Key knowledge points of mathematics in grade three
1. Three points that are not on the same straight line determine a circle.
2. The vertical diameter theorem bisects the chord perpendicular to its diameter and bisects the two arcs opposite the chord.
Inference 1
(1) bisecting the chord is not perpendicular to the diameter of the chord, and bisecting the two arcs opposite to the chord.
(2) The perpendicular line of the chord passes through the center of the circle and bisects the two arcs opposite to the chord.
③ bisect the diameter of an arc opposite to the chord, bisect the chord vertically, and bisect another arc opposite to the chord.
Inference 2 The arcs between two parallel chords of a circle are equal.
3. A circle is a central symmetrical figure with the center of the circle as the symmetrical center.
A circle is a point whose distance from a point to a fixed point is equal to a fixed length.
5. The interior of a circle can be regarded as the * * * of a point whose center is smaller than the radius.
6. The excircle of a circle can be regarded as a * * * circle with the center distance greater than the radius.
7. The same circle or the same circle has the same radius.
8. The distance to a fixed point is equal to the trajectory of a fixed-length point, which is a circle with the fixed point as the center and the fixed length as the radius.
9. Theorem In the same circle or in the same circle, the isocentric angle has equal arc, chord and chord center distance.
10. It is inferred that in the same circle or equal circle, if one set of quantities in two central angles, two arcs, two chords or the chord-center distance between two chords is equal, the corresponding other set of quantities is also equal.
1 1 Theorem The diagonals of the inscribed quadrilateral of a circle are complementary, and any external angle is equal to its internal diagonal.
12.① intersection point d of straight line l and ⊙O
(2) the tangent of the straight line l, and ⊙ o d = r.
③ Lines L and ⊙O are separated from each other d>r.
13. The judgment theorem of tangent is that the outer end of the radius and the straight line perpendicular to this radius are the tangents of the circle.
14. The property theorem of tangent. The tangent of a circle is perpendicular to the radius passing through the tangent point.
15. Inference 1 A straight line passing through the center of the circle and perpendicular to the tangent must pass through the tangent point.
16. Inference 2 A straight line that crosses the tangent point and is perpendicular to the tangent must pass through the center of the circle.
17. The tangent length theorem leads to two tangents of a circle from a point outside the circle. Their tangents are equal in length, and the connecting line between the center of the circle and this point bisects the included angle between the two tangents.
18. The sum of two opposite sides of the circumscribed quadrangle of a circle is equal, and the outer angle is equal to the inner diagonal.
19. If two circles are tangent, then the tangent point must be on the line.
20.① Two circles are separated by d>. R+r
(2) circumscribed circle d d = r+r.
(3). The intersection of two circles
④ inscribed circle d = R-rR & gt；; R ⑤ Two circles contain Dr.
2 1. Theorem The intersection line of two circles bisects the common chord of two circles vertically.
22. Theorem divides a circle into nn≥3:
(1) The polygon obtained by connecting the points in turn is the inscribed regular N polygon of this circle.
(2) The tangent of a circle passing through each point, and the polygon whose vertex is the intersection of adjacent tangents is the circumscribed regular N polygon of the circle.
23. Theorem Any regular polygon has a circumscribed circle and an inscribed circle, which are concentric circles.
24. Each inner angle of a regular N-polygon is equal to n-2×180/n.
25. Theorem The radius and vertex of a regular N-polygon divide the regular N-polygon into 2n congruent right triangles.
26. The area of a regular N-polygon Sn=pnrn/2 p represents the perimeter of the regular N-polygon.
27. The regular triangle area √3a/4 a indicates the side length.
28. If there are k positive N corners around a vertex, since the sum of these corners should be 360, then k× n-2 180/n = 360 becomes n-2k-2=4.
29. Calculation formula of arc length: L = nσR/ 180.
30. Sector area formula: s sector =n r 2/360 = LR/2.
3 1. Inner common tangent length = d-R-r outer common tangent length = D-R+R.
32. Theorem The angle of an arc is equal to half of its central angle.
33. Inference 1 is equal to the circumferential angle of the same arc or equal arc; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.
