Summary of knowledge points of mathematical theorems in junior high school of Beijing Normal University Edition
Grade 8 (Volume 2)
Chapter 1 One-dimensional linear inequalities and one-dimensional linear inequalities.
I. Unequal relations
1. Generally, formulas connected by symbols ""(or "≥") are called inequalities.
2. Distinguish between equality and inequality: equality represents an equal relationship; Inequality represents an unequal relationship.
3. Accurately "translate" inequalities and correctly understand mathematical terms such as "non-negative number" and "not less than". ※ 。
nonnegative number
Nonpositive number
Second, the basic properties of inequality
1. Master the basic properties of inequalities and use them flexibly. ※:
Add (or subtract) the same algebraic expression on both sides of inequality (1), and the direction of inequality remains unchanged, namely:
If a>b, then A+C > b+c，a-c & gt； b-c。
(2) Both sides of the inequality are multiplied by (or divided by) the same positive number, and the direction of the inequality remains unchanged, that is,
If a>b and c>0, then ac> BC,
(3) When both sides of the inequality are multiplied by (or divided by) the same negative number, the direction of the inequality changes, namely:
If a>b and c < 0, AC
2. Comparison size: (A and B represent two real numbers or algebraic expressions respectively). ※
Generally speaking:
If a>b, then a-b is a positive number; On the other hand, if a-b is positive, then a >；; b；
If a=b, then a-b is equal to 0; On the other hand, if a-b is equal to 0, then a = b；;
If a
Namely:
a & gtb & lt= = = & gta-b & gt； 0
a = b & lt= = = & gta-b=0
a & ltb & lt= = = & gta-b & lt； 0
It can be seen that to compare the sizes of two real numbers, just look at their differences.
3. Solution set of inequality;
1. The value of the unknown quantity that can make the inequality hold is called the solution of the inequality. All the solutions of an inequality constitute the solution set of this inequality. ※: The process of finding the solution set of inequality is called solving inequality.
2. There are countless solutions to inequality, generally all numbers in a certain range, which are different from the solutions of equations. ※ 。
3. Representation of inequality solution set on the number axis;
When using the number axis to represent the solution set of inequality, we should determine the boundary and direction:
① Boundary: a solid circle with an equal sign and a hollow circle without an equal sign;
② Direction: large on the right and small on the left.
4. One-dimensional linear inequality;
1. A formula containing only one unknown is an algebraic expression, and the degree of the unknown is 1. Inequalities like this are called one-dimensional linear inequalities. ※ 。
2. The process of solving a linear inequality with one variable is similar to that of solving a linear equation with one variable, especially when both sides of the inequality are multiplied by a negative number, the sign of the inequality will change direction. ※ 。
3. Steps to solve linear inequality of one variable. ※:
1 naming;
(2) the bracket is removed;
③ shifting items;
(4) merging similar projects;
⑤ Change the coefficient to 1 (variable inequality problem)
4. The basic situation of one-dimensional linear inequality is ax > ※ b (or axe.
① when a >; 0, the solution is;
② when a=0, b
When a=0 and b≥0, there is no solution;
③ when a
5. Exploration of inequality application (using inequality to solve practical problems)
The basic steps of solving application problems with column inequalities are similar to those of solving application problems with column equations, namely:
(1) Examination: Carefully examine the questions, find out the unequal relations in the questions, and grasp the key words in the questions, such as "greater than", "less than", "not greater than" and "not less than";
(2) setting: setting appropriate unknowns;
③ Column: list inequalities according to the inequality relations in the question;
④ Solution: Solve the solution set of the listed inequalities;
Answer: Write the answer and check whether the answer conforms to the meaning of the question.
Verb (abbreviation of verb) One-dimensional linear inequality and linear function
One-dimensional linear inequality system of intransitive verbs
1. Definition: An inequality group consisting of several linear inequalities with the same unknown number is called a linear inequality group. ※ 。
2. The common part of the solution set of each inequality in one-dimensional linear inequality group is called the solution set of inequality group. If the solution set of these inequalities has no common part, it is said that this inequality group has no solution. ※ 。
The common part of several inequality solution sets is usually determined by the number axis.
3. Steps to solve linear inequalities. ※:
(1) Find the solution set of each inequality in the inequality group;
(2) Use the number axis to find the common part of these solution sets, that is, the solution set of this inequality group.
Four cases of the solution set of two linear inequalities (A and B are real numbers, A
Graphic narrative language expression of solution set of one-dimensional linear inequality
The larger of x> is adopted.
X> takes the smallest of the two.
A<x<b size cross middle search
There is no solution for size separation.
(it's an empty set)
Chapter II Factorization
I. Factorization
1. Converting a polynomial into the product of several algebraic expressions is called decomposing this polynomial. ※ 。
2. Factorization and algebraic expression multiplication are reciprocal. ※ 。
Differences and relations between factorization and algebraic expression multiplication;
(1) Algebraic expression multiplication is to multiply several algebraic expressions into polynomials;
(2) Factorization is to multiply a polynomial by several factors.
