1. Correctly understand the concepts of straight line, ray and line segment and their differences;

The name graph represents the endpoint length of the method.

Straight line AB (or BA)

The end point of the straight line l cannot be measured.

X-ray OM 1 unmeasurable

Line AB (or BA)

Measurable length of L2 line segment

2. axiom of straight line: there is only one straight line after two o'clock.

2. Compare the lengths of line segments

1. line segment axiom: the line segment between two points is the shortest; The length of the line segment between two points is called the distance between these two points.

2. Two methods to compare the length of line segments:

① Compass interception comparison method;

② Scale measurement comparison method.

3. The midpoint of the line segment, the sum, difference, time and minute of the line segment can be drawn with scales;

The sum, difference and degree of line segments can be drawn with compasses.

3. Angle measurement and representation

1. Angle: A graph composed of two rays with a common endpoint is called an angle;

This common endpoint is called the vertex of the angle;

These two rays are called the edges of the angle.

2. Representation of angle: the symbol of angle is "∞"

The operation of finding the product of n identical factors is called power, and the result of power is called power. In the n power of a, a is called the base and n is called the exponent. The odd power of a negative number is negative, and the even power of a negative number is positive (negative odd negative, negative even positive). Any power of a positive number is a positive number, and any power of 0 is 0.

② The even power of two values is equal to a positive number (two opposite numbers), such as a2=4, a=2 or a=-2.

Note: |a|+b2=0: a=0, b=0.

Note: A0 =1(a ≠ 0); (- 1)2= 1 ; - 12=- 1; (- 1)3=- 1;

- 13=- 1; (-2)2 =4; -22=-4; (-2)3 =-8; -23=-8

③ Mixed arithmetic of rational numbers: multiply first, then divide, and finally add and subtract; Operation at the same level, from left to right; If there are brackets, do the operation in brackets first, and then follow the brackets, brackets and braces in turn. Note: 12-4×5 = 12-20(- cannot be changed to+).

(4) Numbers greater than 10 are expressed by the n power of a× 10, and scientific counting method is used. Note that the range of A is 1≤a n minus 1, starting from the original integer bit. (Pay attention to scientific counting methods and original numbers.

(5) where the rounding is accurate. When rounding, look back at a place and use rounding. For example: 3.5449 is accurate to 0.0 1 is 3.54 instead of 3.55. (Another example: 24,000: accurate to 100; 6.5× 104 accurate to thousands. If there is an order of magnitude and scientific counting method, it should be restored to the original number, depending on the order of magnitude and the last number of scientific counting methods).

Three elements of a 3-digit axis: origin, positive direction and unit length (all three are indispensable).

Any rational number can be represented by a point on the number axis. (Conversely, it cannot be said that all points on the number axis represent rational numbers. )

If two numbers differ only in sign, then we call one of them the reciprocal of the other number, and we also call these two numbers the reciprocal of each other. (The reciprocal of 0 is 0)

On the number axis, two points representing the opposite number are located on one side of the origin, and the distance from the origin is equal.

The number represented by two points on the number axis is always larger on the right than on the left. Positive numbers are on the right of the origin, and negative numbers are on the left of the origin.

Definition of absolute value: The absolute value of a number A is the distance between the point representing the number A on the number axis and the origin. The absolute value of the number a is recorded as |a|.

The absolute value of a positive number is itself; The absolute value of a negative number is its number; The absolute value of 0 is 0.

or

Nature of absolute value: In addition to 0, there are two numbers whose absolute values are positive and the two numbers are opposite;

The absolute values of two mutually opposite numbers (except 0) are equal;

The absolute value of any number is always non-negative, that is |a|0.

Comparing the sizes of two negative numbers, the larger absolute value is smaller. The steps to compare the sizes of two negative numbers are as follows:

① Find the absolute values of two negative numbers first;

② Compare two absolute values;

(3) According to two negative numbers, the absolute value is larger but smaller to make a correct judgment.

Nature of absolute value:

① For any rational number A, there is |a|0.

② If |a|=0, |a|=0, and vice versa.

③ If |a|=b, then a = b..

④ For any rational number A, there is |a|=|-a|.

Rational number addition rule:

① Add two numbers with the same symbol, take the same symbol, and add the absolute values.

② Two numbers with different signs are added, and when the absolute values are equal, the sum is 0; When the absolute values are not equal, take the sign of the number with larger absolute value, and subtract the absolute value of the smaller number from the absolute value of the larger number.

