Knowledge points in the first volume of seventh grade mathematics of Beijing Normal University Edition
The first chapter is a rich graphic world.
1, geometry
Various graphics abstracted from objects, including three-dimensional graphics and plane graphics.
2. Points, lines, surfaces and bodies
Synthesis of (1) Geometry
Point: The point where straight lines intersect is the point, which is the most basic figure in geometry.
Line: The intersection of faces is a line, which can be divided into straight lines and curves.
Face: Surrounding the body is the face, which is divided into plane and curved surface.
Volume: Geometry is also called volume for short.
(2) inching into a line, the line moves into a plane, and the plane moves into an adult.
3, the three-dimensional graphics in life
Column: prism: triangular prism, quadrangular prism (cuboid, cube), pentagonal prism, ...
Chapter II Rational Numbers and Their Operations
1. rational number
A number that can be expressed as the ratio of two integers.
positive integer
Rational number zero rational number
Responsible fraction
2. Inverse number: Only two numbers with different signs are called inverse numbers, and the inverse number of 0 is 0.
3. Number axis: The straight line defining the origin, positive direction and unit length is called number axis (when drawing number axis, all three elements are indispensable). Any rational number can be represented by a point on the number axis.
4. Reciprocal: If A and B are reciprocal, there is ab= 1, and vice versa. The numbers whose reciprocal equals itself are 1 and-1. Zero has no reciprocal.
5. Absolute value: the distance between the point corresponding to a number and the origin on the number axis is called the absolute value of the number, |a|≥0. If |a|=a, then a ≥ 0; If |a|=-a, then a≤0.
The absolute value of a positive number is itself; The absolute value of a negative number is its reciprocal; The absolute value of 0 is 0. The absolute values of two opposite numbers are equal.
6. Rational number comparison size: positive number is greater than 0, negative number is less than 0, and positive number is greater than negative number; The number represented by two points on the number axis is always larger on the right than on the left; Two negative numbers, the larger one has the smaller absolute value.
7. Rational number operation
(1) Five operations: addition, subtraction, multiplication, division and multiplication.
Multiply multiple numbers, and the sign of the product is determined by the number of negative factors. When there are odd negative factors, the sign of the product is negative. When there are even negative factors, the sign of the product is positive. As long as one number is 0, the product is 0.
Rational number addition rule:
Add two numbers with the same sign, take the same sign, and then add the absolute values.
Two numbers with different signs are added, and the sum is 0 when the absolute values are equal; When the absolute values are not equal, take the sign of the addend with the larger absolute value and subtract the smaller absolute value from the larger absolute value.
Add a number to 0 and you still get the number.
The sum of two opposite numbers is 0.
Rational number subtraction rule:
Subtracting a number equals adding the reciprocal of this number!
Rational number multiplication rule:
Multiply two numbers, the same sign is positive, the different sign is negative, and then multiply by the absolute value.
Multiply any number by 0, and the product is still 0.
Rational number division rule:
Divide two rational numbers, the same sign is positive, the different sign is negative, and divide by the absolute value.
Divide 0 by any number except 0 to get 0.
Note: 0 cannot be divided.
Power of rational number: the operation of finding the product of n identical factors a is called power.
Any power of a positive number is positive, even power of a negative number is positive and odd power of a negative number is negative.
(2) Operation sequence of rational numbers
Calculate the power first, then multiply and divide, and finally add and subtract. If there are brackets, count them first.
(3) Operation law
Additive commutative law, additive associative law, multiplicative commutative law, multiplicative associative law, distributive law from multiplication to addition.
8. Scientific symbols
Generally speaking, numbers greater than 10 can be expressed in the form, where n is a positive integer. This notation is called scientific notation. (n= integer digits-1)
Chapter III Algebraic Expressions and Their Addition and Subtraction
1, algebraic expression
Connect numbers or letters representing numbers with operation symbols (addition, subtraction, multiplication, division, multiplication, root, etc.) to form an algebraic expression. ) A single number or letter is also algebraic.
note:
① Besides numbers, letters and operators, algebraic expressions can also have brackets;
② Algebraic expressions do not contain symbols such as "=, >,<, ≦". Equality and inequality are not algebraic, but the formulas on both sides of equal sign and unequal sign are generally algebraic;
(3) The number represented by letters in algebraic expressions must make algebraic expressions meaningful, which is a practical problem and should conform to the meaning of practical problems.
Algebra writing format: ※:
(1) multiplication symbols appear in algebraic expressions and are usually omitted, such as vt;
(2) When the number is multiplied by the letter, the number should be written in front of the letter, such as 4a;
(3) When multiplying the band score by letters, the band score should be turned into a false score first;
(4) the number multiplier, generally still use the "x" sign, that is, do not omit the "x" sign;
(5) when there is a division operation in algebraic expressions, it is generally written in the form of fractions, such as 4÷(a-4) should be written as 4/(a-4); Note: Fractions have the dual functions of "∫" and brackets.
⑥ If there is a unit name after the algebraic expression of sum (or) difference, you must enclose the algebraic expression and then write the unit name after the expression, such as square meters.
