The first volume of seventh grade mathematics in People's Education Press mainly includes four chapters: rational number, addition and subtraction of algebraic expression, linear equation of one yuan and preliminary understanding of graphics.

Chapter 1 Rational Numbers

I. Knowledge framework

Two. The concept of knowledge

1. rational number:

(1) Any number that can be written in form is a rational number. Positive integers, 0 and negative integers are collectively referred to as integers. Positive and negative scores are collectively called scores; Integers and fractions are collectively called rational numbers. Note: 0 is neither positive nor negative; -a is not necessarily negative, and +a is not necessarily positive; P is not a rational number;

(2) Classification of rational numbers: ① ②

2. Number axis: The number axis is a straight line that defines the origin, positive direction and unit length.

3. The opposite number:

(1) There are only two numbers with different signs, and we say that one of them is opposite to the other; The antonym of 0 is still 0;

(2) Is the sum of opposites 0? a+b=0？ A and b are opposites.

4. Absolute value:

(1) The absolute value of a positive number is itself, the absolute value of 0 is 0, and the absolute value of a negative number is its inverse; Note: the absolute value means the distance between the point representing a number on the number axis and the origin;

(2) The absolute value can be expressed as: or; The problem of absolute value is often discussed in categories;

5. Rational number ratio: (1) The greater the absolute value of a positive number, the greater the number; (2) Positive numbers are always greater than 0 and negative numbers are always less than 0; (3) Positive numbers are greater than all negative numbers; (4) The absolute values of two negative numbers are larger than the size, but smaller; (5) Of the two numbers on the number axis, the number on the right is always greater than the number on the left; (6) large number-decimal number > 0, decimal number-large number < 0.

6. Reciprocal: Two numbers whose product is 1 are reciprocal; Note: 0 has no reciprocal; If a≠0, the reciprocal is; If ab= 1? A and b are reciprocal; If ab=- 1? A and b are negative reciprocal.

7. The rational number addition rule:

(1) Add two numbers with the same symbol, take the same symbol, and add the absolute values;

(2) Add two numbers with different symbols, take the symbol with larger absolute value, and subtract the one with smaller absolute value from the one with larger absolute value;

(3) Adding a number to 0 still gets this number.

8. Arithmetic of rational number addition:

The commutative law of (1) addition: a+b = b+a; (2) The associative law of addition: (a+b)+c=a+(b+c).

9. Rational number subtraction rule: subtracting a number is equal to adding the reciprocal of this number; That is, a-b=a+(-b).

10 rational number multiplication rule:

(1) Multiply two numbers, the same sign is positive, the different sign is negative, and the absolute value is multiplied;

(2) Multiply any number by zero to get zero;

(3) When several numbers are multiplied, one factor is zero and the product is zero; Each factor is not zero, and the sign of the product is determined by the number of negative factors.

1 1 rational number multiplication algorithm;

(1) The commutative law of multiplication: AB = BA(2) The associative law of multiplication: (AB) C = A (BC);

(3) Distribution law of multiplication: a(b+c)=ab+ac.

12. rational number division rule: dividing by a number is equal to multiplying the reciprocal of this number; Note: Zero cannot be divisible.

13. Power Law of Rational Numbers:

(1) Any power of a positive number is a positive number;

(2) The odd power of a negative number is a negative number; Even the power of negative numbers is positive; Note: When n is positive odd number: (-a)n=-an or (a -b)n=-(b-a)n, when n is positive even number: (-a)n =an or (a-b) n = (b-a) n. 。

14. Definition of power:

(1) The operation of seeking common ground factor product is called power;

(2) In power, the same factor is called base, the number of the same factor is called exponent, and the result of power is called power;

15. Scientific notation: Write numbers greater than 10 in the form of a× 10n, where a is a number with only one integer digit. This notation is called scientific notation.

16. Approximation precision: a divisor rounded to that bit, that is, the divisor is accurate to that bit.

17. Significant digits: All digits from the first non-zero digit on the left to the exact digit are called significant digits of this approximation.

