Final outline of mathematics in the second volume of the eighth grade
Fraction and its basic properties
First, the concept of score
1. Definition of fraction: If A and B represent two algebraic expressions, and B contains letters, then this formula is called a fraction.
2. To understand the concept of score, we should grasp the following points:
The (1) fraction is the quotient of the division of two algebraic expressions. The numerator is divided, the denominator is divided, and the fractional line plays the role of divisor and bracket; (2) The numerator of a fraction may or may not contain letters, but the denominator of a fraction must contain letters to be a fraction; (3) The denominator cannot be zero.
3. Conditions for meaningful and meaningless scores.
The meaningful condition of (1) score: the denominator of the score is not equal to 0;
(2) The condition that the score is meaningless: the denominator of the score is equal to 0.
4. The condition that the score is 0:
When the numerator of a fraction is equal to 0 and the denominator is not equal to 0, the value of the fraction is 0. That is, the conditions of make = 0 are: A=0, B≠0.
Step 5 be rational
Algebraic expressions and fractions are collectively called rational forms. Algebraic expressions are divided into monomials and polynomials.
Classification: rational expression
Monomial: an algebraic expression consisting of the product of numbers and letters;
Polynomial: An algebraic expression consisting of the sum of several monomials.
Second, the basic nature of the score
The basic properties of 1. Fraction: both the numerator and denominator of the fraction are multiplied (or divided) by the same algebraic expression that is not equal to zero, and the value of the fraction remains unchanged.
Expressed as: = =, where M(M≠0) is an algebraic expression.
2. General Fraction: Using the basic properties of fractions, the numerator and denominator are multiplied by an appropriate algebraic expression, and several fractions with different denominators are converted into fractions with the same denominator without changing the value of fractions. This kind of fractional deformation is called the general fraction of a fraction.
The key to general division is to determine the simplest common denominator of several fractions. The general method to determine the simplest common denominator is: (1) If all denominators are monomials, then the simplest common denominator is the least common multiple of all coefficients, the power of the same letter, and the product of all different letters and exponents. (2) If each denominator has a polynomial, then the denominator should first be the factorization factor of the polynomial, and then the simplest common denominator can be determined from three aspects: coefficient, same factor and different factor.
3. Simplification: According to the basic properties of the fraction, the common factor of the numerator and denominator of the fraction is removed without changing the value of the fraction. This fractional distortion is called fractional reduction.
Note: (1) If both the numerator and denominator are monomials, the common factor of the numerator and denominator can be directly eliminated, that is, the common factor of the numerator and denominator coefficients and the lowest power of the same letter can be eliminated; (2) If there is at least one polynomial in the numerator and denominator, first decompose the factors, then find their common factors, and then reduce them; (3) The divisor must be finished.
Third, the symbolic law of fractions:
( 1)==-; (2)=; (3)-=
Fractional operation
A, fractional multiplication and division method
1, rule:
(1) multiplication rule: the fraction is multiplied by the fraction, the product of the numerator is the numerator of the product, and the product of the denominator is the denominator of the product. (i.e. fractional multiplication, molecular multiplication, denominator multiplication).
Expressed by the formula:
(2) Division rule: Divide a fraction by a fraction, then reverse the numerator and denominator of the divisor and multiply it by the divisor.
Expressed by the formula:
2. When applying the law, we should pay attention to: (1) The law of the sign in the fraction is the same as that in the rational number multiplication and division method, that is, "the same sign is positive, the different sign is negative, and there are multiple negative signs according to the number, and the odd negative is even positive"; (2) When the denominator of the numerator is a polynomial, factorization should be carried out first to reduce the score; (3) The result of fractional multiplication and division should be simplified to the simplest form.
Second, the power of scores.
1, Law: According to the meaning of power and the law of fractional multiplication, the power of a fraction is to multiply the numerator and denominator respectively and then divide them.
Expressed by formula: (where n is a positive integer, a≠0)
2. Note: When multiplying (1), the score must be placed in brackets; (2) When a formula contains both power and multiplication and division, the power should be calculated first, and then the multiplication and division should be calculated. If there is a polynomial, it must be factorized first and then simplified. (3) The final result should be simplified.
Third, the addition and subtraction of scores.