34. Inference 2 The circumferential angle of a semicircle or diameter is a right angle; A chord with a circumferential angle of 90 is a diameter.
35. the arc length formula l=a*r a is the radian number of the central angle r >; 0 sector area formula s= 1/2*l*r
Review skills of mathematics in grade three
Pay attention to textbook knowledge
We have completed the first stage of reviewing basic knowledge and strengthening basic skills training. In the second stage of review, we will reflect and summarize the omissions and deficiencies in the last round of review, and we will find that some knowledge has not been mastered well and there is no idea when solving problems, so we should further classify the knowledge and deepen our memory while reviewing. We should further understand the connotation and extension of concepts, firmly grasp the derivation or proof of laws, formulas and theorems, and further strengthen the ideas and methods of solving problems; At the same time, we should also ask some similar questions for intensive training, fill in the blanks in a timely and targeted manner, and never give up easily until we truly understand and do it.
At this stage, it is particularly important to review textbooks, because the examples and exercises in textbooks are important components of textbooks and the main carriers of mathematical knowledge. Only by thoroughly understanding the examples and exercises in the textbook can we master the basic knowledge of mathematics and master the basic methods of mathematics comprehensively and systematically, so as to keep constant and change. Therefore, when reviewing, we should learn to examine these examples from multiple directions and angles, from which we can further clearly grasp the basic knowledge, review the thinking process, consolidate various solutions and understand the mathematical thinking method. There are various forms of review, especially to improve review efficiency.
In addition, at present, the proposition of the senior high school entrance examination is still based on basic questions, some of which are original or modified questions in the textbook, and some big questions are "higher than the textbook", but the prototype is generally an example or exercise in the textbook, which is an extension, deformation or combination of the questions in the textbook. Examples, exercises and homework in textbooks should not only be understood, but also done. At the same time, we should also pay attention to reading textbooks, researching topics, doing some things and thinking about things in textbooks.
Pay attention to classroom learning
Under the guidance of teachers, through classroom teaching, students are required to master the internal relationship between knowledge points, clarify the knowledge structure and form an overall understanding. Through the systematic induction of basic knowledge and the classification of problem-solving methods, they can deepen their memory on the basis of forming knowledge structure. At the very least, they should accurately grasp the meaning of each concept, sort out the vague concepts in their usual study, master the knowledge more firmly, and let themselves know the position of each knowledge point in the whole junior high school mathematics. If you want to attend classes and take notes, you should grasp the key points of knowledge in each class, solve problems, improve learning efficiency, and timely check for leaks and fill gaps according to your own specific situation.
Consolidate basic knowledge
In mathematics examination questions over the years, basic scores account for the most, plus basic scores in some intermediate and difficult questions, so scores account for a larger proportion. We must lay a solid foundation. Through systematic review, we can meet the requirements of "understanding" and "mastering" junior high school mathematics knowledge, and we can skillfully, correctly and quickly apply basic knowledge.
Some questions will create new question situations for the knowledge and methods to be examined, especially for some questions that need to be highly discriminated; Each math test with medium or above difficulty usually involves multiple knowledge points and multiple mathematical thinking methods, or skillfully designs the test questions at the intersection of knowledge. Therefore, each of our classmates should learn to think. What the teacher teaches us in class is the angle, method and strategy of thinking. We should use the methods and strategies we have learned to understand how to think correctly in the process of solving problems in the new situation.
Pay attention to the transfer of knowledge
Some examples and exercises in the textbook are not isolated, but closely related. The knowledge of other disciplines is also inextricably linked with mathematics. We should learn to discover, study and show the internal relations of these knowledge from the closest point of thinking development, which will not only help us to deeply understand the textbook knowledge, but also help us to strengthen the knowledge focus. More importantly, it can effectively promote the construction of our own mathematical knowledge network and method system. Knowledge and ability can be transferred in a benign way, so as to achieve the effect of drawing inferences from others. By exploring the internal relationship between typical examples and exercises in textbooks, we can form a knowledge network and method system more effectively while deeply understanding the knowledge of textbooks. For example, the discriminant of the root of a quadratic equation can not only solve the problems of determining the root and finding the letter coefficient when the root is known, but also solve the factorization of the quadratic trinomial, the determination of the root of the equation group and the coordinates of the intersection of the quadratic function image and the horizontal axis.