Two. Improve the public's factorial method.
1. If each term of a polynomial contains a common factor, then this common factor can be proposed, so that the polynomial can be transformed into the product of two factors. This factorization method is called extraction of common factors. ※ 。
For example:
※ 2. Concept connotation:
The final result of factorization of (1) should be "product";
(2) The common factor can be a monomial or polynomial;
(3) The theoretical basis of common factor method is the distribution law of multiplication to addition, namely:
3.※ Comments on error-prone points:
(1) Pay attention to whether the sign of the power exponent term is wrong;
(2) Whether the common factor formula is "clean";
(3) One term in the polynomial is only a common factor. After it is put forward, this item in brackets is+1, and there is no leakage.
Three. Using formula method
1. If the multiplication formula is reversed, it can be used to factorize some polynomials. This factorization method is called formula method. ※ 。
※ 2. Main formula:
(1) square difference formula:
(2) Complete square formula:
3. Error-prone comments:
Factorization should be decomposed to the end. If we don't break it down.
4. Use the formula method. ※:
(1) square difference formula:
(1) Binomial or polynomial should be regarded as binomial;
(2) Every term of binomial (unsigned) is the square of monomial (or polynomial);
Binomials are different symbols.
(2) Complete square formula:
(1) should be a trinomial;
(2) where two numbers are the same and each is the square of an algebraic expression;
(3) There is another term that can be positive or negative, which is twice the base product of the first two terms.
5. Thinking and solving 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.
4. Grouping decomposition method:
1. Grouping decomposition method: The method of grouping factors is called grouping decomposition method. ※ 。
For example:
※ 2. Concept connotation:
The key of grouping decomposition method is how to group, whether there are common factors to be extracted after grouping, whether it can continue to decompose, and whether it can continue to decompose factors by formula after grouping.
3. Note: Pay attention to the change of symbols when grouping. ※ 。
Verb (abbreviation for verb) cross multiplication:
1. For a quadratic trinomial, A and C are decomposed into the product of two factors,,, and are satisfied. ※ 。
For example:
2. Quadratic trinomial decomposition. ※:
※ 3. Connotation of law:
(1) Understanding: When factorizing a factor, if the constant term q is positive, it is decomposed into two factors with the same sign, and their signs are the same as those of the coefficient p of the first term.
(2) If the constant term q is negative, it is decomposed into two factors with different signs, in which the factor with larger absolute value has the same sign as the coefficient p of the first term. For the decomposed two factors, it depends on whether their sum is equal to the coefficient p of the first term.
4.※ Comments on error-prone points:
(1) cross multiplication is easy to make mistakes when decomposing coefficients;
(2) The result of decomposition is different from the original formula, so polynomial multiplication is usually used to test whether the decomposition is correct.
Chapter III Scores
I. Scores
1. When two integers are not divisible, a fraction will appear. Similarly, when two algebraic expressions are not divisible, a fraction appears. ※.
Algebraic expression a divided by algebraic expression b can be expressed as. If division b contains letters, it is called a fraction. For any fraction, the denominator cannot be zero.
2. Algebraic expressions and fractions are collectively called rational expressions, namely:
3. When simplifying and calculating the score, it is often necessary to simplify and divide the score, which is mainly based on the basic properties of the score. ※:
Both the numerator and denominator of a fraction are multiplied (or divided) by the same algebraic expression that is not equal to zero, and the value of the fraction remains unchanged.
4. When the numerator and denominator of a fraction have common factors, we can divide the numerator and denominator of the fraction by their common factors at the same time by using the basic properties of the fraction, that is, we can omit the common factors of the numerator and denominator, which is called reduction. ※ 。
2. Multiplication and division of fractions
1. The fraction is multiplied by the fraction, the product of the numerator is the numerator of the product, and the product of the denominator is the denominator of the product. Fraction divided by fraction, numerator and denominator of divisor multiplied by divisor in turn. ※.
That is,
2. Fractional power, numerator and denominator are powers respectively. ※ 。
Namely:
The reverse application, when n is an integer, still holds.
3. The fraction with no common factor between numerator and denominator is called simplest fraction. ※ 。
3. Addition and subtraction of fractions
1. Scores are similar to scores, and scores can also be divided. According to the basic properties of fractions, several fractions with different denominators are converted into fractions with the same denominator and equal to the original fraction, which is called the general fraction of fractions. ※ 。
2. Addition and subtraction of fractions. ※:
The addition and subtraction of fractions, like the addition and subtraction of fractions, are divided into addition and subtraction of fractions with the same denominator and addition and subtraction of fractions with different denominators.