2. Algebraic expression
Monomial and polynomial are collectively called algebraic expressions.
(1) monomial: Algebraic expressions that are all products of numbers and letters are called monomials. In a monomial, the sum of the indices of all letters is called the number of times of the monomial; This numerical factor is called the coefficient of this single term.
note:
1. A single number or letter is also a monomial;
2. The number of times of a single non-zero number is 0;
3. When the single coefficient is 1 or-1, this "1" should be omitted, such as -ab coefficient is-1 and a3b coefficient is 1.
② Polynomial: The sum of several monomials is called polynomial. In polynomials, each monomial is called a polynomial term; The degree of the term with the highest degree is called the degree of polynomial.
3. Similar items: items with the same letters and the same letter index are called similar items.
note:
Similar items have two conditions: they contain the same letters; The index of the same letter is the same.
(2) Similar terms have nothing to do with the arrangement order of coefficients and letters;
③ Several constant terms are similar.
4. Rules for merging similar items: Add up the coefficients of similar items, and the indexes of letters and letters remain unchanged.
5. Rules for removing brackets
(1) According to the rules of brackets:
There is a "+"before the brackets. Remove the brackets and the "+"in front, and nothing in brackets will change its symbol; There is a "-"before the brackets. Remove the brackets and the "-"in front, and change the symbols of everything in brackets.
(2) Give brackets according to the distribution law:
The+sign before the brackets is regarded as+1, and the-sign before the brackets is regarded as-1. According to the distribution law of multiplication, each item in brackets is multiplied by+1 or-1 to remove the brackets.
6. Parenthesis rule
Add "+"and brackets, and all symbols added in brackets remain unchanged; Add "-"and brackets, and all the symbols in brackets should be changed.
7, algebraic expression operation:
Addition and subtraction of algebraic expressions: (1) bracket removal; (2) Merge similar items.
Basic plane figure
1, line segments, rays and lines
2, the nature of the line
(1) axiom of straight line: There is only one straight line passing through two points. (Two points define a straight line)
(2) There are countless straight lines passing by little by little.
(3) The straight line extends infinitely in two directions, without points, unmeasurable and incomparable in size.
3, the nature of the line segment
(1) Axiom of Line Segment: Of all the connecting lines between two points, the line segment is the shortest. (The line segment between two points is the shortest)
(2) Distance between two points: The length of the line segment between two points is called the distance between these two points.
(3) The relationship between the size of a line segment and its length is consistent.
4, the midpoint of the line segment:
The point M divides the line segment AB into two equal line segments AM and BM, and the point M is called the midpoint of the line segment AB. AM = BM = 1/2AB (or AB=2AM=2BM).
5. Angle
A graph composed of two rays with a common endpoint is called an angle, the common endpoint of the two rays is called the vertex of the angle, and the two rays are called the edges of the angle. Or: a corner can also be regarded as a light that rotates around its endpoint.
6. Representation of angle
There are four ways to express the angle:
① Use numbers to represent individual angles, such as ∠ 1, ∠2, ∠3, etc.
② Use lowercase Greek letters to represent a single angle, such as ∠ α, ∠β, ∠ γ, ∠ θ, etc.
③ An independent angle (a vertex has only one angle) is represented by capital English letters, such as ∠B, ∠C, etc.
④ Use three capital letters to represent any corner, such as ∠BAD, ∠BAE, ∠CAE, etc.
Note: When using three capital letters to represent a corner, be sure to write the letter of the vertex in the middle and the letter of the edge on both sides.
7. Angle measurement
The measurement of angle has the following provisions: divide a flat angle 180 into equal parts, each part is an angle of 1 degree, and the unit is 0, with 1 degree marked as 1 degree and n degree marked as "n".
Divide the angle of 1 into 60 equal parts, each part is called the angle of 1, and 1 is marked as "1'".
Divide the angle of 1' into 60 equal parts, each part is called the angle of 1 sec, and 1 sec is marked as "1".
1 =60', 1'=60"。
8. bisector of an angle
The ray from the vertex of an angle divides the angle into two equal angles. This ray is called the bisector of an angle.
9, the nature of the angle
The size of the (1) angle has nothing to do with the side length, but only with the amplitude of the two rays that make up the angle.
(2) the size of the angle can be measured and compared, and the angle can participate in the operation.
10, straight angle and rounded corner: a ray rotates around its endpoint. When the ending edge and the starting edge are on a straight line, the angle formed is called a straight angle. The ending edge continues to rotate, and when it coincides with the starting edge, the angle formed is called fillet.
1 1. Polygon: A closed plane figure composed of several line segments that are not on the same line is called a polygon. A line segment connecting two nonadjacent vertices is called the diagonal of a polygon.
Starting from the same vertex of an N-polygon, we can draw (n-3) diagonal lines, and divide the N-polygon into (n-2) triangles by connecting this vertex with other vertices respectively.
12. Circle: On the plane, a line segment rotates around one endpoint, and the figure formed by the other endpoint is called a circle. The fixed endpoint O is called the center of the circle, and the length of the line segment OA is called the length of the radius (usually referred to as radius for short).
