Chapter 1 Rational Numbers

1. 1 positive and negative numbers

Books that have been studied before, except 0, are called negative numbers with the minus sign "-"in front of them.

Numbers other than 0 that I learned before are called positive numbers.

The number 0 is neither positive nor negative, it is the dividing line between positive and negative numbers.

In the same question, positive numbers and negative numbers have opposite meanings.

1.2 rational number

1.2. 1 rational number

Positive integers, 0 and negative integers are collectively called integers, and positive and negative fractions are collectively called fractions.

Integers and fractions are collectively called rational numbers.

1.2.2 axis

The straight line that defines the origin, positive direction and unit length is called the number axis.

Function of number axis: All rational numbers can be represented by points on the number axis.

Note: The origin, positive direction and unit length of (1) axis are indispensable.

⑵ The unit length of the same shaft cannot be changed.

Generally speaking, if it is a positive number, the point representing a on the number axis is on the right side of the origin, and the distance from the origin is a unit length; The point representing the number -a is on the left of the origin, and the distance from the origin is one unit length.

1.2.3 reciprocal

Numbers with only two different symbols are called reciprocal.

Two points representing the opposite number on the number axis are symmetrical about the origin.

Add a "-"sign before any number, and the new number represents the antonym of the original number.

1.2.4 absolute value

Generally speaking, the distance between the point representing the number A on the number axis and the origin is called the absolute value of the number A.

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.

Rational numbers are represented on the number axis, and the order from left to right is from small to large, that is, the number on the left is smaller than the number on the right.

Compare the sizes of rational numbers: (1) Positive numbers are greater than 0, 0 is greater than negative numbers, and positive numbers are greater than negative numbers.

(2) Two negative numbers, the larger one has the smaller absolute value.

Addition and subtraction of rational number 1.3

1.3. 1 addition of rational numbers

Law of rational number addition:

(1) Add two numbers with the same sign, take the same sign, and add the absolute values.

⑵ Add two numbers with different symbols with unequal absolute values, take the sign of the addend with larger absolute value, and subtract the number with smaller absolute value from the number with larger absolute value. Two opposite numbers add up to 0.

(3) When a number is added to 0, the number is still obtained.

When two numbers are added, the positions of the addend are exchanged and the sum is unchanged.

Additive commutative law: A+B = B+A.

Add three numbers, first add the first two numbers, or add the last two numbers first, and the sum remains the same.

Additive associative law: (a+b)+c = a+(b+c)

1.3.2 subtraction of rational numbers

The subtraction of rational numbers can be converted into addition.

Rational number subtraction rule:

Subtracting a number is equal to adding the reciprocal of this number.

a-b=a+(-b)

Multiplication and division of rational number 1.4

The rational number multiplication of 1.4. 1

Rational number multiplication rule:

Multiply two numbers, the same sign is positive, the different sign is negative, and then multiply by the absolute value.

Any number multiplied by 0 is 0.

Two numbers whose product is 1 are reciprocal.

Multiply several numbers that are not 0. When the number of negative factors is even, the product is positive. When the number of negative factors is odd, the product is negative.

When two numbers are multiplied, the exchange factor and the product are in the same position.

ab=ba

Multiply three numbers, first multiply the first two numbers, or multiply the last two numbers first, and the products are equal.

C=a (BC)

Multiplying a number by the sum of two numbers is equivalent to multiplying this number by these two numbers respectively, and then adding the products.

a(b+c)=ab+ac

Writing specification for multiplication of numbers and letters;

(1) Multiplies a number with a letter, omitting the multiplication sign or using "".

(2) Numbers multiplied by letters. When the coefficient is 1 or-1, 1 should be omitted.

(3) The band score is multiplied by letters, and the band score becomes a false score.

If any rational number is represented by the letter X, the product of 2 and X is 2x, and the product of 3 and X is 3x, then the formula 2x+3x is the sum of 2x and 3x, 2x and 3x are the terms of this formula, and 2 and 3 are the coefficients of these two terms respectively.

Generally speaking, when combining formulas with the same letter factor, it is only necessary to combine their coefficients, and the obtained results are used as coefficients, and then multiplied by the letter factor, that is,

ax+bx=(a+b)x

In the above formula, X is the letter factor, and A and B are the coefficients of ax and bx respectively.

Support removal rules:

There is a "+"before the brackets. Remove brackets and the "+"in front of brackets, and nothing in brackets will change its sign.