③ A number with 0 still gets this number.

The commutative law and associative law of addition are also applicable to rational number operation.

Flexible use of algorithms and simplification of operations usually have the following rules:

① You can add two opposite numbers first;

Two numbers with the same sign can be added first;

Numbers with the same mother and daughter can be added first;

(4) Several numbers can be added to get an integer, which can be added first.

Rational number subtraction rule:

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

Pay attention to two changes of rational number subtraction;

① Change the operation symbol;

(2) Change the sign of the nature of reduction (become the opposite number)

When doing rational number subtraction, pay attention to a constant: the position of the minuend and the minuend cannot be changed, that is to say, there is no commutative law in subtraction.

Steps of rational number addition and subtraction mixed operation:

(1) is written as an algebraic sum with the plus sign omitted. In an equation, if there is subtraction, it is necessary to convert the subtraction rule of rational numbers into addition, and then omit the plus sign and brackets;

② The calculation is simplified by using the law of addition, additive commutative law and the law of association.

(Note: Subtracting a number equals adding the reciprocal of this number. When subtraction is unified into addition, subtraction should become its own inverse. )

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 0, and the product is still 0.

If two numbers are reciprocal, their product is 1. (such as -2 and so on. )

The commutative law, associative law and distributive law of multiplication are also applicable to rational number operation.

Rational number multiplication operation steps:

(1) first determine the product logo;

② Find the product of the absolute values of each factor.

Two rational numbers whose product is 1 are reciprocal. note:

There is no reciprocal of zero.

(2) To find the reciprocal of a fraction is to reverse the numerator and denominator of the fraction. A band's score must first become a fake score.

③ The reciprocal of a positive number is a positive number, and the reciprocal of a negative number is a negative number.

Rational number division rule:

① Divide two rational numbers, the same sign is positive and the different sign is negative, and divide by the absolute value.

② Divide 0 by any number other than 0 to get 0. 0 cannot be used as a divisor, otherwise it is meaningless.

Power of rational number

note:

(1) A number can be regarded as its own power, such as 5 = 51;

(2) When the radix is negative or fractional, the radix should be enclosed in brackets first, and then the index should be written in the upper right corner.

Operating characteristics of power supply:

① Any degree of a positive number is a positive number;

② The odd power of a negative number is negative and the even power of a negative number is positive;

③ Even powers of any number are nonnegative;

④ Any power of 1 gets1,and any power of 0 gets 0;

⑤ The even power of-1is1; The odd power of-1 is-1;

⑥ In the process of operation, we must first determine the sign of the power, and then calculate the absolute value of the power.

Rational number hybrid algorithm;

① Calculate the power first, then multiply and divide, and finally add and subtract.

(2) If there are parentheses, calculate the parentheses first.

Numbers greater than 0 are called positive numbers.

② Numbers with "-"in front of positive numbers are called negative numbers.

③0 is neither positive nor negative. 0 is the boundary between positive and negative numbers and is the only neutral number.

④ Find out the quantities with opposite meanings: North and South; Things; Up and down; Left and right; Rise and fall; High and low; Growth decline, etc.

⑤ Positive integers, 0 and negative integers are collectively called integers (combined with the number axis and a linear equation), and positive and negative fractions are collectively called fractions. Integers and fractions are collectively called rational numbers.

6. Non-negative numbers are positive numbers and zero; Non-negative integers are positive integers and 0.

⑦ "benchmark" problem: there is a fixed benchmark number, and the solution of sum is: benchmark number × number+algebraic sum compared with benchmark number; Solution of average: Comparative algebra of reference number+number and reference number and number ? (write the original number, or answer it with primary school knowledge); "Non-benchmark" problem: There is no fixed benchmark number, such as tomorrow versus today and the day after tomorrow versus tomorrow.

The first volume of the first grade, knowledge points at the end of mathematics, the fifth chapter, algebraic multiplication;

(1) Multiply the monomial with the monomial, multiply them by their coefficients and the power of the same letter respectively, and the remaining letters, together with their exponents, are the factors of the product.

(2) Multiplying polynomial by monomial means multiplying each term of polynomial by monomial according to the distribution law, and then adding the products.

(3) Polynomial multiplied by polynomial. Multiply each term of one polynomial by each term of another polynomial, and then add the products.

The first volume of the first day of junior high school, the final knowledge points of mathematics, the sixth real number:-rational numbers and irrational numbers are collectively called real numbers.

Rational Numbers: Integers and fractions are collectively called rational numbers.