Mathematics review plan for grade three
The first stage: combing knowledge and forming knowledge network.
1, the form of the first round of review, with the explanation of the senior high school entrance examination as the main line, pays attention to combing the basic knowledge.
The first round of review should "go through three levels":
1 Pass the memory pass. We must remember all the formulas, theorems, etc.
Through basic methods. For example, use the undetermined coefficient method to find the analytical expression of quadratic function.
3 basic skills. For example, the combination of numbers and shapes requires drawing and making.
2. Several problems that should be paid attention to in the first round of review.
1 Be sure to lay a solid foundation. Generally, the ratio of easy: easy: medium: difficult is 4: 3: 2: 1, which requires proficiency, correctness and rapidity in the application of basic knowledge.
Some basic questions in the senior high school entrance examination are original or modified in the textbooks and explanations, so we must dig deep into the textbooks and explanations, and we must not aim too high.
3 don't engage in sea tactics, concentrate on perfection, draw inferences from one another, and touch the analogy. "A lot of practice" is relative, and intensive practice should be targeted, typical, hierarchical and to the point.
4 more induction, more summary.
The second stage: special review
1, the form of the second round of review is no longer based on sections, chapters and units, but on topics.
On the basis of a round of review, teachers should be promoted, concentrated and classified, focusing on key points, difficulties and hot spots, and paying attention to the formation of mathematical ideas and the mastery of mathematical methods, which requires giving full play to teachers' leading role.
2. Several problems that should be paid attention to in the second round of review.
1 For the second round of review, you can set special topics for the difficulties and delays you usually encounter.
The division of special topics should be reasonable and representative, and it is forbidden to cover all aspects; Focus on hot spots, difficulties and key points, work hard at important points, and do not hesitate to "waste" time and be willing to invest energy.
Substituting questions for knowledge, students are far away from the basic knowledge to a certain extent, which will lead to different degrees of knowledge forgetting. The best way to solve this problem is to replace knowledge with problems. You can appropriately insert past little knowledge points to evoke memories.
4. The special review can be appropriately heightened. Without certain difficulty, it is difficult to improve one's ability. Improving one's learning ability is the task of the second round of review. But not too much and too difficult.
The third stage: comprehensive training
1, the form of the third round of review is to simulate the comprehensive practice of the senior high school entrance examination, check for missing parts, commonly known as pre-test training. Train answering skills, examination room mentality, improvisation ability, etc.
2. Several problems that should be paid attention to in the third round of review.
1 The simulation questions must have the characteristics of simulation. The arrangement of time, the number of questions and the proportion of low, medium and high questions should be close to the senior high school entrance examination model.
2 collect wrong questions, check for leaks and fill in gaps.
3 "Liberate" yourself appropriately, especially in the time arrangement. However, it should be noted that liberation is not relaxation, and the amount of questions in the later period should not be too large. Solving problems should be easy, condescending, and jumping out of the review circle to see problems.
4 adjust the biological clock. Try to adjust the time of study and thinking to be consistent with the time of the answer sheet for the senior high school entrance examination.
5 mentality and confidence adjustment. Keep a normal heart.
The fourth stage: check for leaks and fill gaps.
Return the knowledge that you are still vague or forgotten to the textbook, further consolidate and deepen, and meet the senior high school entrance examination.
In short, in the general review of mathematics in grade three, it is fundamental to explore teaching materials and lay a solid foundation; * * * With participation, the focus process is the premise; Selecting exercises, improving quality and reducing burden are the core; Strengthening training and developing ability is the goal. Only in this way can we change with the same, and get twice the result with half the effort with the Belt and Road Initiative.
1. Summary of mathematical knowledge points in the first volume of Grade Three.
2. Summary of mathematics knowledge points in senior high school entrance examination
3. Key knowledge points of junior high school mathematics
4. The third grade mathematics knowledge arrangement
5. Mathematics in Grade Three always reviews knowledge points.