The inductive circle of mathematical knowledge points in the second volume of the ninth grade
★ Key points ① Important properties of the circle; (2) the positional relationship between straight lines and circles, and between circles; ③ Angle theorem related to circle; ④ Theorem of proportional line segment related to circle.
☆ Summary ☆
First, the basic properties of the circle
Definition of 1. circle (two kinds)
2. Related concepts: chord and diameter; Arc, equal arc, upper arc, lower arc, semicircle; Distance from chord to center; Equal circle, same circle, concentric circle.
3. "Three-point circle" theorem
4. Vertical Diameter Theorem and Its Inference
5. "Equivalence" theorem and its inference
6. Angle related to circle: (1) Definition of central angle (equivalence theorem)
(2) the definition of the angle of circle (the theorem of the angle of circle, the relationship with the angle of center)
⑶ Definition of chord angle (chord angle theorem)
Second, the positional relationship between a straight line and a circle
1. Properties of Tangents (Key Points)
2. Tangent (key point) judgment theorem
3. Tangent length theorem
Third, the position relationship between circles.
1. Five positional relationships and their judgments and properties: (emphasis: tangency)
2. The property theorem of the tangent (intersection line) connecting two circles.
3. Common tangent of two circles: (1) Definition (2) Property
Four, proportional line segment related to the circle
1. Intersecting chord theorem
2. Cutting line theorem
Verb (abbreviation for verb) and regular polygon
The inscribed and circumscribed polygons of 1. circle (triangle, quadrilateral)
2. The circumscribed circle, inscribed circle and properties of triangle.
3. The properties of circumscribed quadrangles and inscribed quadrangles of a circle
4. Regular polygon and its calculation
Central corner: review outline of junior high school mathematics
Half of the Inner Corner: Review Outline of Junior High School Mathematics (right)
(Solve Rt△OAM to find out the relevant elements, junior high school mathematics review outline, junior high school mathematics review outline, etc. )
Six, a set of calculation formulas
1. circumference formula
2. Formula of circular area
3. Sector area formula
4. Arc length formula
5. Calculation method of arch area
6. The side development diagram of cylinder and cone and related calculation.
The third day of the second volume of mathematics knowledge points summary I. Acute trigonometric function
Sine equals the hypotenuse of the opposite side.
Cosine is equal to the ratio of adjacent side to hypotenuse.
The tangent is equal to the opposite side of the adjacent side.
Cotangent equals the comparison of adjacent edges.
Secant is equal to the hypotenuse than the adjacent edge.
Second, the calculation of trigonometric function
power series
c0+c 1x+c2x2+...+cnxn+...=∑cnxn(n=0..∞)
c0+c 1(x-a)+c2(x-a)2+...+cn(x-a)n+...=∑cn(x-a)n(n=0..∞)
Their terms are power functions of positive integer powers, where c0, c 1, c2, ... communication network (abbreviation of Communicating Net) ... and A are constants, and this series is called power series.
Taylor expansion (power series expansion method)
f(x)=f(a)+f'(a)/ 1！ . (x-a)+f''(a)/2！ . (x-a)2+...f(n)(a)/n！ . (x-a)n+ ...
Third, solve the right triangle
1. The two acute angles of a right triangle are complementary.
2. The three high intersections of a right triangle are at the same vertex.
3. Pythagorean theorem: the sum of squares of two right angles is equal to the square of hypotenuse.
Fourthly, measure the height with trigonometric function.
1, the application of solving right triangle
(1) Many related measurement problems in practical problems can be solved by solving right triangles.
For example, to measure the height and river width of an object that is difficult to measure directly, the key is to construct a right triangle. By measuring the degree of the angle and the length of the side, the required height or length of the object can be calculated.
(2) The general process of solving a right triangle is:
(1) abstract practical problems into mathematical problems (draw a plane figure, construct a right triangle, and transform it into solving right triangle problems).
(2) According to the known characteristics of the topic, choose the appropriate acute trigonometric function or angular relationship to solve the right triangle, get the answer to the mathematical problem, and then turn it into the answer to the actual problem.
Math learning methods in grade three 1. What should be remembered, what should be remembered. Don't think you can understand it.