18. Mixed algorithm: multiply first, multiply then divide, and finally add and subtract.

This chapter requires students to correctly understand the concept of rational numbers, and understand the meanings of positive and negative numbers, antonyms and absolute values on the basis of real life and learning the number axis. Focus on solving practical problems with the algorithm of rational numbers.

An important reason for the development of experiential mathematics is the actual needs of life. Stimulate students' interest in learning mathematics, teachers cultivate students' ability of observation, induction and generalization, and enable students to establish a correct sense of numbers and the ability to solve practical problems. When teaching this chapter, teachers should create more situations to fully reflect the main position of students' learning.

Chapter II Addition and subtraction of algebraic expressions

I. Knowledge framework

Two. The concept of knowledge

1. monomial: in algebraic expressions, if only multiplication (including power) operations are involved. Or algebraic expressions that contain division but do not contain letters in division are called monomials.

2. The coefficient and times of single item: the non-zero numerical factor in single item is called the numerical coefficient of single item, which is simply referred to as the coefficient of single item; When the coefficient is not zero, the sum of all the letter indexes in a single item is called the degree of the item.

3. Polynomial: The sum of several monomials is called polynomial.

4. Number and degree of polynomials: the number of monomials contained in a polynomial is the number of polynomial terms, and each monomial is called a polynomial term; In polynomial, the degree of the term with the highest degree is called the degree of polynomial.

Through the study of this chapter, students should achieve the following learning objectives:

1. Understand and master the concepts of monomial, polynomial and algebraic expressions, and find out the differences and connections between them.

2. Understand the concept of similar items, master the method of merging similar items, master the changing law of symbols when removing brackets, and be able to merge and remove brackets correctly. On the basis of accurate judgment and correct combination of similar items, add and subtract algebraic expressions.

3. Understand that the letters in the algebraic expression represent numbers, and the addition and subtraction operations of the algebraic expression are based on numbers; Understanding the basis of merging similar items and removing brackets is the distribution law; Understanding the operation rules and properties of numbers is still effective in the addition and subtraction of algebraic expressions.

4. Be able to analyze the quantitative relationship in practical problems and express it with a formula with letters.

In the study of this chapter, teachers can experience the formation process of concepts through group discussion and cooperative learning, and initially cultivate students' thinking ability and application consciousness such as observation, analysis, abstraction and generalization.

Chapter III One-variable Linear Equation

I. Knowledge framework

Two. The concept of knowledge

1. One-dimensional linear equation: An integral equation with only one unknown number and a degree of 1 and a non-zero coefficient is a one-dimensional linear equation.

2. The standard form of one-dimensional linear equation: ax+b=0(x is unknown, a and b are known numbers, a≠0).

3. General steps for solving a linear equation with one variable: sorting out the equation ... removing the denominator ... dismantling the bracket ... changing the terminology ... merging similar terminology ... and converting the coefficient into 1 ... (testing the solution of the equation).

4. Set up a linear equation of one variable to solve application problems:

(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, For, 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 the graph is the key to solve the problem, so as to get 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.

1 1. Common formulas for solving application problems with column equations:

(1) Travel problem: distance = speed time;

(2) Engineering problems: workload = work efficiency and working time;

(3) ratio: part = total ratio;

(4) Downstream problem: Downstream velocity = still water velocity+water velocity, and countercurrent velocity = still water velocity-water velocity;

(5) Commodity price: selling price = pricing discount, profit = selling price-cost;

(6) Perimeter, area and volume: C circle =2πR, S circle =πR2, C rectangle =2(a+b), S rectangle =ab, C square =4a,

S square =a2, S ring =π(R2-r2), V cuboid =abc, V cube =a3, V cylinder =πR2h, V cone =πR2h.

The content of this chapter is the core of algebra and the basis of all algebraic equations. Colorful problem situations and happiness in solving problems can easily arouse students' interest in mathematics, so we should pay attention to the study of problems around us, guide students to carry out effective mathematical activities and cooperative exchanges, and let students acquire knowledge, improve their ability and experience mathematical thinking methods in the process of active learning and inquiry learning.

The first chapter is a preliminary understanding of graphics.

