1. Algebraic formula: A formula that uses operation symbols+-connection numbers and letters to represent numbers is called an algebraic formula. Note: There are certain restrictions on using letters to represent numbers. Firstly, the number obtained by letters should ensure that the formula in which they are located is meaningful, and secondly, the number obtained by letters should also make real life or production meaningful; A single number or letter is also algebraic.
2. Some points for attention in column algebra:
(1) number multiplied by letters, or letters multiplied by letters are usually multiplied, or writing is omitted;
(2) Multiplication is still needed when multiplying numbers, and multiplication is not needed, and the multiplication sign cannot be omitted;
(3) When a number is multiplied by a letter, the number is usually written before the letter in the result, for example, a5 should be written as 5A;
(4) When the band score is multiplied by letters, the band score should be changed to a false score, for example, A should be written as A;
(5) When there is a division operation in the algebraic expression, the division method and the division method are generally connected by fractional lines, such as those written in 3a;
(6) The difference between A and B should be written in alphabetical order; If we only talk about the difference between two numbers, when we set two numbers as A and B respectively, we should classify them and write them as a-b and B-A. 。
3. Several important algebraic expressions: (m and n represent integers)
(1) The square difference between A and B is: A2-B2; The square of the difference between a and b is: (a-b) 2;
(2) If a, b and c are positive integers, the two-digit integer is 10a+b and the three-digit integer is10a+10b+c;
(3) If both m and n are integers, the quotient m is divided by 5, and the remainder n is 5m+n; Even number is 2n, and odd number is 2n+1; Three consecutive integers are: n- 1, n, n+1;
(4) If b0, positive number is: a2+b, negative number is: -a2-b, non-negative number is: a2, and non-positive number is: -a2.
Summary of knowledge points in the first volume of Math 2 in Grade One. Related concepts of equation
1. Equation: An equation with an unknown number is called an equation.
2. One-dimensional linear equation: contains only one unknown (element) X, and the exponents of the unknown X are all 1 (degree). This equation is called one-dimensional linear equation. For example,1700+50x =1800,2 (x+1.5x) =1
3. Solution of the equation: The value of the unknown that makes the left and right sides of the equation equal is called the solution of the equation.
Note: The solution of the (1) equation and the solution of the equation are different concepts. The solution of the equation is essentially the result of the solution, which is a numerical value (or several numerical values), and the meaning of solving the equation refers to the process of finding the solution of the equation or judging that the equation has no solution. (2) The test method of the equation solution is to substitute the unknown value into the left and right sides of the equation to calculate its value, and then compare the values on both sides to draw a conclusion.
Second, the nature of the equation.
Properties of equation (1): Add (or subtract) the same number (or formula) on both sides of the equation, and the results are still equal.
The properties of the equation (1) are expressed in the form of a formula: if a=b, then a c = b c
Property of the equation (2): If both sides of the equation are multiplied by the same number, or divided by the same number that is not 0, the results are still equal. The property (2) of the equation is expressed in the form of a formula: if a=b, then ac = bc; if a=b(c≠0), then ca=cb.
Third, the law of shift term: moving the symbol on one side of the equation to the other side is called shift term.
Fourth, the rule of removing brackets.
1. The factors outside the brackets are positive numbers, and the symbols of the items after removing the brackets are the same as those of the corresponding items in the original brackets.
2. The factor outside the bracket is negative, and the sign of each item is changed by the sign of the corresponding item in the original bracket after the bracket is removed.
Five, the general steps to solve the equation
1. denominator (least common multiple of denominator on both sides of the equation)
2. Parenthesis deletion (according to the rules of parenthesis deletion and distribution)
3. Move the term (move the term containing the unknown to one side of the equation, and all other terms will be moved to the other side of the equation. Moving the term will change the sign).
4. Merge (transform the equation into ax = b (a≠0))
5. Convert the coefficient into 1 (divide the coefficient a of the unknown quantity on both sides of the equation to get the solution of equation x=a(b)).
Sixth, the general steps to solve practical problems with equation thought.
1. Examination: Examination of questions, analysis of what is known and what is sought in questions, and clarification of the relationship between quantity and quantity.
