Knowledge points in the first volume of junior high school mathematics
1. rational number:
(1) Any number that can be written in form is a rational number. Positive integers, 0 and negative integers are collectively referred to as integers. Positive and negative scores are collectively called scores; Integers and fractions are collectively called rational numbers. Note: 0 is neither positive nor negative; -a is not necessarily negative, and +a is not necessarily positive; π is not rational number;
(2) Note: among rational numbers, 1, 0 and-1 are three special numbers with their own characteristics; These three numbers divide the numbers on the number axis into four areas, and the numbers in these four areas also have their own characteristics;
2. Number axis: The number axis is a straight line that defines the origin, positive direction and unit length.
3. The opposite number:
(1) There are only two numbers with different signs, and we say that one of them is opposite to the other; The antonym of 0 is still 0;
(2) Note: The inverse of a-b+c is-A+B-C; The inverse of a-b is b-a; The inverse of a+b is-a -a-b；;
4. Absolute value:
(1) The absolute value of a positive number is itself, the absolute value of 0 is 0, and the absolute value of a negative number is its inverse; Note: the absolute value means the distance between the point representing a number on the number axis and the origin;
(2) The absolute value can be expressed as:
The problem of absolute value is often discussed in categories;
(3)a| is an important non-negative number, that is | A | ≥ 0; Note: |a|? |b|=|a？ b|，
5. Rational number ratio: (1) The greater the absolute value of a positive number, the greater the number; (2) Positive numbers are always greater than 0 and negative numbers are always less than 0; (3) Positive numbers are greater than all negative numbers; (4) The absolute values of two negative numbers are larger than the size, but smaller; (5) Of the two numbers on the number axis, the number on the right is always greater than the number on the left; (6) large number-decimal number >; 0, decimal-large number < 0.
Mathematics knowledge points in the second volume of the seventh grade
Probability; possibility
I. Events:
1. Events are divided into inevitable events, impossible events and uncertain events.
2. Inevitable events: events that will definitely happen in advance. In other words, the event must happen every time, and it is impossible not to happen, that is, what may happen is 100% (or 1).
3. Impossible events: events that will definitely not happen in advance. In other words, there is no chance at all, that is, the possibility of occurrence is zero.
4. Uncertain event: it is impossible to determine whether it will happen in advance, that is, the event may or may not happen, that is, the probability of occurrence is between 0 and 1.
Second, equal possibility: refers to the equal possibility of several events.
1. probability: it is a quantity that reflects the possibility of an event. It is a proportional number, generally expressed by p, and P(A)= the number of possible outcomes of event A/all possible outcomes.
2. The probability of the inevitable event is 1, and it is recorded as p (inevitable event) =1;
3. The probability of an impossible event is 0, and it is recorded as p (impossible event) = 0;
4. The probability of uncertain events is between 0 and 1, and it is recorded as 0.
Third, geometric probability.
1, the probability of the occurrence of event A is equal to the area of the possible result of this event A (expressed by SA) divided by the area of the graph of all possible results (expressed by S total), so the geometric probability formula can be expressed as P(A)=SA/S total, because the probability of the occurrence of events in each unit area is the same.
2. Find the geometric probability:
(1) Firstly, analyze the relationship between the area occupied by events and the total area;
(2) Then calculate the area of each part;
(3) Finally, the geometric probability is obtained by substituting into the formula.
Review materials at the end of the seventh grade mathematics volume I
-3. 1 unary linear equation and its solution.
① Equation is an equation with unknown numbers.
② All equations contain only one unknown (element) X, and the exponent of the unknown X is 1 (times). Such an integral equation is called a linear equation with one variable.
(3) Pay attention to three points when judging whether an equation is linear:
1) The formula of unknown quantity is algebraic expression (equation is integral equation);
2) The simplified equation contains only one unknown number; (coefficient cannot be zero when letters are included)
3) The number of unknowns in the ranking equation is 1.
(4) Solving the equation is to find the value of the unknown quantity that makes the left and right sides of the equation equal, and this value is the solution of the equation. The solution of the equation is satisfied and the equation is established.
⑤ Properties of the equation:
1) When the same number or the same formula (algebraic expression or fraction) is added or subtracted from both sides of the equation, the equation remains unchanged (the results are still equal). A=b: A+(-) C = B+(-) C。
2) Both sides of the equation are multiplied or divided by the same non-zero number at the same time, and the equation remains unchanged.
A=b: a×c=b×c or a÷c=b÷c(c≠0).
Note: When using attributes, you must pay attention to both sides of the equal sign+,-,×, ⊙; We must pay attention to the number 0 when using property 2.
