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Outline of knowledge points of mathematics circle in the second volume of ninth grade
Thinking method of mathematics learning
Mathematics learning method
Outline of knowledge points of mathematics circle in the second volume of ninth grade
1. A circle is a set of points whose distance from a fixed point is equal to a fixed length.
2. The interior of a circle can be regarded as a collection of points whose center distance is less than the radius.
3. The outside of a circle can be regarded as a collection of points whose center distance is greater than the radius.
4. The same circle or the same circle has the same radius.
5. The distance to the fixed point is equal to the trajectory of the fixed-length point, which is a circle with the fixed point as the center and the fixed length as the radius.
6. The locus of a point whose distance is equal to the two endpoints of a known line segment is the median vertical line of this line segment.
7. The locus of a point with equal distance to both sides of a known angle is the bisector of this angle.
8. The locus of a point with equal distance to two parallel lines is a straight line parallel to these two parallel lines and with equal distance.
9. Theorem Three points that are not on the same straight line determine a circle.
10, the vertical diameter theorem bisects the chord perpendicular to the diameter of the chord and bisects the two arcs opposite the chord.
1 1, inference 1:
(1) bisects the diameter (not the diameter) of the chord perpendicular to the chord and bisects 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.
12, Inference 2: The arcs sandwiched by two parallel chords of a circle are equal.
13. A circle is a centrosymmetric figure with the center of the circle as the symmetry center.
14, Theorem: In the same circle or in the same circle, the isocentric angle has equal arc, chord and chord center distance.
15, inference: If one set of quantities in the same circle or equal circle, two central angles, two arcs, two chords or the distance between two chords are equal, the corresponding other set of quantities are also equal.
16, theorem: the angle of an arc is equal to half of its central angle.
17, inference: the circumferential angles of the same arc or equal arc are equal; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.
18, inference: the circumference angle (or diameter) of a semicircle is a right angle; A chord with a circumferential angle of 90 is a diameter.
19, inference: If the median line of one side of a triangle is equal to half of this side, then this triangle is a right triangle.
Theorem: Diagonal lines of inscribed quadrangles of a circle are complementary, and any external angle is equal to its internal angle.
2 1, ① intersection of straight line l and ⊙O D R
(2) the tangent of the straight line l, and ⊙ o d = r.
③ straight lines l and ⊙O are separated by d r.
22. Judgment theorem of tangent: The straight line passing through the outer end of the radius and perpendicular to this radius is the tangent of the circle.
23. Tangent Theorem: The tangent of a circle is perpendicular to the radius passing through the tangent point.
24. Inference: A straight line passing through the center of the circle and perpendicular to the tangent must pass through the tangent point.
25. Inference: A straight line passing through the tangent point and perpendicular to the tangent line must pass through the center of the circle.
26. Tangent Length Theorem: Two tangents leading from a point outside the circle are equal in length, and the connecting line between the center of the circle and this point bisects the included angle of the two tangents.
27. The sum of two opposite sides of a circle's circumscribed quadrilateral is equal.
28. Chord tangent angle theorem: the chord tangent angle is equal to the circumferential angle of the arc pair it clamps.
29. Inference: If the arc enclosed by two chord tangent angles is equal, then the two chord tangent angles are also equal.
30. Theorem of intersecting chords: The length of two intersecting chords in a circle divided by the product of the intersection point is equal.
3 1, inference: if the chord intersects the diameter vertically, then half of the chord is the average of the ratio of the two line segments formed by dividing it by the diameter.
32. Secant theorem: The tangent and secant of a circle are drawn from a point outside the circle, and the length of the tangent is the middle term in the ratio of the lengths of the two lines from this point to the intersection of the secant and the circle.
33. Inference: The product of two secant lines leading from a point outside the circle to the intersection of each secant line and the circle is equal.
34. If two circles are tangent, then the tangent point must be on the connecting line.
35. ① The distance between two circles is d-r+r.
(2) circumscribed circle d d = r+r.
③ Two circles intersect R-r﹤d﹤R+r(R﹥r).
④ inscribed circle D = r-r (r-r)
⑤ Two circles contain d￠R-R(R￠R).
Theorem: The intersection line of two circles bisects the common chord of two circles vertically.
37. Theorem: Divide the circle into n (n ≥ 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.
38, theorem:
Any regular polygon has a circumscribed circle and an inscribed circle, which are concentric circles.
39. Each inner angle of a regular N-polygon is equal to (n-2) ×180/n.
40. Theorem: The radius and vertex of a regular N-polygon divide the regular N-polygon into 2n congruent right triangles.