There is a "-"before the brackets. Remove brackets and the "-"sign in front of brackets, and change all the symbols in brackets.

The factors outside brackets are positive numbers, and the symbols of the items in the formula after removing brackets are the same as those of the corresponding items in the original brackets; The factor outside the bracket is negative, and the sign of each item in the formula after the bracket is opposite to that of the corresponding item in the original bracket.

1.4.2 division of rational numbers

Rational number division rule:

Dividing by a number that is not equal to 0 is equal to multiplying the reciprocal of this number.

a÷b=a？ (b≠0)

Divide two numbers, the same sign is positive, the different sign is negative, and divide by the absolute value. Divide 0 by any number that is not equal to 0 to get 0.

Because the division of rational numbers can be converted into multiplication, the operation can be simplified by using the operational nature of multiplication. The mixed operation of multiplication and division often turns division into multiplication first, then determines the sign of the product, and finally calculates the result.

1.5 power of rational number

1.5. 1 power

The operation of finding the product of n identical factors is called power, and the result of power is called power. In, a is called the base and n is called the exponent. When an is regarded as the result of the n power of a, it can also be read as the n power of a. ..

The odd power of a negative number is negative and the even power of a negative number is positive.

Any power of a positive number is a positive number, and any power of a positive integer is 0.

Operation sequence of rational number mixed operation:

(1) first power, then multiply and divide, and finally add and subtract;

(2) the same layer operation, from left to right;

(3) If there are brackets, do the operation in brackets first, and then press brackets, brackets and braces in turn.

1.5.2 scientific counting method

Numbers greater than 10 are expressed in the form of a× 10n (where a is a number with only one integer and n is a positive integer), and scientific notation is used.

Use scientific notation to represent n-bit integers, where the exponent of 10 is n- 1.

1.5.3 divisors and significands

A number that is close to the actual number but still different from the actual number is called a divisor.

Accuracy: an approximate value is rounded to the nearest place, so it is accurate to the nearest place.

From the first non-zero digit to the last digit on the left of a number, all digits are valid digits of this number.

For the number a× 10n expressed by scientific notation, its effective number is specified as the effective number in A. ..

Chapter II One-variable Linear Equation

2. 1 From Formula to Equation

2. 1. 1 linear equation

Equations with unknowns are called equations.

There is only one unknown (yuan), and the exponent of the unknown is 1 (degree). Such an equation is called a one-dimensional linear equation.

It is a method to solve practical problems by analyzing the quantitative relations in practical problems and listing equations by using their equal relations.

Solving the equation is to find the value of the unknown quantity that makes the left and right sides of the equation equal, and this value is the solution of the equation.

2. Properties of1.2 equation

The nature of the equation 1 Add (or subtract) the same number (or formula) on both sides of the equation, and the results are still equal.

Properties of Equation 2 Multiply both sides of the equation by the same number, or divide by the same number that is not 0, and the results are still equal.

2.2 from the ancient algebra books-discussion of linear equations (1)

Moving the sign of the term on one side of the equation to the other side is called moving the term.

2.3 From the "problem of buying cloth"-the discussion of the linear equation of one yuan (2)

When there are brackets in the equation, the method of removing brackets is similar to that in rational number operation.

Solving an equation is to find an unknown number (such as x). By removing the denominator, brackets, shifting the term, merging and converting the coefficient into 1, the linear equation can be gradually transformed into the form of X = A, which mainly depends on the properties and operation rules of the equation.

Denominator:

(1) Specific method: multiply both sides of the equation by the least common multiple of each denominator.

⑵ Basis: Equation Property 2

⑶ Precautions: ① Put brackets around the molecules.

② Items without denominator should also be multiplied.

2.4 Re-explore practical problems and linear equations of one variable.

The third chapter is the preliminary understanding of graphics.

3. 1 color graphics

In real life, we only care about the shape, size and position of objects, and graphics are called geometric graphics.

3. 1. 1 stereogram and plan view

Cuboid, cube, sphere, cylinder and cone are all three-dimensional figures. In addition, prisms and pyramids are also common three-dimensional figures.

Rectangular, square, triangle and circle are all plane figures.

Many three-dimensional graphics are surrounded by some plane graphics, which can be expanded into plane graphics by proper cutting.

3. 1.2 Point, line, surface and body

Geometry is also called volume for short. Cuboid, cube, cylinder, cone, sphere, prism and pyramid are all geometric bodies.