(1) Addition and subtraction of fractions with the same denominator
1, rule: add and subtract with denominator fraction, denominator unchanged, numerator added and subtracted.
Expressed by the formula:
2. Note: (1) "Addition and subtraction of molecular phase" refers to the addition and subtraction of all "whole molecules", and each molecule should have brackets; When the numerator is a monomial, parentheses can be omitted, but when the denominator is a polynomial, parentheses cannot be omitted. (2) The result of fractional addition and subtraction must be converted into the simplest fractional or algebraic expression.
(2) Addition and subtraction of fractions with different denominators
1, rule: add and subtract fractions with different denominators, divide first, convert fractions with the same component, and then add and subtract. Use the formula:.
2. Precautions: (1) In addition and subtraction of scores with different denominators, dividing points first is the key. Change addition and subtraction of scores with different denominators into addition and subtraction of scores with the same denominator. (2) If the algebraic expression is included in the fractional addition and subtraction operation, it should be regarded as the denominator of 1, and then the division operation is performed. (3) When the number of numerator is higher than or equal to the number of denominator, it should be decomposed into algebraic sum and the sum of true fraction to participate in the operation, which can make the operation simple.
Fourth, the mixed operation of fractions
1, operation rules: addition, subtraction, multiplication, division and multiplication of fractions are mixed operations. Multiply first, then multiply and divide, and finally add and subtract. When you encounter brackets, you must first count what is in the brackets.
2. Note: (1) The key of fractional mixing operation is to find out the operation order; (2) The operation order and law of rational numbers are also suitable for fractional operation, and the exchange law, association law and distribution law should be used flexibly; (3) The result of fractional operation must be reduced to the simplest and reducible offer point, so as to ensure that the operation result is the simplest fractional or algebraic expression.
Fractional equation that can be reduced to one-dimensional linear equation
First, the basic concept of fractional equation
1. Definition: An equation with fractions and unknowns in the denominator is called a fractional equation.
2. To understand the fractional equation, two points should be made clear: (1) equation contains fractions; (2) The denominator of the score contains an unknown number.
The difference between fractional equation and integral equation lies in whether there are unknowns in denominator.
Second, the solution of fractional equation
1. The basic idea of solving fractional equation is to turn fractional equation into integral equation. Method: "denominator".
The method is: multiply both sides of the equation by the simplest common denominator of each clause, remove the denominator and become an integral equation to solve.
2, the general steps to solve the fractional equation:
(1) denominator. That is, multiply both sides of the equation by the simplest common denominator of each fraction, remove the denominator, and turn the original fractional equation into an integral equation;
(2) solving the whole equation;
(3) Root inspection. Root-checking method: substitute the root of the whole equation into the simplest common denominator, so that the root with the simplest common denominator not equal to 0 is the root of the original fractional equation, and the root with the simplest common denominator 0 is the additional root of the original fractional equation, which must be discarded. This root test method can't check the calculation errors in the process of solving the equation, so another root test method can be used, that is, the unknown values obtained are substituted into the original equation for testing, and whether there are calculation errors in the process of solving the equation can be found.
3. Increase the root of the fractional equation. The significance is that after the fractional equation is transformed into an integral equation, the root of the solved integral equation is sometimes only the root of the algebraic expression equation, not the root of the original fractional equation, and this root is an increased root, so it is necessary to test the root to solve the fractional equation.
Third, the application of fractional equation
1, meaning: the application of fractional equation is to solve application problems with fractional equation, which is basically the same as the method, steps and thinking of solving application problems with linear equation of one variable. The difference is that because of the concept of fraction, the relationship between listed algebras is no longer limited by algebraic expressions. The listed equations contain fractions and the denominator contains unknowns, which need to be tested after solving the equations.
2. The general steps of solving application problems with column fractional equations are as follows:
(1) review the questions. Understand the meaning of the question and find out the known conditions and unknown quantities;
(2) Set an unknown number. There are two ways to reasonably set an unknown to represent an unknown: direct method and indirect method.
(3) Find out the equivalence relation in the topic and write the equation;
(4) The statements on both sides of the equation are expressed by algebraic expressions containing known quantities and unknowns, and the equations are listed;
(5) Solve the equation. Seek the unknown value;
(6) inspection. It is necessary to check not only whether the value of the unknown is the root of the original equation, but also whether the value of the unknown conforms to the practical significance of the topic. "Double root test".
