Summary of eighth grade mathematics knowledge points
Isosceles triangle judgment
median
1, the median line on the bottom of the isosceles triangle is perpendicular to the bottom, and the vertex angles are equally divided;
2. The median lines of the two waists of an isosceles triangle are equal, and the intersection points are equidistant from both ends of the bottom.
1, the triangle with equal median lines on both sides is an isosceles triangle;
2. If the median line of one side of a triangle is perpendicular to this side (the diagonal line bisecting this side), then this triangle is an isosceles triangle.
internal bisector
1, the bisector of the top angle of the isosceles triangle bisects the bottom vertically;
2. The bisectors of the two base angles of an isosceles triangle are equal, and the distance between their intersection points and the two endpoints of the base is equal.
1. If the bisector of the vertex of a triangle is perpendicular to the opposite side of this angle (bisecting the opposite side), then this triangle is an isosceles triangle;
The bisectors of two angles in a triangle are equal, so this triangle is an isosceles triangle.
altitude
1, higher vertex angle and higher bottom of isosceles triangle;
2. The two waists of an isosceles triangle are equal in height, and their intersection points are equal in distance from both ends of the bottom edge.
1, if the height of one side of a triangle bisects this side (bisecting the diagonal of this side), then this triangle is an isosceles triangle;
2. Two equal-height triangles are isosceles triangles.
Summary of eighth grade mathematics knowledge points II
Function and its related concepts
1, variables and constants
In a certain change process, the quantity that can take different values is called a variable, and the quantity whose value remains unchanged is called a constant.
Generally speaking, there are two variables, X and Y, in a certain change process. If for each value of X, Y has a definite value corresponding to it, then X is an independent variable and Y is a function of X. ..
2. Resolution function
The mathematical formula used to express functional relationship is called resolution function or functional relationship.
The whole set of values of independent variables that make a function meaningful is called the range of independent variables.
3. Three representations of functions and their advantages and disadvantages.
(1) analysis method
The functional relationship between two variables can sometimes be expressed by an equation containing these two variables and the symbols of digital operations. This representation is called analytical method.
(2) List method
A series of values of the independent variable x and the corresponding values of the function y are listed in a table to represent the functional relationship. This representation is called list method.
(3) Image method
The method of expressing functional relations with images is called image method.
4. General steps of drawing images with resolution function.
(1) List: List gives some corresponding values of independent variables and functions.
(2) Point tracking: Take each pair of corresponding values in the table as coordinates, and track the corresponding points on the coordinate plane.
(3) Connection: according to the order of independent variables from small to large, connect the tracked points with smooth curves.
Summary of eighth grade mathematics knowledge points III
factoring
1. Factorization: decomposing a polynomial into the product of several algebraic expressions is called decomposing this polynomial; Note: Factorization and multiplication are two opposite transformations.
2. Factorization method: Common methods include common factor extraction, formula method, grouping decomposition method and cross multiplication.
3. Determination of common factor: common factor of coefficient? The lowest power of the same factor.
Pay attention to the formula: a+b = b+a; a-b =-(b-a)； (a-b)2 =(b-a)2； (a-b)3=-(b-a)3。
4. The formula of factorization:
(1) square difference formula: A2-B2 = (a+b) (a-b);
(2) Complete square formula: A2+2ab+B2 = (a+b) 2, A2-2ab+B2 = (a-b) 2.
5. Matters needing attention in factorization:
(1) The general order of selecting factorization methods is: one extraction, two formulas, three grouping and four crossover;
(2) When using factorization formula, special attention should be paid to the integrity of letters in the formula;
(3) The final result of factorization needs factorization until every factorization cannot be decomposed;
(4) The final result of factorization requires the first sign of each factor to be positive;
(5) The final result of factorization needs to be sorted;
(6) The final result of factorization requires that the same factor be written as a power.
6. Problem solving skills of factorization: (1) transposition arrangement, bracket arrangement or bracket arrangement; (2) negative sign; (3) Total symbol change; (4) exchange RMB; (5) formula; (6) treating the same formula as a whole; (7) flexible grouping; (8) extracting the fractional coefficient; (9) expand some or all of the brackets; (10) Dismantle or supplement.
7. Completely flat mode: A polynomial that can be reduced to (m+n)2 is called completely flat mode; For quadratic trinomial x2+px+q, is "x2+px+q completely flat?" .
mark
1. Fraction: Generally, two algebraic expressions are represented by A and B, and A÷B can be represented as one form. If b contains letters, the formula is called a fraction.
2. Rational formula: Algebraic formula and fractional formula are collectively called rational formula; Namely.
3. Two important judgments about the score: (1) If the denominator of the score is zero, the score is meaningless, and vice versa; (2) If the numerator of a fraction is zero and the denominator is not zero, the value of the fraction is zero; Note: If the numerator of a fraction is zero and the denominator is zero, the fraction is meaningless.