What surrounds the body is the surface. There are two kinds of face shapes: flat and curved.

Lines are formed at the intersection of faces.

The intersection of lines is a point.

Geometric figures are all composed of points, lines, surfaces and bodies, and points are the basic elements of figures.

3.2 Lines, rays and line segments

There is a straight line through two points, and there is only one straight line.

Two points define a straight line.

The line segment AB at point C is divided into two equal line segments AM and MB, and point M is called the midpoint of line segment AB. Similarly, line segments also have bisectors and quartiles.

The point of a straight line and the part next to it are called rays.

Among the connecting lines between two points, the line segment is the shortest. To put it simply: between two points, the line segment is the shortest.

3.3 Angle measurement

Angle is also a basic geometric figure.

Degrees, minutes and seconds are commonly used units of angle measurement.

Divide a fillet into 360 equal parts, each equal part is an angle of one degree, and record it as1; Divide the angle of 1 degree into 60 equal parts, each part is called the angle of 1 minute, and it is recorded as1; Divide the angle of 1 into 60 equal parts, each part is called 1 sec, and it is recorded as 1.

3.4 Angle comparison and operation

3.4.1angle comparison

Starting from the vertex of an angle, the ray that divides the angle into two equal angles is called the bisector of the angle. Similarly, there is the so-called bisector.

3.4.2 Complementary Angle and Complementary Angle

If the sum of two angles is equal to 90 degrees (right angle), they are said to be complementary angles.

If the sum of two angles is equal to 180 (flat angle), the two angles are said to be complementary.

The complementary angles of equal angles are equal.

The complementary angles of equal angles are equal.

Knowledge structure diagram in this chapter

Chapter IV Data Collection and Arrangement

Collecting, sorting, describing and analyzing data is the basic process of data processing.

4. 1 What kind of animals do students like best-an example of comprehensive investigation

Data are recorded by strokes, and each stroke of the word "positive" represents a piece of data.

The survey of all subjects is a comprehensive survey.

4.2 primary and secondary school students' eyesight survey-taking sampling survey as an example

Sampling survey is a survey that takes samples from the investigated population and estimates the population according to the samples.

Statistical survey is a common method to collect data, which generally includes comprehensive survey and sampling survey, and sampling survey is often used in practice. In the process of investigation, data can be obtained in different ways. Besides questionnaires and interviews, consulting literature and experiments is also an effective way to obtain data.

Organizing data with tables can help us find the distribution law of data. Using statistical charts to represent the sorted data can reflect the data law more intuitively.

4.3 Project research survey "How do you dispose of waste batteries?"

The investigation activities mainly include the following five steps:

First, design a questionnaire.

(1) Steps to design a questionnaire

① Determine the purpose of the investigation;

(2) Select the survey object;

③ Design survey questions.

2. When designing the questionnaire, we should pay attention to:

(1) Questions cannot involve the personal opinions of the questioner;

Don't ask questions that others don't want to answer;

③ The choice answers provided should be as comprehensive as possible;

④ Ask questions concisely;

⑤ The questionnaire should be short.

Second, the implementation of the survey.

Copy a sufficient number of questionnaires and send them to the respondents.

Please note:

(1) Explain to the respondent as the object of investigation why he became the respondent;

(2) Tell the interviewee the purpose of your data collection.

Third, processing data.

Collate, describe and analyze the collected data according to the collected questionnaires.

Fourth, communication.

According to the survey results, discuss what findings and suggestions your group has.

Verb (short for verb) Write a simple investigation report.

Second book

Chapter V Intersecting Lines and Parallel Lines

5. 1 intersection line

5. 1. 1 intersection line

One vertex has a common * * *, one side has a common * * *, and the other side is an extension line opposite to each other. Such two angles are called adjacent complementary angles.

There are four pairs of adjacent complementary angles when two straight lines intersect.

There is a vertex with a common * * *, and both sides of the corner are opposite extension lines. These two angles are called antipodal angles.

Two straight lines intersect and have two opposite angles.

The vertex angles are equal.

5. 1.2

Two straight lines intersect, and one of the four corners is a right angle, so the two straight lines are 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.

Note: (1) The vertical line is a straight line.

⑵ The four angles formed by two straight lines with vertical relationship are all 90.

(3) Verticality is a special case of intersection.

(4) Vertical symbols: a⊥b, AB⊥CD.

There are countless vertical lines that draw known straight lines.