2. Assumptions: Assumptions about the unknown (which can be divided into direct and indirect ways).
3. Column: List the equations according to the meaning of the question.
4. Solution: Solve the listed equations.
5. Check: Check whether the solution meets the meaning of the problem.
6. Answer: Write the answer (some units should indicate the answer)
Summary of knowledge points in the first volume of junior one mathematics 3 (1) All numbers that can be written in form are rational numbers. 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; P is not a rational number;
(2) Classification of rational numbers: ① Integer ② Fraction
(3) 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;
(4) Natural number 0 and positive integer; A0 a is a positive number; A0 a is a negative number;
A≥0 a is positive or 0 a is non-negative; a≤ 0？ A is negative or 0 a is not positive.
Rational number ratio size:
(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 0, decimal-large number 0.
Summary of Knowledge Points in Chapter 4 of Book 1 of Junior 1 Mathematics: Rich Graphic World
1, geometry
Various graphics abstracted from objects, including three-dimensional graphics and plane graphics.
2. Points, lines, surfaces and bodies
Composition of (1) geometric figure
Point: The point where straight lines intersect is the point, which is the most basic figure in geometry.
Line: The intersection line between 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) Point-to-line, opposite to the line and facing the body.
3, the three-dimensional graphics in life
Three-dimensional graphics in life (by name)
Column:
① cylinder
② Prisms: triangular prism, quadrangular prism (cuboid, cube), pentagonal prism, ...
Cone:
① Cone
② Pyramid
ball
4, prism and related concepts:
Edge: In a prism, any intersection of two adjacent faces is called an edge.
Side: The intersection of two adjacent sides is called a side.
N prism has two bottom faces, n side faces and ***(n+2) faces; 3n sides and n sides; 2n vertices.
5. Plane expansion diagram of the cube:
1 1 species (frequent test: test form: whether the unfolded figure can form a cube; Pattern on opposite sides of that cube)
6. Cut a cube:
Cut a cube with a plane, and the section can be triangular, quadrilateral, pentagonal or hexagonal.
7. Three views:
Three views of an object refer to the front view, the top view and the left view.
Front view: The view seen from the front is called the front view.
Left view: The picture seen from the left is called the left view.
Top view: The view from above is called top view.
Chapter two: rational numbers and their operations.
1, the classification of rational numbers
① Positive rational number
Rational number {② zero
③ Negative rational number
Rational Number {① Integer
② score
2, the opposite number:
Only two numbers with different signs are called reciprocal, and the reciprocal of zero is zero.
3. Number axis:
The straight line defining the origin, positive direction and unit length is called the number axis (when drawing the number axis, the three elements are indispensable). Any rational number can be represented by a point on the number axis.
4. Countdown:
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:
On the number axis, the distance between the point corresponding to a number and the origin 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 numbers are greater than 0, negative numbers are less than 0, and positive numbers are greater than negative numbers;
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 and division.
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 equals 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 Arithmetic rules (5 kinds)
Additive commutative law
associative law of addition
Commutative law of multiplication
Multiplicative associative law
Distribution law of multiplication to addition
8. Scientific symbols
Generally, numbers greater than 10 can be expressed as ×
10n, where 1 ≦ n
Chapter 3: Algebraic expression and its 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.
Algebraic 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 by letters, the band score should be turned into a false score first.
(4) the number multiplier, generally still use the "x" sign, that is, do not omit the "x" sign;
⑤ When there is division operation in algebraic expression, it is generally written in the form of fraction; Note: Fractions have the dual functions of "∫" and brackets.
⑥ If there is a company name after the algebraic expression of sum (or) difference, you must enclose the algebraic expression and then write the company name after the expression.
2. Algebraic expressions: monomials and polynomials are collectively referred to as algebraic expressions.
① Single item:
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:
A single number or letter is also a monomial;
The number of times of a single non-zero number is 0;
When the single coefficient is 1 or-1, this "1" should be omitted. For example, the coefficient of -AB is-1 and the coefficient of a3b 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 term with the highest degree is called the degree of polynomial.
③ Similar projects:
Items with the same letter and the same letter index are called similar items.
note:
There are two conditions for similar items: a. They contain the same letters; B the index of the same letter is the same.