I. Knowledge framework

The main content of this chapter is a preliminary understanding of graphics. Starting from the familiar objects in our life, the understanding of the shape of objects gradually rose from perceptual to abstract geometric figures. Looking at the three-dimensional graphics from different directions and expanding the three-dimensional graphics, we can initially understand the relationship between the three-dimensional graphics and the plane graphics. On this basis, we can understand some simple plane figures-straight lines, rays, line segments and angles.

Second, the mathematical ideas involved in this chapter:

1. Discuss ideas by classification. When drawing a straight line through several points on the plane, we should pay attention to the discussion of these points; When drawing graphics, we should pay attention to the possibility of graphics.

2. Equation idea. When dealing with the calculation of angle size and line segment size, it is often necessary to solve it by column equation.

3. Graphics conversion ideas. When learning the concept of angle, we should fully understand the understanding of light rotation. When dealing with graphics, we should pay attention to the application of transformation ideas, such as the mutual transformation between three-dimensional graphics and plane graphics.

4. Turn to thinking. When calculating straight lines, line segments, angles and related figures, it always belongs to the concrete application of formula n(n- 1)/2.

Mathematics knowledge points of grade seven (part two)

The second volume of seventh grade mathematics published by People's Education Press mainly includes six chapters: intersection and parallel lines, plane rectangular coordinate system, triangle, binary linear equation, inequality and inequality group, data collection, arrangement and expression.

Chapter II Intersecting Lines and Parallel Lines

I. Knowledge framework

Two. Knowledge concept

1. Adjacent complementary angles: among the four angles formed by the intersection of two straight lines, two angles with a common vertex and a common edge are adjacent complementary angles.

2. Diagonal: Two sides of one angle are relative extension lines of another angle, and two angles like this are diagonal to each other.

3. Vertical line: When two lines intersect at right angles, they are called perpendicular to each other, and one of them is called the vertical line of the other.

4. Parallel lines: In the same plane, two disjoint lines are called parallel lines.

5. Isostatic angle, internal angle and internal angle of the same side:

Isomorphism angle: ∠10 and ∠ 5. Diagonal lines with the same positional relationship like this are called isomorphism angles.

Internal angles: ∠2 and ∠6 A pair of angles like this is called an internal angle.

The diagonal lines such as ∠ 2 and ∠ 5 are called ipsilateral internal angles.

6. Proposition: A statement that judges a thing is called a proposition.

7. Translation: In a plane, a figure moves a certain distance in a certain direction. This movement of graphics is called translation transformation, or translation for short.

8. Corresponding points: Every point in the new image obtained after translation is obtained by moving a point in the original image. Such two points are called corresponding points.

9. Theorems and properties

The nature of antipodal angle: antipodal angle is equal.

10 attributes of vertical line:

Property 1: There is one and only one straight line perpendicular to the known straight line.

Property 2: Of all the line segments connecting a point outside the straight line and a point on the straight line, the vertical line segment is the shortest.

1 1. Parallelism axiom: One and only one straight line passes through a point outside the straight line and is parallel to the known straight line.

Inference of the axiom of parallelism: If two straight lines are parallel to the third straight line, then the two straight lines are also parallel to each other.

12. Properties of parallel lines:

Property 1: Two straight lines are parallel and equal to the complementary angle.

Property 2: Two straight lines are parallel and the internal dislocation angles are equal.

Property 3: Two straight lines are parallel and complementary.

13. Determination of parallel lines:

Judgment 1: congruent angles are equal and two straight lines are parallel.

Decision 2: The internal dislocation angles are equal and the two straight lines are parallel.

Judgment 3: The internal angles on the same side are equal and the two straight lines are parallel.

This chapter enables students to understand the two positional relationships between the intersection and parallelism of two non-overlapping straight lines on the plane, to study the characteristics of the angle formed by the intersection of two straight lines, the characteristics of the two straight lines being perpendicular to each other, the long-term existence conditions of the two straight lines being parallel and all their characteristics, and the properties of graphic translation transformation, and to design some beautiful patterns through translation. Emphasis: vertical lines and their properties, judgment methods of parallel lines and their properties, translation and their properties, as well as their organization and application. Difficulties: explore the conditions and characteristics of parallel lines, the differences between the conditions and characteristics of parallel lines, and explore the translation relationship between graphics and pattern design by using translation properties.