4 1, the area of the regular N-polygon Sn=pr/2p represents the perimeter of the regular N-polygon, and r is apothem.
42. The area of a regular triangle √3a2/4a indicates the side length.
43. If there are K positive N corners around a vertex, the sum of these corners should be 360, so
K (n-2) 180/n = 360 was changed to (n-2)(k-2)=4.
44. Calculation formula of arc length: L = nσR/ 180.
45, fan area formula:
South sector = North R2/360=LR/2
Outer common tangent length =d-(R+r)
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Thinking method of mathematics learning
Comparative method.
By comparing the similarities and differences between mathematical conditions and problems, we study the reasons for the similarities and differences, so as to find a solution to the problem, which is the comparative method.
Comparative law should pay attention to:
(1) Finding similarities means finding differences, and finding differences means finding similarities, and being indispensable means being complete.
(2) Find the connection and difference, which is the essence of comparison.
(3) Comparison must be conducted under the same relationship (same standard), which is the basic condition of "comparison".
(4) To compare the main contents, try to use the "exhaustion method" as little as possible, which will make the key points less prominent.
(5) Because of the rigor of mathematics, comparison must be meticulous, and often a word or a symbol determines the right or wrong conclusion of comparison.
2. Formula method
Methods to solve problems by using laws, formulas, rules and rules. It embodies the deductive thinking from general to special. Formula method is simple and effective, and it is also a method that children must learn and master when learning mathematics. But children must have a correct and profound understanding of formulas, laws, rules and rules, and can use them accurately.
3. Logical method
Logic is the foundation of all thinking. Logical thinking is a thinking process in which people observe, compare, analyze, synthesize, abstract, generalize, judge and reason things with the help of concepts, judgments and reasoning in the process of cognition. Logical thinking is widely used to solve logical reasoning problems.
4. Reverse thinking method
Reverse thinking, also known as divergent thinking, is a way of thinking about common things or opinions that seem to have become conclusive. Dare to "do the opposite", let thinking develop in the opposite direction, conduct in-depth exploration from the opposite side of the problem, establish new concepts and shape new images.
Step 5 classify
According to the similarities and differences of things, things are divided into different categories, which is called classification. Classification is based on comparison. According to the * * * similarity between things, they are grouped into larger classes, and the larger classes are subdivided into smaller classes according to differences.
Classification is to pay attention to the different levels between categories and subcategories to ensure that subcategories in categories are not duplicated, omitted or crossed.
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Mathematics learning method
1. Pay attention to preview and cultivate self-study ability.
When previewing, we should separate theorems, laws, formulas, constants and specific symbols, and copy them every time to deepen our impression. In class, when the teacher talks about these places, he should compare his understanding in preview with that of the teacher to see if there is any misunderstanding. Preview can adopt the preview method of "one stroke, two batches, three trials and four points".
One stroke: it is to circle the main points of knowledge and basic concepts.
The second batch is to annotate the experiences, opinions and contents that you can't understand for the time being in the blank space of the book.
Test 3: Try to do some simple exercises to test the effect of your preview.
Four points: that is, list the main points of this section of knowledge that you have previewed, and distinguish which knowledge is mastered through preview and which is incomprehensible through preview, which needs further study in classroom learning.
2. Grasp the classroom and improve the learning effect.
Classroom learning is the most basic and important link in the learning process, and we should adhere to the "five arrivals", that is, listening, watching, speaking, feeling and reaching.
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;
Ears: Listen attentively, listen to the teacher how to lecture, how to analyze and how to summarize. In addition, we should listen to the students' answers to see if they are enlightening, especially the questions that we 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.
3. 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 realize 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), to develop a good habit of thinking first, then solving, and then checking, you can't blindly practice when you encounter problems, and the calculation is invalid. You should first deeply understand the meaning of the question, think carefully, grasp the key, and then answer. After you answer, you should also check it.
4. 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 and strike while the iron is hot. You must review the lessons you have learned that day, no matter how difficult the homework is, you must consolidate the review.
(2) Adopting the comprehensive review method, that is, by finding out the left-right relationship and criss-crossing internal relations of knowledge, the overall improvement can be divided into "three steps": first, looking at the overall situation, browsing all the contents, and initially forming an impression on the knowledge system by evoking memories; Second, deepen understanding, comprehensively analyze what you have learned, and finally, consolidate and form a complete knowledge system.
(3) Review methods and 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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