Hunan education edition eighth grade first volume mathematics outline
(A) the use of formula method
We know that algebraic multiplication and factorization are inverse deformations of each other. If the multiplication formula is reversed, the polynomial is decomposed into factors. So there are:
a2-b2=(a+b)(a-b)
a2+2ab+b2=(a+b)2
a2-2ab+b2=(a-b)2
If the multiplication formula is reversed, it can be used to factorize some polynomials. This factorization method is called formula method.
(2) Variance formula
formula for the difference of square
Equation (1): a2-b2=(a+b)(a-b)
(2) Language: the square difference of two numbers is equal to the product of the sum of these two numbers and the difference of these two numbers. This formula is the square difference formula.
(3) Factorization
1. In factorization, if there is a common factor, first raise the common factor and then decompose it further.
2. Factorization must be carried out until each polynomial factor can no longer be decomposed.
(4) Complete square formula
(1) Reversing the multiplication formula (a+b)2=a2+2ab+b2 and (a-b)2=a2-2ab+b2, we can get:
a2+2ab+b2=(a+b)2
a2-2ab+b2=(a-b)2
That is to say, the sum of squares of two numbers, plus (or minus) twice the product of these two numbers, is equal to the square of the sum (or difference) of these two numbers.
Equations a2+2ab+b2 and a2-2ab+b2 are called completely flat modes.
The above two formulas are called complete square formulas.
(2) the form and characteristics of completely flat mode
① Number of projects: three projects.
② Two terms are the sum of squares of two numbers, and the signs of these two terms are the same.
A term is twice the product of these two numbers.
(3) When there is a common factor in the polynomial, the common factor should be put forward first, and then decomposed by the formula.
(4) A and B in the complete square formula can represent monomials or polynomials. Here as long as the polynomial as a whole.
(5) Factorization must be decomposed until every polynomial factor can no longer be decomposed.
(5) Grouping decomposition method
Let's look at the polynomial am+an+bm+bn. These four terms have no common factor, so we can't use the method of extracting common factor, and we can't use the formula method to decompose the factors.
If we divide it into two groups (am+an) and (bm+bn), these two groups can decompose the factors by extracting the common factors respectively.
Original formula =(am+an)+(bm+bn)
=a(m+n)+b(m+n)
Doing this step is not called factorization polynomial, because it does not conform to the meaning of factorization. But it is not difficult to see that these two terms have a common factor (m+n), so they can be decomposed continuously, so
Original formula =(am+an)+(bm+bn)
=a(m+n)+b(m+n)
=(m+n)×(a+b)。
The key to learning mathematics well is to summarize and classify it in time and properly. Next, I'll sort out this article about congruent triangles's knowledge points in eighth grade mathematics published by PEP, hoping to help everyone.
Congruent triangles's property: congruent triangles's corresponding edges are equal, and its corresponding angles are equal.
Congruent triangles's judgment: three sides are equal (SSS), two sides are equal to their included angle (SAS), two angles are equal to their sandwiched edge (ASA), two angles are equal to the opposite side of an angle (AAS), and two right-angled triangles are equal to their hypotenuse and right-angled edge (HL).
The nature of the angle bisector: the angle bisector bisects the angle, and the distance between points on the angle bisector is equal to both sides of the angle.
Inferred from the bisector of the angle: the points from the inside of the angle to both sides of the angle are on the bisector.
The basic method steps to prove the congruence of two triangles or to prove the equality of line segments or angles with it are as follows: ①. Determine the known conditions (including implicit conditions, such as common edge, common angle, diagonal, bisector of angle, median line, height and isosceles triangle, etc.). ); 2. Review the triangle judgment and find out what we need; (3) Write the proof format correctly (the order and correspondence are derived from the known.
This method of decomposing factors by grouping is called grouping decomposition. As can be seen from the above example, if the terms of a polynomial are grouped and their other factors are exactly the same after extracting the common factor, then the polynomial can be decomposed by group decomposition.
(6) Common factor method
1. When decomposing a polynomial by extracting the common factor, first observe the structural characteristics of the polynomial and determine the common factor of the polynomial. When the common factor of each polynomial is a polynomial, it can be converted into a monomial by setting auxiliary elements, or the polynomial factor can be directly extracted as a whole. When the common factor of the polynomial term is implicit, the polynomial should be deformed or changed in sign appropriately until the common factor of the polynomial can be determined.