4. The basic nature and application of scores;
(1) If both the numerator and denominator of a fraction are multiplied (or divided) by the same non-zero algebraic expression, the value of the fraction remains unchanged;
(2) Note: In a fraction, if any two symbols of numerator, denominator and fraction itself are changed, the value of the fraction remains unchanged;
that is
(3) When simplifying the complex fraction, it is relatively simple to multiply the numerator denominator by the least common multiple of the decimal denominator.
5. Fraction: The divisor of the numerator and denominator of a fraction is called a fraction; Note: Factorization is often needed before fractional reduction.
6. simplest fraction: There is no common factor between the numerator and denominator of a fraction. This fraction is called simplest fraction; Note: The final result of score calculation requires simplification to the simplest score.
7. The law of multiplication and division of scores:
8. the power of the score:.
9. The calculation rules of negative integral index:
(1) formula: a0= 1(a≠0), a-n = (a ≠ 0);
(2) The algorithm of positive integer exponent can be used to calculate negative integer exponent;
(3) Formula:
(4) Formula: (-1)-2= 1, (-1)-3=- 1.
10. General fraction of fractions: according to the basic properties of fractions, several fractions with different denominators are converted into fractions with the same denominator equal to the original fraction, which is called general fraction of fractions; Note: The simplest common denominator should be determined before the general division of fractions.
1 1. Determination of the simplest common denominator: the least common multiple of the coefficient? Power of the same factor.
12. Rules for addition and subtraction of fractions with the same denominator and different denominator:.
13. One-variable linear equation with letter coefficient: In the equation ax+b=0(a≠0), X is unknown, and A and B are known numbers expressed by letters. For X, the letter A is the coefficient of X, which is called the letter coefficient, and the letter B is a constant term, which is called a one-dimensional linear equation with letter coefficient. note:
14. Formula deformation: transforming a formula from one form to another is called formula deformation; Note: The essence of formula deformation is to solve equations with letter coefficients. It is particularly important to note that when both sides of the letter equation are multiplied by the algebraic expression with letters at the same time, it is generally necessary to confirm that the value of this algebraic expression is not 0 first.
15. Fractional equation: the equation with unknown number in denominator is called fractional equation; Note: previously learned equations with unknown denominator are integral equations.
16. Rooting of fractional equation: When solving fractional equation, in order to remove the denominator, both sides of the equation have to be multiplied by algebraic expressions containing unknowns, so root growth may occur, so the fractional equation must be tested for root growth; Note: When solving the equation, don't divide both sides of the equation by algebraic expressions containing unknowns at the same time, because the roots may be lost.
17. Method of testing the root of fractional equation: substitute the root of fractional equation into the simplest common denominator (or each denominator of fractional equation). If the value is zero, the root is the root, and then the original equation has no solution; If the value is not zero, the root is the solution of the original equation; Note: From this, it can be judged that the unknown value that makes the denominator zero may be the root of the original equation.
18. Application of fractional equation: Fractional equation can solve application problems in the same way as integral equation, except that the program of "checking and increasing roots" needs to be added.
The fourth summary of eighth grade mathematics knowledge points
1 congruent triangles has equal sides and angles.
2 Angular Axiom (SAS) has two triangles with equal angles.
The Axiom of Triangle (ASA) has the congruence of two triangles, which have two angles and their sides correspond to each other.
4 Inference (AAS) has two angles, and the opposite side of one angle corresponds to the congruence of two triangles.
The pentagonal axiom (SSS) has congruences of two triangles and three corresponding equilateral sides.
6 Axiom of hypotenuse and Right Angle (HL) Two right angle triangles with hypotenuse and right angle are congruent.
Theorem 7 1 The distance between a point on the bisector of an angle and both sides of the angle is equal.
Theorem 2: The points with equal distance on both sides of an angle are on the bisector of this angle.
The bisector of angle 9 is the set of all points with equal distance to both sides of the angle.
The property theorem of 10 isosceles triangle: the two base angles of isosceles triangle are equal (that is, equilateral and equilateral).
2 1 Inference 1 The bisector of the vertices of an isosceles triangle bisects the base and is perpendicular to the base.
The bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of the isosceles triangle coincide with each other.
Inference 3 All angles of an equilateral triangle are equal, and each angle is equal to 60.
24 Judgment Theorem of an isosceles triangle If a triangle has two equal angles, then the opposite sides of the two angles are also equal (equal angles and equal sides).
Inference 1 A triangle with three equal angles is an equilateral triangle.
Inference 2 An isosceles triangle with an angle equal to 60 is an equilateral triangle.
In a right triangle, if an acute angle is equal to 30, the right side it faces is equal to half of the hypotenuse.
The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.
Theorem 29 The distance between the point on the vertical line of a line segment and the two endpoints of this line segment is equal.
