The eighth grade mathematics knowledge points summary People's Education Edition Volume I Chapter 1 1- 12
Chapter 11 congruent triangles
Knowledge concept
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
Knowledge concept
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 an 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.
The first volume of the eighth grade mathematics knowledge points summary People's Education Edition Chapter 13- 14
Chapter 13 Real Numbers
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
Knowledge concept
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.
Eighth grade mathematics knowledge points summary People's Education Edition Volume I 15 chapter
Chapter 15 multiplication, division and factorization of algebraic expressions.
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< is 0, the value of a-p can be positive or negative.
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.
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