a gold mine. Only by digging and refining can we gain something.
(4) Ask and discuss questions that you don't understand.
Find problems you don't understand and actively ask others for advice. This is a very common truth. But this is what many students can't do. There may be two reasons: first, insufficient attention has been paid to this issue; Second, I'm sorry, I'm afraid of asking teachers to be trained and asking students to be looked down upon by them. With this mentality, you can't learn anything well. ? Close the door? It will only make your problems more and more. Knowledge itself is coherent, the previous knowledge is unclear, and it will be more difficult to understand later. When these problems accumulate to a certain extent, you will gradually lose interest in the subject. Until I can't keep up.
Discussion is a very good learning method. A difficult topic, after discussion with classmates, may get good inspiration and learn good methods and skills from each other. It should be noted that it is best to discuss with your classmates at the same level, and everyone can learn from each other.
Our suggestion is: study hard? Is the foundation? Good question? Is the key.
(5) Pay attention to the cultivation of actual combat (examination) experience.
Examination itself is a science. Some students usually get good grades. Teachers ask questions in class, and they can do anything. I can also do problems after class. But when it comes to the exam, the results are not ideal. There are two main reasons for this: first, the test mentality is not bad, and it is easy to be nervous; Second, the examination time is tight and it can never be completed within the specified time. Bad mentality, on the one hand, we should pay attention to our own adjustment, but at the same time we also need to exercise through large-scale exams. Every exam, everyone should find a suitable adjustment method and gradually adapt to the rhythm of the exam with the passage of time. The problem of slow problem solving needs students to solve in their usual problem solving. Doing homework at ordinary times can limit time and gradually improve efficiency. In addition, in the actual exam, we should also consider the completion time of each part to avoid unnecessary panic.
Our suggestion is: release? Do your homework? As an exam, right? Exam? As homework.
Suggestions on mathematics learning in senior two.
1, preview method
Preview is to read the upcoming math content before class, so as to have a good idea and grasp the initiative in class. This is conducive to improving learning ability and forming the habit of self-study, so it is an important part of mathematics learning.
(1) Read and write. (No pen and ink, no reading)
(1) Reading, thinking and writing are generally used to draw out or mark the main points, levels and connections of the content, write down your own opinions or mark the places and problems that you don't understand;
(2) Once you find that you don't master the old knowledge well or even understand it, you should turn over the books in time and take measures to make up for it, so as to create conditions for learning new content smoothly.
(3) Understand the basic content of this lesson, that is, know what to talk about, what problems to solve, what methods to adopt, where the key points are, and so on.
Take out the chapters corresponding to a workbook and read them roughly to see which questions can be read at once and which questions can't be understood at all, and then go to class with questions.
(2) Determine the main points of the lecture. Grasping the main problems you want to solve can improve the efficiency of class.
2, the method of listening to lectures
Listening to classes is the main form of learning mathematics. With the guidance, inspiration and help of teachers, we can make fewer detours, reduce difficulties and acquire a large number of systematic mathematical knowledge in a short time, otherwise we will get twice the result with half the effort and it is difficult to improve efficiency. So attending classes is the key to learning math well.
(1) Keep an eye on the teacher. In addition to the clear tasks in the preview, we should also solve our own problems in a targeted manner and keep up with the teacher's lectures, such as how the theorem was discovered or produced, how the idea of proof was worked out, and what key places need to be broken through. How to use formulas and theorems? Many mathematicians emphasize it? You should not only see what is written, but also see what is behind the book. ?
(2) Dare to speak. When listening to the class, on the one hand, we should understand what the teacher said, think or answer the questions raised by the teacher, on the other hand, we should think independently. If you have any questions or new problems, you should be brave enough to put forward your own opinions.
(3) take notes. Write down the main points, supplementary contents and methods of the teacher's lecture in class.
Step 3 review methods
Review is to learn the learned mathematical knowledge again, so as to achieve the purpose of in-depth understanding, mastery, refinement and generalization, and firm grasp. Review should be closely linked with lectures, and the contents of lectures should be recalled while reading textbooks or checking class notes, so as to solve the existing knowledge defects and problems in time.
(1) Review notes and roll paper. Try to understand the content of learning and really understand and master it. We should not stop at the requirements of reviewing and memorizing what we have learned, but should think hard about how new knowledge is produced, how it is developed or proved, what its essence is, and how to expand and broaden it through application. Be diligent in reviewing (knowledge points, typical questions, etc.). ), and often watch it repeatedly-this is the truth revealed by Ebbinghaus forgetting curve in psychology. Students are advised to show movies. After finishing your homework, close your books and notes, recall the contents of the class, such as rules, formulas, problem-solving ideas and methods, and reproduce them as completely as possible in your mind. Then open the textbook and compare the notes, and focus on reviewing the missing knowledge points. This not only consolidated the content of the class that day, but also checked the missing items.
(2) Do the questions moderately. Prepare a mistake book, record the mistakes you have made and practice again. For the topic I did wrong, recall why and where I was wrong. Where I make mistakes is often my weakness. It is not enough to correct it for a while, but also to carry out appropriate intensive training.
(3) Dare to question and enhance the initiative of learning. Always study with classmates or ask teachers, and don't accumulate too many questions. Don't put all the questions you can't know in class and wait for the teacher to tell you.
4, the method of doing homework
Mathematics learning is often to consolidate knowledge, deepen understanding and learn to use it by doing homework, thus forming skills and developing intelligence and mathematical ability. Because the homework is done independently on the basis of review, you can check your mastery of the mathematics knowledge you have learned, examine your ability level, and find out the existing problems and difficulties. When there are many wrong questions, it often indicates that there are defects or problems in the understanding and mastery of knowledge, which should arouse vigilance and need to find out the reasons and solve them as soon as possible.
(1) Review before you do your homework. You need to review before you do your homework, and then do it on the basis of having a basic understanding and mastery of the textbooks you have learned. Otherwise, you will get twice the result with half the effort, take time and get the desired result.
(2) Must be done independently. Develop good habits, do your homework neatly, and pay attention to the problem-solving format. Writing norms. Homework must be done independently. High-quality homework can cultivate a sense of responsibility for independent thinking and correct problem solving.
(3) short time and high efficiency. Set a specific time during which nothing is allowed except doing homework. Loose thinking and unfocused work habits are harmful to improving math ability.
(4) Check carefully. Prepare a red pen, tick it correctly, do it again if it is different, check whether you have done it right or not, and ask teachers and classmates some questions that you can't or can't scream.
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