30 inverse theorem and the point where the distance between the two endpoints of a line segment is equal is on the middle vertical line of this line segment.
Five Summary of Eight Grade Mathematics Knowledge Points
Chapter 11 congruent triangles
I. Knowledge framework
Two. The concept of knowledge
1. congruent triangles: When two triangles have the same shape and size, one of them can be translated, rotated and symmetrical to make it coincide with the other. These two triangles are called congruent triangles.
2. The nature of congruent triangles: the corresponding angles and sides of congruent triangles are equal.
3. The axiom and inference of triangle congruence are:
(1) "corner" is abbreviated as "SAS"
② The abbreviation of "corner" is "ASA"
(3) "Edge" is abbreviated as "SSS"
(4) The abbreviation of "corner edge" is "AAS"
(5) Two right-angled triangles (HL) with equal hypotenuse and right-angled side.
4. Inference from the bisector of the angle: the points with equal distance from the inside of the angle to both sides of the angle are on the bisector.
5. The basic method steps to prove the congruence of two triangles or to prove the equality of line segments or angles with it: ①. Determine the known conditions (including implied conditions, such as common * * * edge, common * * * angle, diagonal, bisector of angle, median line, height, isosceles triangle and other implied angular relations. ); 2. Review the triangle judgment and find out what else we need; ③.
When learning triangle congruence, teachers should start from the real life graphics, lead to congruence graphics, and then lead to congruent triangles. Through intuitive understanding and comparison, we can discover the mystery of congruent triangles. Stimulate students' collective thinking and inspire them. By exploring the bisector and midline of the triangle, students can realize the true charm of the collection.
Chapter 12 Axisymmetric
I. Knowledge framework
Two. The concept of knowledge
1. Symmetry axis: If a graph is folded along a straight line and the parts on both sides of the straight line can overlap each other, then the graph is called an axisymmetric graph; This straight line is called the axis of symmetry.
2. Properties: (1) The symmetry axis of an axisymmetric graph is the median vertical line of any pair of line segments connected by corresponding points.
(2) The distance between the point on the bisector of the angle and both sides of the angle is equal.
(3) The distance between any point on the vertical line in the line segment and the two end points of the line segment is equal.
(4) The point with equal distance from the two endpoints of a line segment is on the middle vertical line of this line segment.
(5) The corresponding line segment and the corresponding angle on the axisymmetric figure are equal.
3. The nature of isosceles triangle: the two base angles of isosceles triangle are equal (equilateral and equiangular).
4. The bisector of the top angle of the isosceles triangle, the height on the bottom edge and the midline on the bottom edge coincide with each other, which is called "three lines in one" for short.
5. Determination of isosceles triangle: equilateral and equilateral.
6. Characteristics of equilateral triangle angles: three internal angles are equal, equal to 60,
7. Determination of equilateral triangle: A triangle with three equal angles is an isosceles triangle.
An isosceles triangle with an angle of 60 is an equilateral triangle.
A triangle with two angles of 60 is an equilateral triangle.
8. In a right triangle, the right side facing an angle of 30 is equal to half of the hypotenuse.
9. The midline of the hypotenuse of a right triangle is equal to half of the hypotenuse.
This chapter requires students to analyze and appreciate the graphics in life, appreciate the beauty of mathematics, correctly understand the properties and judgments of isosceles triangles and equilateral triangles, and use these properties to solve some mathematical problems.
Chapter 13 Real Numbers
I. Knowledge framework
Two. The concept of knowledge
1. arithmetic square root: Generally speaking, if the square of a positive number X is equal to A, that is, x2=a, then this positive number X is called the arithmetic square root of A, and is recorded as. The arithmetic square root of 0 is 0; By definition, A has an arithmetic square root only when a≥0.
2. Square root: Generally speaking, if the square root of a number X is equal to A, that is, x2=a, then this number X is called the square root of A. ..
3. A positive number has two square roots (one positive and one negative), which are in opposite directions; 0 has only one square root, which is itself; Negative numbers have no square root.
4. The cube root of a positive number is a positive number; The cube root of 0 is 0; The cube root of a negative number is a negative number.
5. The inverse of number A is -a, the absolute value of positive real number is itself, the absolute value of negative number is its inverse, and the absolute value of 0 is 0.
The real number part mainly requires students to understand the concepts of irrational numbers and real numbers, know that real numbers correspond to points on the number axis one by one, and estimate the size of irrational numbers; Understand the algorithm and operation law of real numbers, and be able to operate real numbers. The emphasis is on the meaning and classification of real numbers; Arithmetic and arithmetic laws of real numbers.
Chapter 14 Linear Functions
I. Knowledge framework
Two. The concept of knowledge
1. linear function: if the relationship between two variables x and y can be expressed in the form of y=kx+b(k≠0), then y is said to be a linear function of x (x is the independent variable and y is the dependent variable). In particular, when b=0, y is said to be a proportional function of x.