⑥ General steps for solving linear equations with one variable:
Denominator removal (the least common multiple of denominator multiplied by both sides of the equation) → bracket removal → item shift → merger of similar items → coefficient is1;
The above are that five basic step for solving one-dimensional linear equation. In the actual process of solving the equation, there are five steps
Steps may not be fully used, or some steps need to be reused. Therefore, when solving the equation,
According to the characteristics of the equation, choose the method flexibly. When solving equations, pay attention to the following points:
(1) Denominator removal: multiply both sides of the equation by the least common multiple of the denominator, and do not omit the multiplication.
Denominator term; The numerator is a whole, and brackets should be added after removing the denominator;
Note: Denominator removal (the basic property of the equation) and denominator rounding (the basic property of the fraction) are two concepts that cannot be confused;
(2) Remove the brackets: first remove the brackets, then remove the brackets, and finally remove the braces, and do not omit the items in brackets; Don't get the symbols wrong (multiply together);
⑶ Shift the term: shift the term containing the unknown to one side of the equation, and all other terms to the other side of the equation (bounded by =), and the shift term needs to be signed;
(4) Merging similar terms: Don't lose terms, solving equations is homomorphic deformation, and every step is an equation.
You can't write in the form of energy connection like calculating or simplifying problems.
5] Coefficient 1: (divide both sides by unknown coefficient) Convert the equation into ax=b(a≠0).
In the form of, the letter and its exponential invariant coefficient are transformed into 1, and the unknown coefficient A is divided on both sides of the equation, and the solution of the equation is obtained step by step without the inversion of the numerator and denominator.
Knowledge points of the seventh grade mathematics mid-term exam book 1
The first chapter is a rich graphic world.
1, geometry
Various graphics abstracted from objects, including three-dimensional graphics and plane graphics.
2. Points, lines, surfaces and bodies
Synthesis of (1) Geometry
Point: The point where straight lines intersect is the point, which is the most basic figure in geometry.
Line: The intersection of faces is a line, which can be divided into straight lines and curves.
Face: Surrounding the body is the face, which is divided into plane and curved surface.
Volume: Geometry is also called volume for short.
(2) Points move into lines, lines move into planes, and planes move into adults.
3, the three-dimensional graphics in life
Three-dimensional graphics in life
Column: prism: triangular prism, quadrangular prism (cuboid, cube) and pentagonal prism. ...
positive integer
Rational number zero rational number
Responsible fraction
2. Inverse number: Only two numbers with different signs are called inverse numbers, and the inverse of zero is zero.
3. Number axis: The straight line defining the origin, positive direction and unit length is called number axis (when drawing number axis, all three elements are indispensable). Any rational number can be represented by a point on the number axis.
4. Reciprocal: If A and B are reciprocal, there is ab= 1, and vice versa. The numbers whose reciprocal equals itself are 1 and-1. Zero has no reciprocal.
5. Absolute value: the distance between the point corresponding to a number and the origin on the number axis is called the absolute value of the number, (|a|≥0). If |a|=a, then a ≥ 0; If |a|=-a, then a≤0.
The absolute value of a positive number is itself; The absolute value of a negative number is its reciprocal; The absolute value of 0 is 0. The absolute values of two opposite numbers are equal.
6. Rational number comparison size: positive number is greater than 0, negative number is less than 0, and positive number is greater than negative number; The number represented by two points on the number axis is always larger on the right than on the left; Two negative numbers, the larger one has the smaller absolute value.
7, rational number operation:
(1) Five operations: addition, subtraction, multiplication, division and multiplication.
Multiply multiple numbers, and the sign of the product is determined by the number of negative factors. When there are odd negative factors, the sign of the product is negative. When there are even negative factors, the sign of the product is positive. As long as one number is zero, the product is zero.
Rational number addition rule:
Add two numbers with the same sign, take the same sign, and then add the absolute values.
Two numbers with different signs are added, and the sum is 0 when the absolute values are equal; When the absolute values are not equal, take the sign of the addend with the larger absolute value and subtract the smaller absolute value from the larger absolute value.
Add a number to 0 and you still get the number.
The sum of two opposite numbers is 0.
Rational number subtraction rule: subtracting a number is equal to adding the reciprocal of this number!
Rational number multiplication rule:
Multiply two numbers, the same sign is positive, the different sign is negative, and then multiply by the absolute value.
Multiply any number by 0, and the product is still 0.
Rational number division rule:
Divide two rational numbers, the same sign is positive, the different sign is negative, and divide by the absolute value.
Divide 0 by any number except 0 to get 0.
Note: 0 cannot be divided.
Power of rational number: the operation of finding the product of n identical factors a is called power.
Any power of a positive number is positive, even power of a negative number is positive and odd power of a negative number is negative.
(2) Operation sequence of rational numbers
Calculate the power first, then multiply and divide, and finally add and subtract. If there are brackets, count them first.