One and only one straight line is perpendicular to the known straight line.

Of all the line segments connecting points outside the straight line and points on the straight line, the vertical line segment is the shortest. Simply put: the vertical line segment is the shortest.

The length from a point outside a straight line to the vertical section of the straight line is called the distance from the point to the straight line.

5.2 parallel lines

0+0 parallel line

In the same plane, if two straight lines have no intersection, then the two straight lines are parallel to each other, which is marked as: a ∨ b.

There are only two relationships between two straight lines in the same plane: intersecting or parallel.

Parallelism axiom: after passing a point outside a straight line, there is one and only one straight line parallel to this straight line.

If both lines are parallel to the third line, then the two lines are also parallel to each other.

Conditions of parallel lines

Two straight lines are cut by a third line. On the same side of two sections, on the same side of the section, such two angles are called congruent angles.

Two straight lines are cut by a third straight line, and between the two cutting lines, on both sides of the cutting line, such two angles are called inscribed angles.

Two straight lines are cut by a third line, and between the two cut lines, on the same side of the cut line, such two angles are called ipsilateral internal angles.

Judgment method of two parallel lines:

Method 1 Two straight lines were cut by the third straight line. If congruent angles are equal, two straight lines are parallel. To put it simply: the same angle is equal and two straight lines are parallel.

Method 2 Two straight lines are cut by a third straight line. If the internal dislocation angles are equal, two straight lines are parallel. To put it simply: the internal dislocation angles are equal and the two straight lines are parallel.

Method 3 Two straight lines are cut by a third straight line. If they are complementary, then these two straight lines are parallel. To put it simply: the internal angles on the same side are complementary and the two straight lines are parallel.

5.3 Properties of parallel lines

Parallel lines have properties:

Property 1 Two parallel lines are cut by a third line, and the congruence angles are equal. To put it simply: two straight lines are parallel and have the same angle.

Property 2 Two parallel lines are cut by a third straight line, and their internal angles are equal. To put it simply: two straight lines are parallel and their internal angles are equal.

Property 3 Two parallel lines are cut by a third straight line and complement each other. Simply put, two straight lines are parallel and complementary.

The length of a line segment perpendicular to and sandwiched between two parallel lines is called the distance between two parallel lines.

A statement that judges a thing is called a proposition.

5.4 Translation

(1) Move a graphic as a whole in a certain direction, and you will get a new graphic with the same shape and size as the original graphic.

⑵ Every point in the new graph is obtained by moving a point in the original graph. These two points are corresponding points, and the line segments connecting each group of corresponding points are parallel and equal.

This movement of graphics is called translation transformation, or translation for short.

Chapter VI Plane Cartesian Coordinate System

6. 1 plane rectangular coordinate system

6. 1. 1 ordered number pair

A number pair consisting of two consecutive numbers A and B is called an ordered number pair.

6. 1.2 plane rectangular coordinate system

Draw two mutually perpendicular number axes with overlapping origins on the plane to form a plane rectangular coordinate system. The horizontal axis is called the X axis or the horizontal axis, and it is customary to take the right as the positive direction; The vertical axis is called the Y axis or the vertical axis takes 2 as the positive direction; The intersection of the two coordinate axes is the origin of the plane rectangular coordinate system.

Any point on the plane can be represented by an ordered number pair.

After the rectangular coordinate system is established, the coordinate plane is divided into four parts, I, II, III and IV, which are called the first quadrant, the second quadrant, the third quadrant and the fourth quadrant respectively. The points on the coordinate axis do not belong to any quadrant.

6.2 Simple application of coordinate method

6.2. 1 Geographical location is expressed in coordinates.

The process of drawing the distribution plan of some places in the area using the plane rectangular coordinate system is as follows:

(1) Establish a coordinate system, select a suitable reference point as the origin, and determine the positive direction of the X axis and the Y axis;

⑵ Determine the appropriate scale according to specific problems and mark the unit length on the coordinate axis;

(3) Draw these points on the coordinate plane and write down the coordinates of each point and the name of each place.

6.2.2 Coordinate translation.

In the plane rectangular coordinate system, the corresponding point (x+a, y) (or (x-a, y)) can be obtained by translating the point (x, y) to the right (or to the left) by a unit length. The corresponding point (x, y+b) (or (x, y-b)) can be obtained by translating the point (x, y) up (or down) by b unit lengths.