2. The general formula of the proportional function: y=kx(k≠0), which is like a straight line passing through the origin (0,0).
3. The image of the proportional function y=kx(k≠0) is a straight line passing through the origin. When k >: 0, the straight line y=kx passes through the first and third quadrants, and y increases with the increase of X. When k < 0, the straight line y=kx passes through the second and fourth quadrants, and y decreases with the increase of X. In the linear function y=kx+b, when k >; 0, y increases with the increase of x; When k < 0, y decreases with the increase of x.
4. Solving the resolution function with known two-point coordinates: undetermined coefficient method.
Linear function is the beginning of junior high school students' learning function, and it is also the cornerstone of learning other functions in the future. When studying this chapter, teachers should start from practical problems, introduce variables, and understand things from concrete to abstract. Cultivate students' good sense of change and correspondence, and experience the idea of combining numbers with shapes. In the teaching process, we should pay more attention to understanding and application, and solve practical problems at the same time, so that students can appreciate the practical value and fun of mathematics.
Chapter 15 multiplication, division and factorization of algebraic expressions.
I. The concept of knowledge
1. Multiplication rule of the same base number: (m, n are all positive numbers)
2. power law: (m, n is a positive number)
3. Multiplication of algebraic expressions
(1) Multiplication rule of monomial: Multiply the monomial by its coefficient and the same letter respectively. For a letter contained only in a monomial, together with its exponent, it is a factor of the product.
(2) Multiplication of single item and polynomial: Multiplication of polynomial and single item is the distribution law of multiplication and addition, which is converted into multiplication of single item and single item, that is, multiplication of single item and polynomial is to multiply each term of polynomial with single item, and then add the obtained products.
(3) Polynomial and Polynomial Multiplication
Multiply polynomials by multiplying each term in one polynomial by each term in another polynomial, and then add the products.
4. Variance formula:
5. Complete square formula:
6. same base powers's division rule: same base powers divides, the base is unchanged, and the exponent is subtracted, that is, (a≠0, m, n is a positive number, m >；; n)。
Pay attention to the following points when applying:
(1) The prerequisite for using the rule is "divisible by same base powers" and 0 is not divisible, so a≠0 is included in the rule.
② Any number that is not equal to 0, whose power of 0 is equal to 1, that is, if (-2.50= 1), 00 is meaningless.
(3) The power of any number not equal to 0 is -p (p is a positive integer) which is equal to the reciprocal of the power of this number, that is, (a≠0, p is a positive integer), 0- 1, 0-3 is meaningless; When a>0, the value of a-p must be positive; When a<0, the value of a-p can be positive or negative, for example,
④ Pay attention to the operation sequence.
7. Division in algebraic expressions
Monomial division monomial: monomial division, which divides the coefficient and the same base separately as the factor of quotient. For letters only included in the division formula, they are used as factors of quotient together with their indices;
Polynomial divided by monomial: Polynomial divided by monomial, first divide each term of this polynomial by monomial, and then add the obtained quotients.
8. Factorization: The factorization of a polynomial is called the product of several algebraic expressions.
General method of factorization: 1. Factor method to improve the public. Formula 3. Cross multiplication.
Steps of factorization: (1) First, check whether each item has a common factor, and if so, extract the common factor first;
(2) See if the formula method can be used;
(3) Using the grouping decomposition method, that is, extracting the common factors of each group after grouping or using the formula method to achieve the purpose of decomposition;
(4) The final result of factorization must be the product of several algebraic expressions, otherwise it is not factorization;
(5) The results of factorization must be carried out until every factorization can no longer be decomposed within the scope of rational numbers.
There are many knowledge points in this chapter of algebraic multiplication and division and factorization. On the surface, there are many fragmentary concepts and properties, which are actually inseparable whole. When studying this chapter, we should prepare more group cooperation and exchange activities to cultivate students' reasoning ability and computing ability. Experience the beauty of simplicity and harmony of mathematical rules and formulas in doing problems, and improve the efficiency of doing problems.
Summarize the related articles of eighth grade mathematics knowledge points;
1. Summarize the knowledge points of eighth grade mathematics.
2. Summary of knowledge points in the first volume of Mathematics in the second day of junior high school
3. Summary of knowledge points in the first volume of eighth grade mathematics published by People's Education Press
4. Summary of Mathematics Knowledge Points in Volume 1 of Grade 8
5. Summarize the knowledge points of the first volume of eighth grade mathematics.
6. Summary of the first volume of eighth grade mathematics knowledge points and eighth grade mathematics learning skills.
7. Summarize the knowledge points of the first volume of eighth grade mathematics.
8. Grade 8, Book 2 Mathematics Knowledge Points.