Knowledge platform

concept

Monomial (monomial):

1. any algebraic expression in the form of a product of letters and numbers (in division, dividing by a number is equal to multiplying the reciprocal of this number).

2. A letter or number is also called a monomial.

3. The denominator has no letters (the monomial is an algebraic expression, not a fraction).

A, -5, 1x, 2XY and x/2 are all monomials, while 0.5m+n and 2/x are not monomials.

The number of single events refers to the exponential sum of all letter factors in a single event.

This noun was translated into Chinese by Li, a mathematician in Qing Dynasty, according to the concept of the original word.

The monomial is the product of letters and numbers.

Number of times of a monomial: The sum of the indices of all the letters in the monomial is called the number of times of the monomial.

One-way coefficient: the numerical factor in a single item. For example, the coefficient of 2xy is 2; The coefficient of -5zy is -5.

pay attention to

1. Numbers are written before letters, omitting the multiplication sign. [5a、 16xy]

2. The degree of the constant is 0.

3. The denominator of a monomial cannot be a letter. (Otherwise, it is a score, not a single item)

π is a constant, so it can be used as a coefficient.

If the coefficient is a fraction, it should become a false fraction.

5. But when the coefficient of the monomial is 1 or-1, "1" is usually omitted, such as [(- 1)ab] written as [-ab].

Polynomial polynomial

Polynomial A formula consisting of the sum of several monomials is called a polynomial (in subtraction, subtracting a number equals adding its inverse). Each monomial in a polynomial is called a polynomial term, and the highest degree of these monomials is the degree of this polynomial. Items without letters are called constant items. If the degree of the highest term in a formula is 5 and the formula consists of three monomials, it is called a quintic trinomial.

In a broader definition, the sum of 1 or 0 monomials is also a polynomial. According to this definition, polynomials are algebraic expressions. In fact, no theorem is valid only for narrow polynomials, but not for monomials: when 0 is a polynomial, the degree is negative infinity.

Algebraic expressions monomials and polynomials are collectively called algebraic expressions.

A rational expression in algebraic expression. If there is no division or fraction, if there is a division and fraction, but there is no variable in the division or denominator, it is called an algebraic expression. If there is a division operation with letters, then the formula is called fractional decimal. )

Algebraic expressions can be divided into definitions and operations, definitions can be divided into monomials and polynomials, and operations can be divided into addition, subtraction, multiplication and division.

Addition and subtraction involve merging similar items. Multiplication and division include basic operations, rules and formulas. Basic operations can be divided into power operations. Rules can be divided into algebra and division, and formulas can be divided into multiplication formula, zero exponential power and negative integer exponential power.

What is an index? In the power A n, where A is the base, N is the exponent, and the result is the power.

grasp the main point (of)

1. Understand the concept of algebraic expression, and be able to tell the quantitative relationship represented by an algebraic expression.

2. Can use algebra to express simple words related to quantity.

Thinking click

1. When there is a division operation in the algebraic expression, it needs to be expressed by a fraction, for example, ab÷2 should be written as.

2. The algebraic expression in the form of sum and difference, if there is a unit behind it, must be enclosed in brackets. For example, the temperature is t℃, and a drop of 2℃ is (t-2)℃.

3. Column algebra is to convert written language into mathematical symbol language, and the specific conversion should be carried out according to the following requirements.

(1) Grasp key words, such as "big", "small", "many", "little", "sum", "difference", "product", "quotient", "multiple", "reciprocal" and "remainder". For example, twice the sum of x

.

(2) Arrange the operation sequence. For some quantitative relations, usually the operation of reading first comes first, and the operation of reading later comes last.

(3) Enumerate algebraic expressions in practical problems:

① Basic quantitative relationship: such as distance = speed × time.

② Problems related to area: for example, rectangular area = length × width.

③ Number problem: If the unit number is A, the decimal number is B and the hundred number is C, then these three numbers are expressed as 100c+ 10b+a, which must not be written as cba.

Test site browsing

test center

1. Use algebraic expressions to express simple sentences related to quantity.

2. According to the known special quantitative relations, some relations with general laws are explored.

Suppose the number A is A, the number B is B, and the difference between the squares of the numbers A and B is _ _ _ _ _.