(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 projects:
Add up the coefficients of similar projects, and the index of letters and letters remains the same.
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 "+"before the bracket is regarded as+1, and the "-"before the bracket 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.
Chapter IV Basic Plane Graphics
1, line segments, rays and lines
name
Representation method
extreme point
length
straight line
Straight line AB (or BA)
Straight line l
Endless point
unmeasurable
ray
Ray Aum
1
unmeasurable
line segment
Line AB (or BA)
Line segment l
two
Measurable length
2, the nature of the line
① axiom of straight line: there is only one straight line passing through two points. (Two points define a straight line. )
There are countless straight lines passing by little by little
(3) The straight line extends infinitely in two directions, without points, unmeasurable and incomparable in size.
3, the nature of the line segment
① Axiom of line segment: Of all the connecting lines between two points, the line segment is the shortest. (The line segment between two points is the shortest. )
② Distance between two points: The length of the line segment between two points is called the distance between these two points.
(3) The relationship between the size of a line segment and its length is consistent.
4, the midpoint of the line segment:
The point M divides the line segment AB into two equal line segments AM and BM, and the point M is called the midpoint of the line segment AB. AM = BM = 1/2AB (or AB=2AM=2BM).
5. Angle:
A graph composed of two rays with a common endpoint is called an angle, the common endpoint of the two rays is called the vertex of the angle, and the two rays are called the edges of the angle. Or: a corner can also be regarded as a light that rotates around its endpoint.
6. Representation of angle
There are four ways to express the angle:
① Use numbers to represent individual angles, such as ∠ 1, ∠2, ∠3, etc.
② Use lowercase Greek letters to represent a single angle, such as ∠ α, ∠β, ∠ γ, ∠ θ, etc.
③ An independent angle (a vertex has only one angle) is represented by capital English letters, such as ∠B, ∠C, etc.
④ Use three capital letters to represent any corner, such as ∠BAD, ∠BAE, ∠CAE, etc.
Note: When using three capital letters to represent a corner, be sure to write the letter of the vertex in the middle and the letter of the edge on both sides.
7. Angle measurement
The measurement of angle has the following provisions: divide a flat angle 180 into equal parts, each part is an angle of 1 degree, and the unit is 0, with 1 degree marked as 1 degree and n degree marked as "n".
Divide the angle of 1 into 60 equal parts, each part is called the angle of 1, and 1 is marked as "1'".
Divide the angle of 1' into 60 equal parts, each part is called the angle of 1 sec, and 1 sec is marked as "1".
1 =60', 1'=60"
8. bisector of an angle
The ray from the vertex of an angle divides the angle into two equal angles. This ray is called the bisector of an angle.
9, the nature of the angle
① The angle has nothing to do with the length of the side, but only with the amplitude of the two rays that make up the angle.
(2) the size of the angle can be measured and compared, and the angle can participate in the operation.
10, straight corners and rounded corners:
The light rotates around its endpoint. When the ending edge and the starting edge are on a straight line, the angle formed is called a right angle.
The ending edge continues to rotate, and when it coincides with the starting edge, the angle formed is called fillet.
1 1, polygon:
A closed plane figure composed of several line segments that are not on the same line is called a polygon.
A line segment connecting two nonadjacent vertices is called the diagonal of a polygon.
Starting from the same vertex of an N-polygon, we can draw (n-3) diagonal lines, and divide the N-polygon into (n-2) triangles by connecting this vertex with other vertices respectively.
12, circle:
On the plane, a line segment rotates once around one endpoint, and the figure formed by the other endpoint is called a circle.
The fixed endpoint O is called the center of the circle, and the length of the line segment OA is called the length of the radius (usually referred to as radius for short).
The part between any two points A and B on a circle is called arc, which is called "arc AB" or "arc AB" for short.
A figure consisting of an arc AB and two radii OA and OB passing through the end of the arc is called a sector.
The angle of the vertex at the center of the circle is called the central angle.
Chapter 5 One-variable linear equation
1, equation
Equations with unknowns are called equations.
2, the solution of the equation
The value of the unknown that can make the left and right sides of the equation equal is called the solution of the equation.