In the plane rectangular coordinate system, if a positive number A is added (or subtracted) to the abscissa of each point of the graph, the corresponding new graph is to translate the original graph to the right (or left) by a unit length; If a positive number A is added (or subtracted) to the ordinate of each point, the corresponding new figure is to translate the original figure up (or down) by a unit length.

Chapter VII Triangle

7. 1 Line segment related to triangle

7. 1. 1 triangle edge

A figure composed of three line segments that are not on the same line end to end is called a triangle. The angle formed by two adjacent sides is called the inner angle of a triangle, which is called the angle of a triangle for short.

A triangle with vertices A, B and C is marked as △ABC and pronounced as "triangle ABC".

The sum of two sides of a triangle is greater than the third side.

7. 1.2 The bisector of the height, midline and angle of a triangle.

7. Stability of1.3 Triangle

Triangles are stable.

7.2 Angle related to triangle

7.2. 1 triangle inner angle

The sum of the internal angles of a triangle is equal to 180.

7.2.2 External Angle of Triangle

The angle formed by one side of a triangle and the extension line of the other side is called the outer angle of the triangle.

The outer angle of a triangle is equal to the sum of two non-adjacent inner angles.

The outer angle of a triangle is greater than any inner angle that is not adjacent to it.

7.3 sum of polygons and their internal angles

7.3. 1 polygon

On the plane, a figure composed of some end-to-end line segments is called a polygon.

The line segment connecting two nonadjacent vertices of a polygon is called the diagonal of the polygon.

Diagonal formula of n polygon:

An equilateral polygon is called a regular polygon.

7.3.2 Sum of interior angles of polygons

The formula for the sum of internal angles of n polygons: 180 (n-2)

The sum of the outer angles of a polygon is equal to 360 degrees.

7.4 Project Learning Mosaic

Chapter VIII Binary Linear Equations

8. 1 binary linear equations

An equation with two unknowns whose exponents are 1 is called a binary linear equation.

Two binary linear equations with the same unknowns are combined into one binary linear equation group.

The values of two unknowns that make the values on both sides of the binary linear equation equal are called the solutions of the binary linear equation.

The common * * * solution of two equations of binary linear equations is called the solution of binary linear equations.

8.2 elimination

Starting from an equation in binary linear equations, an unknown number is expressed by a formula containing another unknown number, and then it is substituted into another equation to realize elimination, and then the solution of this binary linear equations is obtained. This method is called substitution elimination method, or substitution method for short.

When the coefficients of the same unknown in two binary linear equations are opposite or equal, the unknown can be eliminated by adding or subtracting the two sides of the two equations respectively, thus a univariate linear equation system can be obtained. This method is called addition, subtraction and elimination, or addition and subtraction for short.

8.3 Re-explore practical problems and binary linear equations

Chapter 9 Inequality and Unequal Groups

9. 1 inequality

9. 1. 1 inequality and its solution set

The formula for expressing the relationship between size with ""is called inequality.

The value of the unknown quantity that makes the inequality valid is called the solution of the inequality.

The range of unknowns that can make inequality hold is called inequality solution set, which is called solution set for short.

An inequality with an unknown degree of 1 is called one-dimensional linear inequality.

9. Properties of1.2 inequality

Inequality has the following characteristics:

The nature of inequality 1 Add (or subtract) the same number (or formula) on both sides of inequality, and the direction of inequality remains unchanged.

The nature of inequality 2 Both sides of inequality are multiplied by (or divided by) the same positive number, and the direction of inequality remains unchanged.

The nature of inequality 3 Both sides of inequality are multiplied (or divided) by the same negative number, and the direction of inequality changes.

9.2 Practical Problems and One-dimensional Linear Inequalities

To solve a linear equation with one variable, the equation should be gradually transformed into the form of x = a according to its properties; To solve one-dimensional linear inequality, it is necessary to gradually transform inequality into the form of x < a (or x > a) according to the nature of inequality.

9.3 One-dimensional linear inequality system

When these two inequalities are combined together, a unitary linear inequality group is formed.

The common part of the solution set of several inequalities is called the solution set of inequalities composed of them. Solving inequality is to find its solution set.

All kinds of inequality problems can be solved by inequality groups. When solving a system of linear inequalities with one variable. Generally, the solution set of each inequality is found first, and then the common part of these solution sets is found. The number axis can be used to express the solution set of inequality groups intuitively.

9.4 Project Learning and Application Inequality Analysis Competition