2. Use the formula x2+(p+q)x+pq=(x+q)(x+p) for factorization, and pay attention to:
1) must first decompose the constant term into the product of two factors, and the algebraic sum of these two factors is equal to
Coefficient of linear term.
2) Many attempts have been made to decompose the constant term into the product of two factors that meet the requirements. The general steps are as follows:
(1) lists all possible situations in which a constant term is decomposed into the product of two factors;
(2) try which sum of two factors is exactly equal to the first-order coefficient.
3) decompose the original polynomial into (x+q)(x+p).
(7) Multiplication and division of fractions
1. Dividing the numerator of a fraction by the common factor of the denominator is called the divisor of the fraction.
2. The purpose of score reduction is to reduce this score to the simplest score.
3. If the numerator or denominator of the fraction is a polynomial, we can first consider decomposing it into factors to get the form of factor product, and then reduce the common factor of the numerator and denominator. If the polynomial in the numerator or denominator can't decompose the factor, then we can't separate some items in the numerator and denominator.
4. Pay attention to the correct use of the sign law of power in fractional reduction, such as x-y=-(y-x), (x-y)2=(y-x)2, (x-y)3=-(y-x)3.
5. The numerator or denominator of a fraction is signed to the nth power, which can be changed into the symbol of the whole fraction according to the sign law of the fraction, and then treated as the positive even power and negative odd power of-1. Of course, the numerator and denominator of a simple fraction can be directly multiplied.
6. Pay attention to the parentheses, then the power, then the multiplication and division, and finally the addition and subtraction.
(viii) Addition and subtraction of scores
1. Although general fractions and reduction are aimed at fractions, they are two opposite variants. Reduction is for one score, while general scores are for multiple scores. The approximate fraction is a simplified fraction, and the general fraction is a simplified fraction, thus unifying the denominator of the fraction.
2. Both the general score and approximate score are deformed according to the basic properties of the score, and their common point is to keep the value of the score unchanged.
3. The general denominator is written in the form of unexpanded continuous product, and the numerator multiplication is written in polynomial to prepare for further operation.
4. Total score basis: the basic nature of the score.
5. The key to general division is to determine the common denominator of several fractions.
Usually, the product of the powers of all factors of each denominator is taken as the common denominator, which is called the simplest common denominator.
6. By analogy, get the total score of this score:
Changing several fractions with different denominators into fractions with the same mother equal to the original fraction is called the general fraction of fractions.
7. The rules for adding and subtracting fractions with the same denominator are: adding and subtracting fractions with the same denominator and adding and subtracting numerators with the same denominator.
Addition and subtraction of fractions with the same denominator, denominator unchanged, addition and subtraction of molecules, that is, the operation of fractions is transformed into the operation of algebraic expressions.
8. Fraction addition and subtraction law of different denominators: Fractions of different denominators are added and subtracted, first divided by fractions of the same denominator, and then added and subtracted.
9. Fractions with the same denominator are added and subtracted, and the denominator remains the same. Add and subtract molecules, but pay attention to each molecule as a whole, and put parentheses in due course.
10. For the addition and subtraction between the algebraic expression and the fraction, the algebraic expression is regarded as a whole, that is, it is regarded as a fraction with the denominator of 1, so as to divide.
1 1. For addition and subtraction of fractions with different denominators, first observe whether each formula is the simplest fraction. If the fraction can be simplified, it can be simplified first and then divided, which will simplify the operation.
12. As the final result, if it is a score, it should be the simplest score.
(9) One-dimensional linear equation with letter coefficient
One-dimensional linear equation with letter coefficient
Example: A times (a≠0) of a number is equal to B, so find this number. This number is represented by X. According to the meaning of the question, the equation ax=b(a≠0) can be obtained.
In this equation, X is unknown, and A and B are known numbers in letters. For x, the letter a is the coefficient of x and b is a constant term. This equation is a one-dimensional linear equation with letter coefficients.
The solution of the letter coefficient equation is the same as that of the numerical coefficient equation, but special attention should be paid to: multiply or divide two sides of the equation with a letter, and the value of this formula cannot be equal to zero.