(3) Operation law
Additive commutative law's law of additive association
Multiplicative commutative law multiplicative associative law
Distribution law of multiplication to addition
8. Scientific symbols
Generally speaking, numbers greater than 10 can be expressed in the form, where n is a positive integer. This notation is called scientific notation. (n= integer digits-1)
Chapter III Algebraic Expressions and Their Addition and Subtraction
1, algebraic expression
Connect numbers or letters representing numbers with operation symbols (addition, subtraction, multiplication, division, multiplication, root, etc.) to form an algebraic expression. ) A single number or letter is also algebraic.
Note: ① Besides numbers, letters and operators, algebraic expressions can also have brackets;
② Algebraic expressions do not contain symbols such as "=, >,<, ≦". Equality and inequality are not algebraic, but the formulas on both sides of equal sign and unequal sign are generally algebraic;
(3) The number represented by letters in algebraic expressions must make algebraic expressions meaningful, which is a practical problem and should conform to the meaning of practical problems.
Algebra writing format: ※:
(1) multiplication symbols appear in algebraic expressions and are usually omitted, such as vt;
(2) When the number is multiplied by the letter, the number should be written in front of the letter, such as 4a;
(3) When multiplying the band score with letters, you should first turn the band score into a false score, such as writing;
(4) the number multiplier, generally still use the "x" sign, that is, do not omit the "x" sign;
(5) When there is a division operation in the algebraic expression, it is generally written in fractional form, such as 4 \u( a-4) should be written; Note: Fractions have the dual functions of "∫" and brackets.
⑥ If there is a unit name after the algebraic expression of sum (or) difference, you must enclose the algebraic expression and then write the unit name after the expression, such as square meters.
2. Algebraic expressions: monomials and polynomials are collectively referred to as algebraic expressions.
(1) monomial: Algebraic expressions that are all products of numbers and letters are called monomials. In a monomial, the sum of the indices of all letters is called the number of times of the monomial; This numerical factor is called the coefficient of this single term.
Note: 1. A single number or letter is also a monomial; 2. The number of times of a single non-zero number is 0; 3. When the single coefficient is 1 or-1, this "1" should be omitted, such as -ab coefficient is-1 and a3b coefficient is 1.
② Polynomial: The sum of several monomials is called polynomial. In polynomials, each monomial is called a polynomial term; The degree of the degree term is called the degree of the polynomial.
3. Similar items: items with the same letters and the same letter index are called similar items.
Note: ① There are two kinds of similar items: a. They contain the same letters; B. the same index as the letter.
(2) Similar terms have nothing to do with the arrangement order of coefficients and letters;
③ Several constant terms are similar.
4. Rules for merging similar items: Add up the coefficients of similar items, and the indexes of letters and letters remain unchanged.
5. Rules for removing brackets
(1) According to the rules of brackets:
There is a "+"before the brackets. Remove the brackets and the "+"in front, and nothing in brackets will change its symbol; There is a "-"before the brackets. Remove the brackets and the "-"in front, and change the symbols of everything in brackets.
(2) Give brackets according to the distribution law:
The+sign before the brackets is regarded as+1, and the-sign before the brackets is regarded as-1. According to the distribution law of multiplication, each item in brackets is multiplied by+1 or-1 to remove the brackets.
6. Parenthesis rule
Add "+"and brackets, and all symbols added in brackets remain unchanged; Add "-"and brackets, and all the symbols in brackets should be changed.
7, algebraic expression operation:
Addition and subtraction of algebraic expressions: (1) bracket removal; (2) Merge similar items.
In 2022, the first grade of mathematics must recite articles about knowledge points;
★ Combing the knowledge points of mathematics in the senior high school entrance examination in 2022
★ Summary of Mathematics Knowledge Points of Senior High School Entrance Examination in 2022
★ Collection of knowledge points for reviewing the senior high school entrance examination in 2022.
★ Summary of Required Knowledge Points in Mathematics 2022 College Entrance Examination
★ Summary of Required Knowledge Points of Mathematics in 2022 College Entrance Examination
★ Summarize the latest knowledge points of the 2022 college entrance examination mathematics.
★ The latest work plan for mathematics teaching in senior one in 2022.
★ 2022 seventh grade mathematics knowledge points
★ Summary of Mathematics Knowledge Points of Compulsory Two in Senior One in 2022.
★ Summary and Reflection on Mathematics Teaching in Grade One in 2022
var _ HMT = _ HMT | |[]； (function(){ var hm = document . createelement(" script ")； hm.src = "/hm.js？ 3b 57837d 30 f 874 be 5607 a 657 c 67 1896 b "； var s = document . getelementsbytagname(" script ")[0]； s.parentNode.insertBefore(hm，s)； })();