The squares of A and B are a2 and b2, respectively, and the square difference between A and B is a2-b2.

The answer is: a-B

Online detection

1.n boxes of apples weigh p kilograms, and each box weighs _ _ _ _ _ _ _ _ kilograms.

2. A classmate is 1 cm tall, and B classmate is 6 cm taller than A classmate, so B classmate is _ _ _ _ cm tall.

The total number of students in the school is X, of which 40% are girls, so the number of girls is _ _ _ _ _.

4. A two-digit number, the unit number is X, the decimal number is Y, and this two-digit number is _ _ _ _ _. If one digit is opposite to the decimal number, the two digits are _ _ _ _ _ _ _.

5. Dig a regular triangle with a base of b and a height in a square with a side length of a, and the remaining area is _ _ _ _ _ _.

6. It costs N yuan to buy M workbooks in Wang Jie, so it costs _ _ _ _ _ yuan to buy 2 workbooks.

7. It takes _ _ _ _ _ hours for Chen Xiujuan to walk 9 kilometers at a speed of 5 kilometers per hour.

8. In the process of western development, in order to protect the environment and promote ecological balance, the state plans to plant trees at an annual rate of 65,438+00%. If one hectare is planted in the first year, _ _ _ _ _ _ in the third year.

9. We know:

1+3=4=22;

1+3+5=9=32;

1+3+5+7= 16=42;

1+3+5+7+9=25=52.

According to the previous laws, we can guess:

1+3+5+7+9+…+(2n- 1)= _ _ _ _ _ _。 (where n is a natural number).

10. Explain the meaning of algebraic expression 300-2a.

1 1. If the algebraic expression (4x? + m - Y + n) - (2nx？ The value -3x+5y- 10) has nothing to do with the value of the letter χ, and the algebraic expression 3(m? - 3mn + 5n)- 2n？ The value.

3.2 Algebraic Formula (Answer)

1.2 . a+6 3.40% x 4. 10y+x 10x+y

5.a2- ab 6。

7.8 . a( 1+ 10%)29 . n 2 10。

1 1. Original formula =(4-2n)x? +(m+3)x-6y+n+ 10

The value is independent of the value of the letter X.

So 4-2n = 0, and m+3 = 0.

n=2，m=-3

So the original formula =3m? -9mn+ 15n-2n？

=27+54+30-8

= 103

Teaching plan exercises of the whole class courseware in grade one summarize the historical geography of mathematical English in China.

(New goal) Review the materials of Unit 1 and Unit 2 in the first volume of seventh grade mathematics.

The law of addition and subtraction of rational numbers-formula symbol

Sign first, then calculate,

The same number plus the same number; Different symbols plus "big" MINUS "small", symbols and "big numbers"; There is no confusion between burden reduction and error correction.

2☆ If A and B are antonyms, then A+B = () A.–2A B B C.0d. Any rational number 3★( 1) If A =- 13, then-A = _ _ _ _ _ _; (2) If -A = -a=-5.4, then A = _ _ _ _ (3) If -x =-6, then X = _ _ _ _ (4)-X = 9, then X = _ _ _ _ _. 4★★★ Given that A and B are rational numbers, and |a|=a and |b|=-b, then ab is ().

A. negative numbers; B. positive numbers; C. negative number or zero; D. non-negative number

Fourth, absolute value.

Geometric meaning: Generally speaking, the point on the axis representing the origin of number A is called the absolute value of number A, which is marked as ∣a∣. 1, and the absolute value of positive number is;

2. The absolute value of a negative number is its;

The absolute values of 3 and 0 are.

4. According to the definition of absolute value, |a-b| represents point A on the number axis.

Distance to point B. Positive number is greater than 0, 0 is greater than negative number, and positive number is greater than negative number; Two negative numbers, the larger one has the smaller absolute value. [Basic exercises]

The absolute value of 1 ☆-2 indicates that its distance from the origin is a unit, and it is recorded as .2 ☆ |-8 | =. -|-5|= 。 The number whose absolute value is equal to 4 is _ _ _ _ _. 3☆ The number whose absolute value is equal to its opposite number must be () A. Negative number B. Positive number.