How to improve junior high school math scores
Learn the basic knowledge of mathematics
If you want to learn math well, you must learn this method of memory and understanding. Understanding is a necessary rule of learning. Only by understanding accurately, sticking to the theme and combining methods can it be solved. As long as we can understand the structure of this topic well, we can work out the answer well. In mathematics learning, memory and reasoning should be closely combined. For example, in the chapter of trigonometric function, all formulas are based on the definition and addition theorem of trigonometric function. While memorizing the mathematical formula, you can use some examples for reasoning, which can speed up your understanding and memory of this formula.
Mathematical problem solving
Learning mathematics must be down-to-earth, there are not so many opportunistic methods, and mathematics practice should pay attention to high quality and suit the remedy to the case. For example, we should get into the habit of analyzing first and then doing problems. If you don't understand anything, you can mark it first, and then communicate with your classmates and teachers. Try to combine a variety of problem-solving methods and practice more.
Wrong problem set
For the wrong questions, list all the solutions (you can ask them from the answers or from your classmates and teachers), and there is always one that you can master. Explaining several sets of test papers can have obvious effects. At first, it seems that all the solutions to each problem take a long time to understand, but the effect is very obvious.
homework
Homework is no stranger to many students. Teachers usually assign some homework after class in order to make full use of what they have learned in class. It is not enough to rely only on listening in class, but also to practice after class to consolidate the knowledge learned in class.
How to learn junior high school mathematics well
First, take the initiative to preview before class.
First of all, the knowledge learned in a math class in junior high school is much more than that in primary school. Furthermore, many primary school students can master mathematics by themselves, but junior high school mathematics is completely different, with more knowledge and complex knowledge points, which requires students to learn to preview actively. In the preview before class, take the initiative to master the context of knowledge points and draw what you have mastered and have questions. In order to study in a targeted way, there is a preview context to help you quickly keep up with the rhythm of the teacher's lecture. Secondly, what you don't understand in the preview can help you understand and analyze the teacher's ideas and methods in class, so that the classroom can be efficient and the math class can be prepared. Therefore, preview is one of the important pre-class preparations for learning junior high school mathematics.
Second, learn to think positively.
Many students of the author reflect that they can understand a lot of contents in junior high school math class, why they still can't do it after class. Actually, in my opinion, this problem is caused by students listening more and thinking less in class. Many students only listen to the teacher in class and never take the initiative to think about why the teacher has such a way of thinking. Precision mathematics is to cultivate students' logical thinking ability. Once you only listen and think less, you will only lose the necessary traces of thinking about the logical relevance of knowledge, resulting in you not getting a topic after class.
Third, be good at summing up laws.
Having said that, the author first gives an example of a mistake that many junior high school students will make in math learning. Do many students always make mistakes on the same type of problems, often? I also took notes on the wrong questions. Why did I change this type of question into another form by mistake?
The emergence of this problem is actually that students lack the habit of summarizing laws. A kind of topic is repeatedly wrong, often wrong, indicating that you have not mastered the law of doing this kind of topic. Not only do you have to take notes on the wrong questions, but you also need to take out all the wrong questions, draw inferences by analogy, and find out where you are wrong every time. Is there any problem in mastering any knowledge points? Or other reasons. We should be good at summing up the rules, compare and summarize the same types of problems, sum up our own ideas and methods to solve problems, and then use the summarized rules and methods to solve such problems. So students, you should not only take notes on the wrong questions, but also be good at summing up the rules. Only by constantly summarizing and summarizing can your thinking be continuously improved and your problem-solving methods be continuously enriched.
Articles related to the first volume of the eighth grade mathematics syllabus of Hunan Education Press;
★ Key Notes on Mathematics at the End of Volume I of Grade Eight
★ The first volume of mathematics teaching plan for eighth grade of Hunan Education Press.
★ Xinxiang Education Publishing House eighth grade mathematics first volume teaching plan.
★ Teaching plan of the first volume of eighth grade mathematics in Hunan Education Press.
★ Hunan education printing plate eighth grade mathematics first volume learning method big vision answer
★ Work Plan of Mathematics Teaching in the Last Term of Grade 8 of Hunan Education Press
★ Work Plan for Mathematics Teaching in the First Volume of Grade 8 of Hunan Education Press (2)
★ Eight Grade Mathematics Teaching Plan of Hunan Education Press
★ Review materials of junior high school mathematics by Hunan Education Press.
★ Hunan Education Edition Senior One Mathematics Knowledge Points