Chapter III Plane Cartesian Coordinate System

I. Knowledge framework

Two. The concept of knowledge

1. Ordered number pair: A number pair consisting of two numbers A and B in sequence is called an ordered number pair, and it is recorded as (a, b).

2. Plane rectangular coordinate system: On a plane, two mutually perpendicular number axes with a common origin form a plane rectangular coordinate system.

3. Horizontal axis, vertical axis and origin: the horizontal axis is called X axis or horizontal axis; The vertical axis is called Y axis or vertical axis; The intersection of the two coordinate axes is the origin of the plane rectangular coordinate system.

4. Coordinates: For any point P on the plane, the passing P is perpendicular to the X axis and Y axis respectively, and the vertical foot is on the X axis and Y axis respectively. The corresponding numbers a and b are called the abscissa and ordinate of the point p, respectively.

5. Quadrant: Two coordinate axes divide the plane into four parts, the upper right part is called the first quadrant, and the counterclockwise part is called the second quadrant, the third quadrant and the fourth quadrant. The point on the coordinate axis is not in any quadrant.

Plane rectangular coordinate system is the transition from one-dimensional number axis to two-dimensional number axis, and it is also the basis of learning function, which plays a connecting role. In addition, the plane rectangular coordinate system combines points and numbers in the plane, which embodies the idea of combining numbers with shapes. Mastering this section is of positive significance for future study and life. When teaching this chapter, teachers should proceed from reality and cultivate students' innovative ability and application consciousness by determining the position of points on the plane.

The fourth chapter triangle

I. Knowledge framework

Two. The concept of knowledge

1. triangle: A figure composed of three line segments that are not on the same line and are connected end to end is called a triangle.

2. Trilateral relationship: the sum of any two sides of a triangle is greater than the third side, and the difference between any two sides is less than the third side.

3. Height: Draw a vertical line from the vertex of the triangle to the line where the opposite side is located, and the line segment between the vertex and the vertical foot is called the height of the triangle.

4. midline: in a triangle, the line segment connecting the vertex and its relative midpoint is called the midline of the triangle.

5. Angular bisector: The bisector of the inner angle of a triangle intersects the opposite side of this angle, and the line segment between the intersection of the vertex and this angle is called the angular bisector of the triangle.

6. Stability of triangle: The shape of triangle is fixed, and this property of triangle is called stability of triangle.

6. Polygon: On the plane, a figure composed of some line segments connected end to end is called polygon.

7. Interior Angle of Polygon: The angle formed by two adjacent sides of a polygon is called its interior angle.

8. Exterior angle of polygon: The angle formed by the extension line of one side of polygon and its adjacent side is called the exterior angle of polygon.

9. Diagonal polygon: The line segment connecting two nonadjacent vertices of a polygon is called diagonal polygon.

10. Regular polygon: A polygon with equal angles and sides in a plane is called a regular polygon.

1 1. plane mosaic: covering a part of a plane with some non-overlapping polygons is called covering the plane with polygons.

12. Formulas and attributes

Sum of triangle internal angles: The sum of triangle internal angles is 180.

Properties of the external angle of a triangle:

Property 1: One outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it.

Property 2: The outer angle of a triangle is larger than any inner angle that is not adjacent to it.

The sum formula of polygon internal angles: the sum of n polygon internal angles is equal to (n-2) 180.

Sum of polygon outer angles: the sum of polygon inner angles is 360.

The number of diagonals of a polygon: (1) Starting from a vertex of an n polygon, (n-3) diagonals can be drawn, and the polygon can be divided into (n-2) triangles.

(2)n sides * * * have diagonal lines.

Triangle is the basic figure of geometry in junior high school mathematics. In the process of learning, teachers should encourage students to use their brains to discover and explore the mysteries of knowledge. Pay attention to cultivating students' correct mathematical sentiment and geometric thinking ability.

