The first chapter is a rich graphic world.
1, geometry
Various graphics abstracted from objects, including three-dimensional graphics and plane graphics.
Three-dimensional figures: Some geometric figures are not all on the same plane, but three-dimensional figures.
Plane figure: All parts of some geometric figures are on the same plane. They are plane figures.
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 line between 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) Points move into lines, lines move into planes, and planes move into adults.
3, the three-dimensional graphics in life
column
pillar; mainstay
Spherical prisms of three-dimensional figures in life: triangular prism, quadrangular prism (cuboid, cube), pentagonal prism, …
Conical cone (by name)
pyramid
4, prism and related concepts:
Edge: In a prism, any intersection of two adjacent faces is called an edge.
Side: The intersection of two adjacent sides is called a side.
N prism has two bottom faces, n side faces and ***(n+2) faces; 3n sides and n sides; 2n vertices.
5. Cubic plane development diagram: 1 1 species.
6. Cut a cube: cut a cube with a plane. The cutting surface can be triangular, quadrilateral, pentagonal or hexagonal.
7. Three Views
Three views of an object refer to the front view, the top view and the left view.
Front view: The view seen from the front is called the front view.
Left view: The picture seen from the left is called the left view.
Top view: The view from above is called top view.
8. Polygon: A closed plane figure composed of some line segments that are not on the same straight line is called polygon.
Starting from the same vertex of an n- polygon, the N- polygon can be divided into (n-2) triangles by connecting this vertex with other vertices respectively.
Arc: The part between point A and point B on a circle is called an arc.
Sector: A figure consisting of an arc and two radii passing through the end of the arc is called a sector.
Chapter II Rational Numbers and Their Operations
1, the classification of rational numbers
Positive rational number
Rational number zero
Negative rational number
Or integer
rational number
mark
2. Inverse number: Only two numbers with different signs are called inverse numbers, and the inverse of zero is zero.
3. Number axis: The straight line defining the origin, positive direction and unit length is called the number axis (it should be noted that the above three elements are indispensable when drawing the number axis). Any rational number can be represented by a point on the number axis. When solving problems, we should really master the idea of combining numbers with shapes and use it flexibly.
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). When the absolute value of zero is itself, it can also be regarded as its inverse. If |a|=a, then a ≥ 0; If |a|=-a, then a≤0.
6. Comparison size of rational numbers: positive numbers are greater than zero, negative numbers are less than zero, and positive numbers are greater than all negative numbers; 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.
(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
associative law of addition
Commutative law of multiplication
Multiplicative associative law
Distribution law of multiplication to addition
Letter representation number
1, algebraic expression
Expressions that connect numbers or letters representing numbers with operational symbols are called algebraic expressions. A single number or letter is also algebraic.
2. Similar projects
Items with the same letter and the same letter index are called similar items. Several constant terms are similar.
3. Rules for merging similar items: Add up the coefficients of similar items, and the indexes of letters and letters remain unchanged.
4. Rules for removing brackets
(1) brackets are preceded by "+". After removing the brackets and the "+"sign in front of them, the symbols of the items in the original brackets remain unchanged.
(2) The brackets are preceded by "-"."After the brackets and the"-"in front of them are removed, the symbols of the original brackets will change.
5, algebraic expression operation:
Addition and subtraction of algebraic expressions: (1) bracket removal; (2) Merge similar items.
Chapter IV Plane Figures and Their Positional Relations
1, line segment: tight chord and crosswalk line can be approximately regarded as line segments. A line segment has two endpoints.
2. Ray: A ray is formed by the infinite extension of a line segment in one direction. A ray has an endpoint.
3. Straight line: A straight line is formed by the infinite extension of line segments in two directions. A straight line has no end.
4. Representation of points, lines, rays and line segments
In geometry, we often use letters to represent figures.
A dot can be represented by capital letters.
A straight line can be represented by lowercase letters or by uppercase letters of two points on the straight line.
A ray can be represented by lowercase letters, or it can be represented by an endpoint and another point on the ray (the endpoint letter is written in front).
A line segment can be represented by a lowercase letter or two uppercase letters at its end.
5. There are two positional relationships between points and straight lines:
The point is on a straight line, or a straight line passes through the point.
② The point is outside the straight line, or the straight line does not pass through this point.
6, the nature of the line
(1) axiom of straight line: There is only one straight line passing through two points.
(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.
(4) There are infinitely many points on a straight line.
(5) Two different straight lines have at most one common point.
7, 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.