3. Properties of the equation
① Adding (or subtracting) the same algebraic expression on both sides of the equation at the same time, the result is still an equation.
② When both sides of the equation are multiplied by the same number at the same time (or divided by the same number that is not 0), the result is still an equation.
4. One-dimensional linear equation
An integral equation with only one unknown number and the highest order of the unknown number is 1 is called a linear equation with one variable.
5. Move the project:
After changing the sign of an item in an equation, it will move from one side of the equation to the other. This deformation is called the moving term.
6, the general steps to solve a linear equation:
① Remove denominator
(2) stent removal
(3) Shift term (after changing the sign of an item in the equation, it moves from one side of the equation to the other. This deformation is called displacement term. )
④ Merge similar items.
⑤ Transform the unknown coefficient into 1.
Chapter VI Data Collection and Arrangement
1, Census and Sample Survey
A comprehensive survey of all disciplines for a specific purpose is called a census.
Among them, all the inspected objects are called the whole, and each inspected object that constitutes the whole is called the individual.
Selecting some individuals from the population for investigation is called sampling investigation, and selecting some individuals from the population is called the sample of the population.
2, vermicelli map
Sector statistical chart: the relationship between the whole and the part is represented by circles and sectors, and the size of the sector reflects the percentage of the part in the whole. This kind of statistical chart is called departmental statistical chart. (The sum of percentages of each sector is 1)
Degree of central angle = 360× the percentage. (The sum of the degrees of the central angle of each part is 360)
3. Frequency histogram
Frequency histogram is a special bar chart, which groups data with the horizontal axis as the statistical object, and the vertical axis represents the frequency of each group of data.
4, the characteristics of various statistical charts
Bar chart: The specific figures of each item can be clearly displayed.
Broken line statistical chart: it can clearly reflect the changes of things.
Department statistical chart: it can clearly show the percentage of each part in the total.
Summary of knowledge points in the first volume of senior one mathematics 5 1. We call all kinds of abstract figures in the real object geometric figures.
2. Some geometric figures (such as cuboids, cubes, cylinders, cones, spheres, etc.). ) are all solidfigure, but not all parts are on the same plane.
3. Some geometric figures (such as line segments, angles, triangles, rectangles, circles, etc.). ) are all in the same plane, which is a plane figure.
4. When the surface of the three-dimensional figure surrounded by the plane figure is properly cut, it can be expanded into a plane figure, which is called the expanded figure (net) of the corresponding three-dimensional figure.
5. Geometry is referred to as three-dimensional.
6. Around the main body is a surface, which includes a plane and a curved surface.
7. Lines are formed at the intersection of faces, and points are formed at the intersection of lines.
8, point opposite, facing the line, line to the body.
9. After exploration, we can get a basic fact: there is a straight line after two points, and there is only one straight line. Simply put, two points determine a straight line (axiom).
10. When two different lines have a common point, we call them intersections, and this common point is called their intersections.
1 1, the point m divides the line segment AB into two equal line segments AM and MB, and the point m is called the center of the line segment AB.
12. After comparison, we can get a basic fact about line segments: among all the connecting lines between two points, the line segment is the shortest. Simply put, the line segment between two points is the shortest. (axiom)
13, the length of the line segment connecting two points is called the distance between these two points.
14 and ∞ (angles are also basic geometric figures.
15, divide a fillet into 360 equal parts, and each equal part is an angle of 1 degree, which is recorded as1; Divide an angle of one degree into 60 equal parts, each part is called an angle of 1 minute, and it is recorded as1'; Divide the angle of 1 into 60 equal parts, and each part is called 1 sec, and it is recorded as 1 ".
16. Starting from the vertex of an angle, the ray that divides this angle into two equal angles is called the bisector of this angle.
17, if the sum of two angles is equal to 90 (right angle), that is, these two angles are called complementary angles, that is, each of them is the complementary angle of the other angle.
18. If the sum of two angles is equal to 180 (flat angle), it is said that these two angles are complementary angles, that is, one of them is the complementary angle of the other.
19, the complementary angles of equal angles are equal, and the complementary angles of equal angles are equal.