Irrational number: Irrational number refers to infinite acyclic decimal.

Natural numbers: 0, 1, 2, 3, 4 ~ (including 0) indicate that the number of objects is called natural numbers.

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

Inverse number: Two numbers with different signs are inverse numbers.

Reciprocal: Two numbers whose product is 1 are reciprocal.

Absolute value: the distance between points representing the number A on the number axis is called the absolute value of A. The absolute value of a positive number is itself, the absolute value of a negative number is its reciprocal, and the absolute value of 0 is 0.

The first day of mathematics, the first book, 7 bottom knowledge points, concept combing

⑴ The general steps to solve practical problems by enumerating linear equations with one variable are: examining questions, paying special attention to keywords and their meanings, making clear the relevant quantitative relations, paying attention to the unity of units and setting unknowns;

① Solution: Set the unknown number (pay attention to the unit),

② List the equations according to the equation relationship,

③ Solve this equation,

(4) Answer (including company name, it is best to take the exam).

(2) Equivalence relations in some fixed models:

① Number problem: If it is represented by a three-digit number, it will be = 100a+ 10b+c (number on digit × digit).

② Travel problem: basic formula: distance = time × speed.

When A and B walk in opposite directions at the same time and meet, the distance of A+the distance of B = the total distance.

A departure time = B departure time;

Both parties go in the same direction at the same time to catch up with the time: the distance of Party A-the distance of Party B = the distance of both parties.

③ Engineering problem (whole 1): basic formula: workload = working time × working efficiency.

Sum of workload of each part = total workload;

④ Savings problem: the sum of principal and interest = principal+interest; Interest = principal × interest rate× time

⑤ Commodity sales problem: Commodity profit = selling price-purchase price (cost price)

Commodity profit rate = (selling price-purchase price)/purchase price

⑥ Equal area deformation problem: the area or volume is unchanged.

⑦ Sum, difference, multiple and score: more, less, multiple and score.

⑧ Proportional distribution: Generally, let each copy be X, for example, 2:3:4 is 2x, 3x, 4x.

9 Resource allocation: the allocation of resources and personnel (sometimes an unknown number is indirectly set).

Second, the way of thinking (summarize the mathematical thinking methods commonly used in this unit)

⑴ Model idea: through the analysis of the quantitative relationship in practical problems, abstract it into a mathematical model and establish a linear equation of one variable.

⑵ Equation idea: the idea of solving practical problems with equations (such as proportional distribution, line segment length, angle size, etc. ) is the idea of equation.

⑶ Transformation (inductive) thinking: The process of solving a linear equation with one variable is essentially to transform the coefficient of unknown quantity into 1 by removing the denominator, brackets, shifting terms and similar terms, constantly replacing the original equation with new and simpler equations, and finally gradually transforming the equation into the form of x=a, which embodies the transformation from "unknown" to "known".

⑷ The idea of combining numbers with shapes: For example, when solving travel problems with series equations, the quantitative relationship is analyzed with the help of line segments and charts, so that the quantitative relationship in the problem is displayed intuitively, which embodies the superiority of combining numbers with shapes.

5. The idea of classification (whole): For example, absolute value, even power, points on line segments (on extension lines, outside line segments) and angles (outside angles) often need to be classified and discussed in the process of solving the letter coefficient equation and the absolute value symbol equation, and attention should be paid to the application of classification ideas in the process of solving practical problems related to scheme design.

Numbers greater than 0 are called positive numbers.

② Numbers with "-"in front of positive numbers are called negative numbers.

③0 is neither positive nor negative. 0 is the boundary between positive and negative numbers and is the only neutral number.

④ Find out the quantities with opposite meanings: North and South; Things; Up and down; Left and right; Rise and fall; High and low; Growth decline, etc.

⑤ Positive integers, 0 and negative integers are collectively called integers (combined with the number axis and a linear equation), and positive and negative fractions are collectively called fractions. Integers and fractions are collectively called rational numbers.

6. Non-negative numbers are positive numbers and zero; Non-negative integers are positive integers and 0.

⑦ "benchmark" problem: there is a fixed benchmark number, and the solution of sum is: benchmark number × number+algebraic sum compared with benchmark number; Solution of the average: the benchmark number+the algebraic sum of the numbers compared with the benchmark number (write the original number, or answer it with primary school knowledge); "Non-benchmark" problem: There is no fixed benchmark number, such as tomorrow versus today and the day after tomorrow versus tomorrow.