(2) Distance between two points: The length of the line segment between two points is called the distance between these two points.
(3) The midpoint of the line segment is equal to the distance between the two endpoints.
(4) The relationship between the size of a line segment and its length is consistent.
8, 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.
9. 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.
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, angle representation
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 side on both sides.
12, 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"
13, the nature of 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) Angle can be measured and compared.
(3) Angle can participate in the operation.
14, bisector of 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.
15, parallel lines:
On the same plane, two lines that do not intersect are called parallel lines. Parallelism is represented by the symbol "∨", such as "AB∨CD", which is read as "AB is parallel to CD".
note:
Parallel lines (1) are infinitely extended and do not intersect no matter how they are extended.
(2) When the line segment and the ray are parallel, it means that the line segment and the ray are parallel.
16, axiom of parallel lines and its inference
Parallelism axiom: after passing a point outside a straight line, there is one and only one straight line parallel to this straight line.
Inference: If two straight lines are parallel to the third straight line, then the two straight lines are also parallel to each other.
Determination method of complementary parallel lines;
(1) Two lines parallel to the same line are parallel.
(2) In the same plane, two straight lines perpendicular to the same straight line are parallel.
(3) The definition of parallel lines.
17, vertical:
When two straight lines intersect at right angles, they are said to be perpendicular to each other. One of the straight lines is called the perpendicular of the other straight line, and their intersection point is called the vertical foot.
The straight line AB and CD are perpendicular to each other, which is recorded as "AB⊥CD" (or "CD⊥AB") and read as "AB is perpendicular to CD" (or "CD is perpendicular to AB").
18, the nature of the vertical line:
Property 1: There is only one straight line perpendicular to the known straight line at a point on the plane.
Property 2: Of all the line segments connecting a point outside and a point on the line, the vertical line segment is the shortest. Abbreviation: the vertical segment is the shortest.
19. Distance from point to straight line: the vertical line passing through point A is L, the vertical foot is point B, and the length of line AB is called the distance from point A to straight line L.
20. The positional relationship between two straight lines on the same plane: intersecting or parallel.
Chapter 5 One-variable linear equation
1, equation
Equations with unknowns are called equations.
2, the solution of the equation
The value of the unknown that can make the left and right sides of the equation equal is called the solution of the equation.
3. Properties of the equation
Both sides of the (1) equation add (or subtract) the same algebraic expression at the same time, and the result is still an equation.
(2) Both sides of the equation are multiplied by the same number at the same time (or divided by the same number that is not 0), and the result is still an equation.
4. One-dimensional linear equation
An integral equation with only one unknown number and the highest order of the unknown number is 1 is called a linear equation with one variable.
5, the general steps to solve a linear equation:
(1) Remove the denominator (2) Remove the brackets (3) Move the term (after changing the sign of an item in the equation, move it from one side of the equation to the other, this deformation is called the moving term. (4) Merge similar terms (5) and change the unknown coefficient into 1.
Chapter VI Data in Life
1, scientific counting method
Generally speaking, numbers greater than 10 can be expressed in the form, where n is a positive integer. This notation is called scientific notation.
2, fan statistical chart and its drawing:
Sector statistical chart: A statistical chart is called a sector statistical chart, and circles and sectors are used to represent the relationship between the whole and parts, that is, the circle represents the whole, and each sector in the circle represents different parts of the whole, and the size of the sector reflects the percentage of parts in the whole.
Painting method:
(1) Calculate the percentage of different parts in the whole (in a sector, the percentage of each part in the whole is equal to the ratio of the degree of the central angle of the sector corresponding to this part to 360).
(2) Calculate the degree of the central angle of each sector (the angle of the vertex at the center of the circle is called the central angle).
(3) Draw each sector in the circle and mark the percentage.
3. Advantages and disadvantages of various statistical charts
Bar chart: The specific figures of each item can be clearly displayed.
Broken line statistical chart: it can clearly reflect the changes of things.
Department statistical chart: it can clearly show the percentage of each part in the total.
possibility
1, definite events and uncertain events
(1), determine the event.
Inevitable events: In life, there are some things that we can be sure will happen in advance. These things are called inevitable events.
Impossible events: some things that we can be sure will not happen in advance are called impossible events.
(2), uncertain events:
Some things that we can't be sure in advance will happen are called uncertain events.
(3)、
Certain events
Decisive event
An impossible event
Uncertain event
2. The possibility of uncertain events.
Generally speaking, the possibility of uncertain events is large or small.
The probability of an inevitable event is 1.
The probability of an impossible event is 0.