Zero exponential power and negative integer exponential power
I. Zero exponential power
1. Definition: The zeroth power of any real number that is not equal to zero is equal to 1, that is, a0= 1(a≠0).
2, special attention: the zero power of zero is meaningless. That is, x = _ _ _ _ 00 is meaningless, and (x-2)0 is meaningful. The answer is x≠2.
(2) according to the definition, it is divided into:
Second, negative integer exponential power
1, definition: Any number that is not equal to the power of -n (n is a positive integer) is equal to the reciprocal of the power of this number.
That is, a-n=(a≠0, n is a positive integer)
2. Precautions:
If the radix is not 0, the power of (1) negative integer exponent holds;
(2) All the algorithms of positive integer exponential power are suitable for negative algebraic expression exponential power, that is, the operation of exponential power can be extended to the range of integer exponential power;
(3) In order to avoid errors like 5-2=-2×5=- 10, the correct algorithm is:
Third, the number whose absolute value is less than 1 is expressed by scientific counting method.
1. Rule: The number whose absolute value is less than 1 is expressed as a× 10-n (n is a positive integer), where 1 ≤| a |.
2. Precautions:
(1)n is the number of all zeros before the first non-zero number on the left of the number (including zeros before the decimal point). Such as-0.00021=-2.1×10-4.
(2) Pay attention to the change of digital symbols. If there is a negative sign in front of the number, the result should also be written with a sign.
(3) The key to writing scientific notation is to determine the value of the exponent n of 10n.
Mathematics review method
The process of mathematics learning is the process of thinking development. Candidates can learn math well only by opening their own minds. To open your mind, candidates need to use their brains and think more. When you usually do the problem, don't turn over the answer when you see the problem. On the contrary, candidates should seriously study the topic, think about the characteristics of the topic, and find ideas and methods to solve the problem. Of course, there is also a time limit. Generally speaking, you should think carefully for three minutes. If you still don't have a clue after three minutes, the candidates will give up this question first and look at it later when they have time.
Candidates don't know how well they master the knowledge points told by the teacher in class if they don't pass the exercises. Candidates should do the corresponding exercises. In other words, the general teacher will arrange exercises for the content of the class. The quantity doesn't need to be too much, just two or three. If there are questions that can't be done, candidates should ask them in time and don't stay there. When problems accumulate, it will be difficult to solve them.
Math answering skills
First of all, make a complete and comprehensive understanding of your own mathematics learning. During the exam, we should accurately locate the center of gravity according to our own situation and prevent "picking up sesame seeds and losing watermelon". Therefore, in your mind, you must give a deadline for the finale or several "difficulties". If you exceed the upper limit you set, you must stop. Go back and carefully check the previous questions, try to ensure that you have multiple-choice questions and fill-in-the-blank questions, and try to check the previous solutions.
The second is to solve the mathematical finale problem. The first question is not a problem for most students; If you can't understand the first question, don't give up the second question easily. The process will be written as much as possible, because the mathematical solution is graded step by step, and the writing must be standardized, the handwriting should be neat, and the layout should be reasonable; Write as much as possible in the process, but don't talk nonsense, and try to avoid unnecessary components in the calculation; Try to use more geometric knowledge, less algebraic calculation, try to use trigonometric functions, and less the properties of similar triangles in right triangles.
Third, solving the mathematical finale problem can generally be divided into three steps. Carefully examine the questions, understand the meaning of the questions, explore the ideas of solving problems, and answer correctly. The examination of questions should comprehensively examine all the conditions and answering requirements of the questions, and grasp the characteristics and structure of the questions as a whole, so as to facilitate the selection of solving methods and the design of solving steps. To solve the mathematical finale, we should be good at summarizing the important mathematical ideas implied in the mathematical finale, such as the idea of reduction, the idea of combining numbers with shapes, the idea of classified discussion, the idea of equations and so on. Understand the relationship between conditions and conclusions, the relationship between geometric characteristics of figures and the number and structural characteristics of numbers and formulas, and determine the ideas and methods of solving problems. When thinking is blocked, we should adjust our thinking and methods in time, re-examine the meaning of the question, pay attention to excavating hidden conditions and internal relations, and prevent us from falling into a dead end and giving up easily.
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