Some students think that mathematics is not like English, history and geography. Words, dates, and place names are required. Mathematics depends on wisdom, skill and reasoning. I said you were only half right. Mathematics is also inseparable from memory. Imagine, elementary school addition, subtraction, multiplication and division and Divison, can you operate smoothly without memorizing the multiplication table? Although you understand that multiplication is the operation of the sum of the same addend, when you do 9.9, you add 9 9s to get 8 1, which is too uneconomical. It is much more convenient to use "998 1". Similarly, it is also made with the rules that everyone knows by heart. At the same time, there are many laws in mathematics that need to be memorized, such as law (a≠0) and so on. So, I think mathematics is more like a game. It has many rules of the game (that is, definitions, rules, formulas, theorems, etc. Whoever remembers these rules of the game will be able to play the game smoothly. Whoever violates these rules of the game will be judged wrong and sent off. Therefore, mathematical definitions, rules, formulas, theorems, etc. Must recite, some can recite, catchy. For example, the familiar "Three Formulas of Algebraic Multiplication", I think some of you here can recite it, while others can't. Here, I want to remind the students who can't recite these three formulas. If they can't recite it, it will cause great trouble for future study, because these three formulas will be widely used in future study, especially the factorization of senior two, in which three very important factorization formulas are all derived from these three multiplication formulas, and they are deformations in opposite directions.
Remember the definitions, rules, formulas and theorems of mathematics, and remember those that you don't understand for the time being, and deepen your understanding on the basis of memory and application to solve problems. For example, mathematical definitions, rules, formulas and theorems are just like axes, saws, Mo Dou and planers in the hands of carpenters. Without these tools, carpenters can't make furniture. With these tools, coupled with skilled craftsmanship and wisdom, you can make all kinds of exquisite furniture. Similarly, if you can't remember the definition, rules, formulas and theorems of mathematics, it is difficult to solve mathematical problems. And remember these, plus certain methods, skills and agile thinking, you can be handy in solving mathematical problems, even solving mathematical problems.
Second, several important mathematical ideas
1, the idea of "equation"
Mathematics studies the spatial form and quantitative relationship of things. The most important quantitative relationship in junior high school is equality, followed by inequality. The most common equivalence relation is "equation". For example, uniform motion, distance, speed and time are equivalent, and a related equation can be established: speed and time = distance. In this equation, there are generally known quantities and unknown quantities. An equation containing unknown quantities like this is an "equation", and the process of finding the unknown quantities through the known quantities in the equation is to solve the equation. We were exposed to simple equations in primary school, but in the first year of junior high school, we systematically studied the solution of one-dimensional linear equations and summarized five steps of solving one-dimensional linear equations. If you learn and master these five steps, any one-dimensional linear equation can be solved smoothly. In the second and third day of junior high school, you will also learn to solve one-dimensional quadratic equations, binary quadratic equations and simple triangular equations. In high school, we will also learn exponential equation, logarithmic equation, linear equation, parametric equation, polar coordinate equation and so on. The solution ideas of these equations are almost the same, and they are all transformed into the form of linear equations or quadratic equations in one variable by certain methods, and then solved by the familiar five steps to solve linear equations in one variable or the root formula to solve quadratic equations in one variable. Energy conservation in physics, chemical equilibrium formula in chemistry, and a large number of practical applications in reality all need to establish equations and get results by solving them. Therefore, students must learn how to solve one-dimensional linear equations and two-dimensional linear equations, and then learn other forms of equations.
The so-called "equation" idea means that for mathematical problems, especially the complex relationship between unknown quantities and known quantities encountered in reality, we are good at constructing relevant equations from the viewpoint of "equation" and then solving them.
2. The idea of "combination of numbers and shapes"
In the world, "number" and "shape" are everywhere. Everything, except its qualitative aspect, has only two attributes: shape and size, which are left for mathematics to study. There are two branches of junior high school mathematics-algebra and geometry. Algebra studies "number" and geometry studies "shape". It is a trend to learn algebra by means of "shape" and geometry by means of "number". The more you learn, the more inseparable you are from "number" and "shape". In senior high school, a course called "Analytic Geometry" appeared, which used algebra to study geometric problems. In the third grade, after the establishment of the plane rectangular coordinate system, the learning of functions can not be separated from images. Often with the help of images, the problem can be clearly explained, and it is easier to find the key to the problem, thus solving the problem. In the future mathematics study, we should pay attention to the thinking training of "combination of numbers and shapes" Any problem, as long as it is a little close to the "shape", should draw a sketch to analyze according to the meaning of the problem. This is not only intuitive, but also comprehensive, easy to find the breakthrough point, which is of great benefit to solving problems. Those who taste the sweetness will gradually develop the good habit of "combining numbers with shapes".
The second book of ninth grade mathematics knowledge articles;
★ The arrangement of knowledge points in the second volume of ninth grade mathematics
★ Summary of ninth grade mathematics knowledge points in People's Education Edition
★ The latest summary of mathematics knowledge points in Grade Three.
★ Summarize the test sites of mathematics knowledge points in Grade Three.
★ Summarize the knowledge points of junior high school mathematics.
★ Knowledge points of junior high school mathematics.
★ People's Education Edition third grade mathematics knowledge points induction
★ Math review materials at the end of the ninth grade next semester
★ Summarize the knowledge points of ninth grade mathematics in junior high school.
★ Summary of Basic Knowledge Points of Mathematics in Grade Three
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