The first volume of eighth grade mathematics knowledge points, Shanghai sub-volume
First of all, in a plane, two data are usually needed to determine the position of an object.
Second, the plane rectangular coordinate system and related concepts
1, plane rectangular coordinate system
In a plane, two mutually perpendicular axes with a common origin form a plane rectangular coordinate system. Among them, the horizontal axis is called X axis or horizontal axis, and the right direction is the positive direction; The vertical axis is called Y axis or vertical axis, and the orientation is positive; The x-axis and y-axis are collectively referred to as coordinate axes. Their common origin o is called the origin of rectangular coordinate system; The plane on which the rectangular coordinate system is established is called the coordinate plane.
2. In order to describe the position of a point in the coordinate plane conveniently, the coordinate plane is divided into four parts, namely the first quadrant, the second quadrant, the third quadrant and the fourth quadrant.
Note: The points on the X axis and Y axis (points on the coordinate axis) do not belong to any quadrant.
3. The concept of point coordinates
For any point P on the plane, the intersection point P is perpendicular to the X-axis and Y-axis respectively, and the numbers A and B corresponding to the vertical feet on the X-axis and Y-axis are respectively called the abscissa and ordinate of the point P, and the ordered number pair (A, B) is called the coordinate of the point P. ..
The coordinates of points are represented by (a, b), and the order is abscissa before, ordinate after, and there is a ","in the middle. The positions of horizontal and vertical coordinates cannot be reversed. The coordinates of points on the plane are ordered real number pairs. At that time, (a, b) and (b, a) were the coordinates of two different points.
There is a one-to-one correspondence between points on the plane and ordered real number pairs.
4. Coordinate characteristics of different locations
(1), the coordinate characteristics of the midpoint of each quadrant.
Point P(x, y) is in the first quadrant: x; 0，y； 0
Point P(x, y) is in the second quadrant: x; 0，y； 0
Point P(x, y) is in the third quadrant: x; 0，y； 0
Point P(x, y) is in the fourth quadrant: x; 0，y； 0
(2) Characteristics of points on the coordinate axis
The point P(x, y) is on the x axis, y=0, and x is an arbitrary real number.
The point P(x, y) is on the y axis, x=0, and y is an arbitrary real number.
Point P(x, y) is on both X and Y axes, and both X and Y are zero, that is, the coordinate of point P is (0,0), that is, the origin.
(3) Coordinate characteristics of points on the bisector of two coordinate axes.
Point P(x, y) is on the bisector of the first and third quadrants (straight line y=x), and x and y are equal.
Point P(x, y) is on the bisector of the second and fourth quadrants, and x and y are reciprocal.
(4) Characteristics of the coordinates of points on a straight line parallel to the coordinate axis
The ordinate of each point on the straight line parallel to the X axis is the same.
The abscissa of each point on the straight line parallel to the Y axis is the same.
(5) Coordinate characteristics of points symmetrical about the X axis, Y axis or origin.
The abscissa of point P and point P' is equal to the X axis, and the ordinate is opposite, that is, the symmetrical point of point P(x, y) relative to the X axis is P'(x, -y).
The axisymmetrical ordinate of point P and point P' with respect to Y is equal, and the abscissa is opposite, that is, the symmetrical point of point P(x, y) with respect to Y axis is P'(-x, y).
Point P and point P' are symmetrical about the origin, and the abscissa and ordinate are opposite, that is, the symmetrical point of point P(x, y) about the origin is P'(-x, -y).
The second volume of the second day of junior high school summarizes the knowledge points of mathematics.
1. Equation and Equivalence: An equation connected by "=" is called an equation. Note: "Equivalent value can be substituted"!
2. The nature of the equation:
Properties of equation 1: Add (or subtract) the same number or the same algebraic expression on both sides of the equation, and the result is still an equation;
Property 2 of the equation: both sides of the equation are multiplied (or divided) by the same non-zero number, and the result is still an equation.
3. Equation: An equation with an unknown number is called an equation.
4. Solution of the equation: the value of the unknown quantity that makes the left and right sides of the equation equal is called the solution of the equation; Note: "The solution of the equation can be substituted"!
5. Moving term: after changing the sign, moving the term of the equation from one side to the other is called moving term. The shift term is based on the equality attribute 1.
6. One-dimensional linear equation: An integral equation with only one unknown number, degree 1 and non-zero coefficient is a one-dimensional linear equation.
7. The standard form of one-dimensional linear equation: ax+b=0(x is unknown, a and b are known numbers, a≠0).
8. The simplest form of linear equation with one variable: ax=b(x is unknown, a and b are known numbers, a≠0).
9. General steps for solving a linear equation with one variable: sorting out the equation ... removing the denominator ... dismantling the bracket ... changing the terms ... merging similar terms ... and converting the coefficient into 1 ... (testing the solution of the equation).
10. Solving application problems by listing linear equations of one variable;
(1) reading analysis method: reading analysis method
Read the stem carefully, find out the key words that express the equal relationship, such as "big, small, many, few, yes, * * *, combination, right, completion, increase, decrease, match-",list the literal equations with these key words, and set the unknown number according to the meaning of the question. Finally, using the relationship between quantity and quantity in the question, fill in the algebraic expression and get the equations.
(2) Drawing analysis method
Analyzing mathematical problems with graphics is the embodiment of the combination of numbers and shapes in mathematics. Read the question carefully, and draw the relevant figures according to the meaning of the question, so that each part of the figure has a specific meaning. Finding the equation relationship through graph is the key to solve the problem, so as to obtain the basis of concise equation. Finally, using the relationship between quantity and quantity (unknown quantity can be regarded as known quantity), filling in the relevant algebraic expression is the basis of getting the equation.
Math learning method in grade two of junior high school
Remember what you should remember, remember what you should recite, and don't think you understand.
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.
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.
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.
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.
3. The concept of "correspondence"
The concept of "correspondence" has a long history. For example, we correspond a pencil, a book and a house to an abstract number "1", and two eyes, a pair of earrings and a pair of twins to an abstract number "2". With the deepening of learning, we also extend "correspondence" to a form, a relationship, and so on. For example, when calculating or simplifying, we will correspond the left side of the formula, A, Y and B, and then directly get the result of the original formula with the right side of the formula.
Knowledge points of eighth grade mathematics related articles in Shanghai Science Edition;
★ Knowledge points of Grade 8 Mathematics Shanghai Science Edition
★ Knowledge points of the first volume of eighth grade mathematics in Shanghai Science Edition
★ Review Outline of Grade Eight Mathematics in Shanghai.
★ The first volume of eighth grade mathematics review materials.
★ Shanghai Science Edition Grade 8 Volume I Mathematics Review Outline
★ Summary of Junior High School Mathematics Knowledge Points (Shanghai Science Edition)
★ Grade 8 Mathematics Volume I Final Examination Paper Shanghai Volume
★ Review materials of Volume II of Grade 8 Mathematics in Shanghai Science Edition
★ Shanghai Science Edition Eighth Grade Mathematics Teaching Plan
★ Shanghai Science Edition Grade 8 Mathematics Catalogue