The part between any two points A and B on a circle is called arc, which is called "arc AB" or "arc AB" for short. A figure consisting of an arc AB and two radii OA and OB passing through the end of the arc is called a sector. The angle of the vertex at the center of the circle is called the central angle.
Methods of learning mathematics well
1, adjust your mentality before class. You must not think about it. Hey, it's math class again. I'm in a bad mood when I attend class. Of course I can't learn well!
2, be sure to listen carefully in class, so that you can hear, see and reach! This is very important. You must learn to take notes. If the teacher speaks quickly in class, you must calm down and listen. Don't take notes, write them in the notebook after class! Keep high efficiency!
As the saying goes, interest is the best teacher. When others say the most annoying class, you should tell yourself that I like math!
It is very important to make sure that every problem you encounter will be understood and understood! Don't be embarrassed if you don't ask, learn to draw inferences! That is flexible use! Don't ask too many questions, but be precise!
5, there must be a set of wrong questions, write down the good questions you usually encounter, write down the wrong questions, read more, think more, and don't stumble in the same place! !
In short, learn math, don't be afraid of difficulties, don't be afraid of being tired, and don't be afraid to ask!
Some suggestions on learning mathematics well.
1. Interested in learning mathematics. Interest is the best teacher. Do anything, as long as you are interested, you will take the initiative to do it, and you will try your best to do it well. But the key to cultivate students' interest in mathematics is to master the basic knowledge and skills of mathematics first. Some students always want to do difficult problems, and when they see others taking math classes, they also want to go. If these students can't even master the basic knowledge in class, they can only make it up in class, which will not help them, but will make them lose confidence in learning mathematics. I suggest that students can read some famous stories about mathematics and interesting mathematics to enhance their self-confidence in learning.
2. Have a correct learning attitude. First of all, it must be clear that learning is for yourself, not for teachers and parents. Therefore, we should concentrate, think positively and speak boldly in class. Secondly, after returning home, you should finish your homework carefully, review what you learned that day in time, and then preview what you will learn tomorrow. In this way, you will learn more easily and understand more deeply.
3. Have the spirit of "perseverance". If you want to improve your academic performance, you should do it step by step. Don't expect to learn everything overnight. Even if the progress is slow, as long as you persist, math learning will be successful! We should also have the spirit of "not ashamed to ask questions" and not be afraid of losing face. In fact, no matter how difficult the knowledge is, as long as you learn and understand it, that is the greatest face!
4. Pay attention to learning skills and methods. Some formulas and laws should not be memorized by rote, but should be understood by analysis and applied flexibly. Special attention should be paid to the study of new knowledge and the analysis of exercises in class. We shouldn't be distracted and mind our own business. Attention must be highly focused and think positively. When you don't understand the topic, you should make a good record in time, discuss it with your classmates after class, and do a good job of filling the vacancy.
5. Have a good habit of observing and reading. As long as we pay attention to mathematics and carefully observe and think, we will find that there is mathematics everywhere in our lives. In addition, students can learn mathematics from many aspects and channels. For example, learn mathematics from newspapers and magazines such as TV, Internet, Math Newspaper for Primary School Students, Math PHS, etc., and constantly expand their knowledge.
6. Have your own opinions. At present, most students encounter some difficult or unclear problems and give up easily without thinking, and some simply listen to the opinions of teachers, parents and books. Even teachers, elders, books and other authorities are not without some mistakes. We should attach importance to authoritative opinions, but it does not mean that we agree without thinking.
7. Learn to generalize and accumulate. Summarize the law of solving problems in time, especially accumulate some classic and special problems. In this way, we can study easily and improve the efficiency and quality of learning.
8. Pay attention to the study of other subjects. Because there is a close relationship between disciplines, it can promote the study of mathematics. For example, learning Chinese well is very helpful to understand the purpose of math problems, and so on.
Beijing normal university edition, the first volume, related articles on the knowledge points of seventh grade mathematics;
★ Combing the knowledge points of the first grade mathematics Beijing Normal University Edition
★ Summary of Mathematics Knowledge Points of Grade One in Beijing Normal University Edition
★ Summary of Mathematics Knowledge Points of Grade One in Beijing Normal University Edition
★ Beijing Normal University Edition Mathematics Knowledge Points, Grade One
★ Summary of Mathematics Knowledge Points of Grade One in Beijing Normal University Edition
★ Summary of Mathematics Knowledge Points in Grade 7 of Beijing Normal University Edition
★ Summary of knowledge points in Chapter 4, Volume 1, Grade 7 Mathematics
★ Beijing Normal University Edition Junior High School Mathematics Teaching Plan
★ Beijing Normal University Edition Junior High School Mathematics Knowledge Point Outline
★ Summarize the knowledge points of seventh grade mathematics.
var _ HMT = _ HMT | |[]； (function(){ var hm = document . createelement(" script ")； hm.src = "/hm.js？ 8 a6b 92 a 28 ca 05 1cd 1 a9f 6 beca 8 DCE 12e "； var s = document . getelementsbytagname(" script ")[0]； s.parentNode.insertBefore(hm，s)； })();