C. negative number or zero D. positive number or zero

4★7? X, then _ _ _ _? x； 7x, so _ _ _ _? X 5★ If aa22? The range of a is ()

A.a > ob.a ≥ oc.a ≤ od.a < o.6 ★★★ If 3？ A, then _ _ _ _ _ 3a _ _ _ _ 3a.7★★★ Integers whose absolute values are not greater than 1 1 have ().

a 1 1 b 12

C. Day 22

Verb (abbreviation of verb) rational number operation

1, rational number addition rule

(1) Add two numbers with the same sign, take the same sign, and add the absolute values.

(2) Add two numbers with different absolute values, take the sign of the addend with larger absolute value, and subtract the one with smaller absolute value from the one with larger absolute value. Two opposite numbers add up to get 0. (3) Add a number to 0 or get this number.

2. additive commutative law: rational numbers are added, two numbers are added, the addend positions are exchanged, and the sum is unchanged. Expression: a+b = b+a

3. Law of addition and association: In addition of rational numbers, three numbers are added, the first two numbers are added first or the last two numbers are added first, and the sum is unchanged. Expression: (a+b)+c=a+(b+c)

The absolute algebraic meaning of any rational number A is: (1) When A is a positive number (that is, A >；; 0), ∣ one ∣ =; (2) when a is negative (that is, a

(New goal) Review the materials of Unit 1 and Unit 2 in the first volume of seventh grade mathematics.

4, rational number subtraction rule

Subtracting a number is equal to adding the reciprocal of this number. Expression: a-b=a+(-b) 5. Rational number multiplication rule.

Multiply two numbers, the same sign is positive, the different sign is negative, and then multiply by the absolute value. Any number multiplied by 0 is 0.

6. Multiplication commutative law: Generally speaking, in rational number multiplication, two numbers are multiplied, and the positions of the commutative factor and the product are equal. Expression: ab=ba 7, Multiplication and Association Law: When three numbers are multiplied, the first two numbers are multiplied first, or the last two numbers are multiplied first, and the products are equal. Expression: (ab)c = a(bc)

8. Multiplication and distribution law: Generally speaking, multiplying a number by the sum of two is equivalent to multiplying this number by these two numbers respectively, and then adding the products.

Expression: a(b+c)= ab+ac

1 1, reciprocal: 1 divided by a number (except zero) is called the reciprocal of this number. If two numbers are reciprocal, then the sum of these two numbers

The product equals 1. If A and B are reciprocal, the responsibility ab = 1 12, and the rational number division rule: dividing by a number is equal to multiplying the reciprocal of this number.

Divide two numbers, the same sign is positive, the different sign is negative, and divide by the absolute value. Divide 0 by any number that is not equal to 0, and you will get 0. 13. The operation of finding the product of n identical factors is called the power of rational numbers, and the result of the power is called the power. One; one

Medium, a

It's called the base, and n is called the exponent. Namely: Ann

=aa？ A (multiplied by n A's) is pronounced as: N power of A (or: N power of A). According to the multiplication rule of rational numbers, we can get:

The odd power of a negative number is negative and the even power of a negative number is positive. Any power of a positive number is a positive number, and any power of a positive integer is 0.

3. Rules for removing brackets

Rule 1. There is a "+"in front of the bracket, and both the bracket and the "+"in front are deleted.

Everything in brackets is marked;

Rule number two There is a "-"in front of the brackets, and the brackets and the "-"in front will be deleted.

Everything in brackets is marked.

▲ The basis for removing the bracket rule is actually.

[Note 1] Pay attention to the symbol before brackets, which is the basis for whether the items in brackets change their numbers after removing brackets. [Note 2] When the brackets are deleted, the symbol before the brackets should be deleted together with the brackets.

[Note 3] When there is a "-"before the brackets, the symbols of the items in brackets should be changed after the brackets are removed. You can't just change the symbol of the first item or the first few items in brackets and forget to change other symbols. If the number factor is in front of parentheses, you can multiply the number by the items in parentheses by multiplication and division before removing the parentheses to avoid mistakes.