Chapter VIII Binary Linear Equations

I. Knowledge structure diagram

Two. Knowledge concept

1. Binary linear equation: There are two unknowns whose exponents are both 1. Equations like this are called binary linear equations. Equation, the general form is ax+by=c(a≠0, b≠0).

2. Binary linear equations: two binary linear equations are combined into one binary linear equation.

3. Solution of binary linear equations: Generally, the value of unknown quantity that makes the values on both sides of binary linear equations equal is called the solution of binary linear equations.

4. Solution of binary linear equations: Generally speaking, the common * * * solution of two equations of binary linear equations is called binary linear equations.

5. Elimination method: The idea of solving the unknowns one by one from more to less is called elimination thought.

6. Substitution elimination method: an unknown number is represented 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 equation group is obtained. This method is called substitution elimination method, or substitution method for short.

7. Addition, subtraction and elimination method: When the coefficients of the same unknown in two equations are opposite or equal, the two sides of the two equations can be added or subtracted respectively to eliminate the element. This method is called addition, subtraction and elimination, or addition and subtraction for short.

This chapter introduces the concepts of binary linear equation, binary linear equation and binary linear equation through examples, so as to cultivate students' understanding, completeness and profundity of the concepts and enable students to master two solutions of binary linear equation. Emphasis: the solution of binary linear equations, enumerating binary linear equations to solve practical problems. Difficulties: Binary linear equations for solving practical problems.

Chapter 9 Inequality and Unequal Groups

I. Knowledge framework

Two. Knowledge concept

1. A formula for symbolizing the relationship between size. "

2. Solution of inequality: the value of the unknown quantity that makes inequality valid is called the solution of inequality.

3. Solution set of inequality: All solutions of an unknown inequality constitute the solution set of this inequality.

4. One-dimensional linear inequality: the left and right sides of the inequality are algebraic expressions, and there is only one unknown, and the highest order of the unknown is 1. Inequalities like this are called one-dimensional linear inequalities.

5. One-dimensional linear inequality group: Generally, several one-dimensional linear inequalities about the same unknown quantity are combined to form a one-dimensional linear inequality group.

7. Theorems and properties

The essence of inequality:

The basic property of inequality is 1: add (or subtract) the same number (or formula) on both sides of inequality, and the direction of inequality remains unchanged.

The basic property of inequality 2: both sides of inequality are multiplied (or divided) by the same positive number, and the direction of inequality remains unchanged.

The basic property of inequality 3: when both sides of inequality are multiplied (or divided) by the same negative number, the direction of inequality changes.

This chapter requires students to go through the process of establishing mathematical models such as linear inequalities (groups) and applying them to solve practical problems, understand the characteristics and functions of inequalities (groups), master the general methods of using them to solve problems, improve the ability of analyzing and solving problems, and enhance the innovative spirit and consciousness of applied mathematics.

Chapter 10 Data Collection, Arrangement and Description

I. Knowledge framework

full investigation

Random surprise inspection

gather data

Descriptive data

categorical data

analytical data

come to a conclusion

Two. The concept of knowledge

1. Comprehensive survey: The survey method for all the subjects is called comprehensive survey.

2. Sampling survey: The survey method of investigating some data and estimating the whole according to some data is called sampling survey.

3. Population: All the investigated objects are called population.

4. Individual: Every object that constitutes a whole is called an individual.

5. Sample: All the extracted individuals constitute a sample.

6. Sample size: The number of individuals in a sample is called sample size.

7. Frequency: Generally speaking, the number of times the data falls into different groups is called the frequency of that group.

8. Frequency: The ratio of frequency to total data is frequency.

9. Number of groups and distance between groups: When counting data, the data is divided into several groups according to a certain range, and the number of groups is called the number of groups, and the difference between the two ends of each group is called the distance between groups.

This chapter requires us to experience the general process of statistics, feel the role of statistics in life and production, enhance our interest in learning statistics, initially establish the concept of statistics, and cultivate a good habit and scientific attitude of attaching importance to investigation and research.