rational number

★ Key points and difficulties ★ Relevant concepts and properties of rational numbers, comprehensive mastery of number axis, absolute value and reciprocal, and operations of rational numbers (addition, subtraction, multiplication, division, multiplication and mixing operations).

I. Key concepts

The classification and concept of 1 figure

2. Non-negative number: the collective name of positive real number and zero. (Table: x≥0)

Common non-negative numbers are: 0, 1, 2…

Property: If the sum of several non-negative numbers is 0, then every unburdened number is 0.

3. Reciprocity: ① Definition and characterization

② attribute: a.a ≠1/a (a ≠1); B. 1/a，a≠0； C.0 & lta 1； a & gt 1， 1/a

4. Reciprocal: ① Definition and representation

② Properties: A.a when A.a≠0; The position of a and -a on the number axis; The sum of c is 0 and the quotient is-1.

5. Number axis: ① Definition ("three elements")

② Function: a. Visually compare real numbers; B. clearly reflect the absolute value; C. establish a one-to-one correspondence between points and real numbers.

6. Odd number, even number, prime number and composite number (positive integer-natural number)

Definition and expression: odd number: 2n- 1.

Even number: 2n(n is a natural number)

7. Absolute value: ① Definition (two kinds):

Algebraic definition:

Geometric definition: the geometric meaning of the absolute value top of the number A is the distance from the point corresponding to the real number A on the number axis to the origin.

② A ≥ 0, and the symbol "│ │" is a sign of "non-negative number"; ③ There is only one absolute value of number A; ④ When dealing with any type of topic, as long as "│ │" appears, the key step is to remove the "│ │" symbol.

Second, the operation of rational numbers

1. Arithmetic (addition, subtraction, multiplication, division, power and root)

2. Algorithm (five plus [multiplication] commutative law and associative law; Distribution law of multiplication to addition)

3. Operation sequence: a. Advanced operation to low-level operation; B. (Operation at the same level) From "Left"

To the "right" (such as 5 ÷ 5); C (when there are brackets) from "small" to "medium" to "large".

Integral expression

★ Key points and difficulties ★ Related concepts and properties of algebraic expressions, operations of algebraic expressions, removal of brackets (the most commonly used and basic identity deformation in algebraic operations), similar terms, multiplication formulas, and decomposition factors.

I. Key concepts

1. algebraic expression

Algebraic expressions are formed by connecting numbers or letters representing numbers with operational symbols. independent

Numbers or letters are also algebraic.

Rational expressions without division or division but without letters are called algebraic expressions.

Classification: single item, multiple items

3. Monomial and Polynomial

Algebraic expressions without addition and subtraction are called monomials. (product of numbers and letters-including single numbers or letters)

The sum of several monomials is called polynomial.

4. Coefficients and indices

Difference and connection: ① from the position; (2) In the sense of representation.

5. Similar projects and their combinations

Conditions: ① The letters are the same; ② The indexes of the same letters are the same.

Basis of merger: law of multiplication and distribution

9. Index

(1)(- power supply, power supply operation)

①a & gt； 0, > 0; ②a0(n is an even number), < 0(n is an odd number)

(2) Zero index: = 1(a≠0)

Negative integer index: = 1/ a(a≠0, p is a positive integer)

Second, the law of operation and the law of nature

3. Algebraic expression algorithm (bracket deletion and bracket addition)

4. The essence of power operation: ① =; ② ÷ = ; ③ = ; ④ = ;

5. Multiplication rule: (1) single× single; (2) single × many; 3 more x more.

6. Multiplication formula: (plus or minus)

(a+b)(a-b)= (a b) = 2ab+

7. Division rules: (1) single-single; (2) Too many orders.

8. Factorization: (1) definition; ⑵ Methods: A. Common factor method; B. formula method; C. cross multiplication; D. group decomposition method; E. find the root formula method.

1 1. Scientific notation: (1 ≤ A

Equation (group)

★ Emphasis★ Solution of linear equations of one variable and two variables; Related application problems of the equation (especially travel and engineering problems)

I. Basic concepts

1. equation, its solution (root), its solution, its solution (group)

Second, the basis of solving equation-the nature of equation

1.a=b←→a+c=b+c

2.a=b←→ac=bc (c≠0)

Third, the solution

1. Solution of linear equation with one variable: remove denominator → remove brackets → move terms → merge similar terms →

The coefficient becomes 1→ solution.

2. Solution of linear equations: ① Basic idea: "elimination method" ② Method: ① Replacement method.

② addition and subtraction

Six, column equation (group) to solve application problems

(1) Overview

Solving practical problems by using equations (groups) is an important aspect of integrating mathematics with practice in middle schools. The specific steps are as follows:

(1) review the questions. Understand the meaning of the question. Find out what is a known quantity, what is an unknown quantity, and what is the equivalent relationship between problems and problems.

⑵ Set an element (unknown). ① Direct unknowns ② Indirect unknowns (often both). Generally speaking, the more unknowns, the easier it is to list the equations, but the more difficult it is to solve them.

⑶ Use algebraic expressions containing unknowns to express related quantities.

(4) Find the equation (some are given by the topic, some are related to this topic) and make the equation. Generally speaking, the number of unknowns is the same as the number of equations.

5] Solving equations and testing.

[6] answer.

To sum up, the essence of solving application problems by column equations (groups) is to first transform practical problems into mathematical problems (setting elements and column equations), and then the solutions of practical problems (column equations and writing answers) are caused by the solutions of mathematical problems. In this process, the column equation plays a role of connecting the past with the future. Therefore, the column equation is the key to solve the application problem.

(2) Common equal relations

1. Basic relation of travel problem (uniform motion): s=vt.

(1) encounter problems (start at the same time): (2) catch-up problems (start at the same time): (3) sail in water:

2. batching problem: solute = solution × concentration solution = solute+solvent

3. Growth rate:

4. Engineering problems: Basic relationship: workload = working efficiency × working time (workload is often considered as "1").

5. Geometric problems: Pythagorean theorem, area and volume formulas of geometric bodies, similar shapes and related proportional properties.

(C) Pay attention to the interaction between language and analytical expression

Such as more, less, increase, increase to (to), at the same time, expand to (to), ...

Another example is a three-digit number, where A has 100 digits, B has 10 digits and C has one digit. Then this three-digit number is: 100a+ 10b+c, not abc.

Fourth, pay attention to writing equal relations from the language narrative.

For example, if X is greater than Y by 3, then x-y=3 or x=y+3 or X-3 = Y, and if the difference between X and Y is 3, then x-y=3. Pay attention to unit conversion

Such as the conversion of "hours" and "minutes"; Consistency of s, v and t units, etc.