Note 4: When encountering multi-layer brackets, the brackets should be removed from the inside out or from the outside in. Count the number of "-".

4. Addition and subtraction of algebraic expressions

The process of adding and subtracting algebraic expressions is. If you encounter parentheses, first, then, merge them into the.

The simplest.

5. Several problems that the unit should pay attention to.

(1) Algebraic expression (both single term and polynomial), and the denominator must not contain letters. ② π is not a letter, but a number.

The "rule of removing (adding) brackets" removes brackets and adds brackets, and the symbol change is the most important. There is a plus sign in front of brackets, and the items in brackets are reserved *

. There is a minus sign in front of the brackets, and all items in the brackets are changed with symbols [* "All items are kept in good condition" means that the symbols of the reserved items remain unchanged]

(New goal) Review the materials of Unit 1 and Unit 2 in the first volume of seventh grade mathematics.

(3) When adding (subtracting) polynomials, the polynomials must be enclosed in brackets before calculation can be made. ④ When removing brackets, we should pay special attention to the factors in front of brackets.

Second, this chapter tracking exercises★☆▲π

1，at 3222 1 12，3， 1，，，4，43 xyxxymnxabxx？ ,?

2b, the monomial is:

The polynomial is: 2, a commodity of one yuan each, according to the cost increase of 20% set price; Later, due to the backlog of inventory, it was sold at 15% off the original price, and the current price was RMB; You can still earn RMB for each piece. 3. Given that -7x2ym is a 7-degree monomial, then m=. 4. It is known that -5xmy3 and 4x3yn can be merged, so mn =. 5. Given that -2x2yn and 4x 1+my2 are similar terms, then 3m+2n=.

6、7-2xy-3x2y3

+5x3y2z-9x4y3z2 is a second-order term, in which the highest term is, and the coefficient of the highest term is,

The constant term is, in alphabetical order. 7.-3a+3a =-3()，2a-2a = 2()，-5a-5a =-5()，4a+4a= 4()，8。 If x-y = 5 and xy = 3 are known, 3xy-7x+7y =. 9. If a = 3x+ 1 and b = 6x-3, then 3a-b =. 10, calculating

①(a3-2a2+ 1)-2(3a2-2a+2

1

)②x-2( 1-2x+x2) + 3(-2+3x-x2)

1 1, ab=3, a+b=4, and find the value of 3ab-[2a-(2ab-2b)+3].

12, if (x2

The value of +ax-2y+7)-(bx2-2x+9y- 1) has nothing to do with the value of the letter X. Find the values of A and B.

13, find 5ab-2[3ab- (4ab2+2.

1ab)] -5ab2, where a=2 1 and b=-32.

Mid-term exam of junior high school mathematics

Name _ _ _ _ _ Student ID _ _ _ _ _ _ _ _

1. Multiple choice questions (3 points for each question, 30 points for * * *)

1. Four students draw several axes, as shown in the figure below. What do you think is right ()?

A.B.

C.D.

2. Calculation: The value of is equal to ()

A. 1 1 B

3. In the following categories, the correct one is ()

A.B.

C.D.

4. The purchase price of a commodity is 96 yuan, and if it is sold at a 20% discount, it can still make a profit of 10%, then the price of the commodity is ().

A. 123 yuan B. 105.6 yuan C. 132 yuan D. 140 yuan.

5. In the following figures, the square root is ().

A.B. C. D。

6.① The -A in the following statement must be negative; ②|-a|-a | must be a positive number; ③ The number whose reciprocal equals itself is1; ④ Numbers whose absolute values are equal to themselves are 0 and 1. The correct number is ()

A. 1

7. The number corresponding to the points on the axis is ().

A. rational number B. irrational number C. real number D. fraction

8. Two digits and one digit are, and ten digits are, so this two digit can be expressed as ().

A.B. C. D。

9. The following set of numbers arranged according to the law: 1, 2, 4, 8, 16, ..., the number 2002 should be ().

A, b,-1 C, d, the above answers are incorrect.

10. The wood is 12 meters, and it should be made into a window frame as shown in the figure. If it is assumed that the length of the window frame track is meters,

Then the area of the window frame is ()

(A) (B)

(C) (D)

2. Fill in the blanks (3 points for each small question, 30 points for * * *)

The reciprocal of 1 1. It is _ _ _ _ _.

12. The distance between the point represented on the number axis and the point represented is _ _ _ _ _ _ _.

13. Among the numbers 3.67, 0, 1,-13.48, -6, the score of negative number is _ _ _ _ _ _ _ _ _ _ _; A positive integer is _ _ _ _ _ _ _ _.

14. If and are similar projects, otherwise.

The square root of 15. Yes _ _ _ _ _ _ _;

16. Macao has a population of 430,000, 90% of which live on the peninsula, covering an area of 7 square kilometers. Try to estimate that there are 10 thousand people per square kilometer on the peninsula. (Keep 2 significant figures)

17. The terms of the polynomial are composed of _ _ _ _ _ _ _ _ _ _ _ _ _

18. When a taxi is driving, the relationship between the remaining oil in the fuel tank and the driving distance of the car is shown in the following table:

Mileage n (km) q (l) Fuel consumption per km a (l)

1 0.04 ()

2 0.08 ()

3 0. 12 ()

4 0. 16 ()

............

(1) Write an algebraic expression of n to represent a, then A=,

(2) When n= 150, A=.

Three. Solution questions (* * * 6 questions in this big question, ***60 points)

2 1. (5 points for this question) Show the following numbers and their antonyms on the number axis and use ">;" relationship

Appendix:

The most efficient skills of sprinting in each examination subject of senior high school entrance examination

The senior high school entrance examination is a very critical shunt examination for students, which is related to their future study career. We must review carefully and improve our grades. The following article will introduce the most efficient skills of sprinting for the senior high school entrance examination.

Chinese

Suggestions for Chinese review in senior high school entrance examination: first, grasp the foundation, second, consolidate the objective point, and third, grasp the composition, which is a major breakthrough.

Basic knowledge, we must learn to break it into parts and keep accumulating. For example, reviewing phonetic symbols and glyphs is ineffective once, and most of them will be forgotten the next day. Many a mickle makes a mickle is a better way. At this stage of review, we should start to correct mistakes and narrow the encirclement.

The composition should be good at drawing lessons from and imitating, and adapt to the propositional style of the composition in the senior high school entrance examination. Reading an article every day, the excellent composition of the senior high school entrance examination, as well as some short articles and beautiful articles, and accumulating ideas, languages, skills and materials every day is one of the most efficient skills in the sprint of the senior high school entrance examination.

mathematics

In the last month or so, we should sort out the basic knowledge and adjust the pace of doing the questions.

The basis of sorting out is to read the outline carefully and write down the difficult knowledge points with cards. This is also the most efficient skill in the sprint of the senior high school entrance examination. Learn to adjust the rhythm of doing questions and train the artistic conception of exams. In the middle of review, it is feasible to do the questions crazily, but in the late review, we should reduce the amount appropriately and practice the feeling and situation of the exam to turn the usual senior high school entrance examination into the normal state.

We should also learn to expand our thinking space, learn how to start and break through, and learn to transfer our thinking. The problem is in essence, and after "polishing", you can draw inferences from one another.

English

At present, the most important thing for Chinese review in senior high school entrance examination is to grasp three main lines-listening, cloze, reading and writing.

Insist on listening to at least one group of listening comprehension every day, and complete a cloze and three reading comprehension articles. In addition, write two or three English compositions for the senior high school entrance examination every week.

Cloze and reading comprehension should maintain a sense of language. And composition, there is a lot of room for improvement, we must learn to "grind".

Make a good draft first, then slowly "polish" it, and change the word "Yan", "Yan" sentence, "Yan" paragraph and the beginning and end of "Yan" from the best possible angle. Good sentences make a good composition. There are seventy or eighty words in a composition, so it is more appropriate to use five sentences, three long sentences and two short sentences, and combine the length with the length.

The above content is an introduction to the most efficient skills of sprint in senior high school entrance examination. I hope all the candidates for the senior high school entrance examination can master the methods and make a final sprint. I believe that as long as you work hard, you will certainly get good grades and win the senior high school entrance examination.