(1) Add and subtract fractions with the same denominator, and add and subtract molecules with the same denominator;
The above rules are expressed by the following formula:
(2) Addition and subtraction of fractions with different denominator, first divided by fractions with the same denominator, and then added and subtracted;
The above rules are expressed by the following formula:
※ 3. Concept connotation:
The key to general division is to determine the simplest denominator, and its methods are as follows: take the coefficient of the simplest common denominator as the least common multiple of the coefficient of each denominator; The letter of the simplest common denominator is the product of the highest power of all letters of each denominator. If the denominator is a polynomial, the polynomial is factorized first.
Four. fractional equation
1. General steps for solving fractional order equations. ※:
① Multiply the simplest common denominator on both sides of the equation, remove the denominator and become an integral equation;
② Solve the whole equation;
③ Substitute the root of the whole equation into the simplest common denominator to see whether the result is zero, so that the root of the simplest common denominator is the root of the original equation and must be discarded.
2. The general steps of solving application problems with column fractional equations. ※:
(1) Examine the meaning of the question;
② setting unknown numbers;
③ According to the meaning of the question, find out the equation relationship and list the (fractional) equation;
④ Solve the equation and test the root;
⑤ Write the answer.
Chapter IV Similar Figures
1. the ratio of line segments.
1. If the same length unit is used to measure two line segments AB, and the length of CD is m and n respectively, then the ratio of these two line segments AB, CD = m: n can be said or written. ※ 。
2. Among the four line segments A, B, C and D, if the ratio of A to B is equal to the ratio of C to D, that is, these four line segments are called proportional line segments for short. ※ 。
※ 3. Note:
①a:b=k, which means that A is k times that of B;
(2) Because the lengths of line segments A and B are both positive numbers, k is a positive number;
③ The ratio has nothing to do with the length units of the selected line segments, and the length units of the two line segments should be consistent when solving;
④ Except a=b, a:b≠b:a is reciprocal;
⑤ Basic properties of proportion: If yes, then ad = bc If ad=bc, then
2. The golden section
1. As shown in figure 1, point C divides the line segment AB into two lines, AC and BC. If so, it is said that the line segment AB is divided by the golden section of point C, which is called the golden section of the line segment AB, and the ratio of AC to AB is called the golden section ratio. ※ 。
2. The golden section is the most beautiful and pleasing point. ※ 。
Four. similar polygons
1.Generally, graphics with the same shape are called similar graphics.
2. Two polygons with equal corresponding angles and proportional corresponding sides are called similar polygons. The ratio of corresponding edges of similar polygons is called similarity ratio. ※ 。
Verb (abbreviation of verb) similar triangles
1. Among the similar polygons, similar triangles is the simplest one. ※ 。
2. A triangle with equal corresponding angles and proportional corresponding sides is called similar triangles. The ratio of corresponding edges in similar triangles is called similarity ratio. ※ 。
3. congruent triangles is a special case of similar triangles, when the similarity ratio is equal to 1. Note: Just like two congruent triangles, the letters representing the corresponding vertices should be written in the corresponding positions. ※ 。
4. The similar triangles corresponds to the height ratio, the ratio corresponding to the center line and the ratio corresponding to the angular bisector are all equal to the similarity ratio. ※ 。
5. The ratio of the circumference of similar triangles is equal to the similarity ratio. ※ 。
6. The ratio of similar triangles area is equal to the square of similarity ratio. ※ 。
6. Explore the conditions of triangle similarity
1. similar triangles's judgment method. ※:
Ordinary triangle right triangle
Fundamental Theorem: A straight line parallel to one side of a triangle and intersecting with the other two sides (or extension lines of both sides) is similar to the original triangle.
(1) The two angles are equal;
(2) The two sides are proportional and the included angle is equal;
③ Three sides are proportional. ① An acute angle is equal;
(2) Both sides are proportional:
A. two right-angled sides are proportional;
B. The hypotenuse is proportional to the right angle.
2. Proportional theorem of parallel lines divided into segments: three parallel lines cut two straight lines, and the corresponding segments are proportional. ※ 。
As shown in figure 2, l1/L2//L3, then.
3. The straight line parallel to one side of the triangle intersects with the other two sides (or the extension lines of both sides), and the triangle formed is similar to the original triangle. ※ 。
Eight. Properties of similar polygons
The perimeter of a similar polygon is equal to the similarity ratio. The area ratio be equal to the square of the similarity ratio. ※.
Nine. Magnification and reduction of graphics
1. If two graphs are not only similar graphs, but also the straight lines of each group of corresponding points pass through the same point, then such two graphs are called potential graphs. This point is called potential center. ※: At this time, the similarity ratio is also called similarity ratio.
2. The ratio of the distance between any pair of corresponding points and the center of the potential diagram is equal to the potential ratio. ※ 。
◎3. Potential changes:
① The transformed graph is not only similar to the original graph, but also the connecting lines of the corresponding vertices intersect at one point, and the distance between the corresponding points and this intersection point is proportional. This special similarity transformation is called potential transformation, and this intersection point is called potential center.
(2) A graph is transformed into another graph by potential, and these two graphs are called potential shapes.
(3) Using analogy method, we can enlarge or reduce the figure.
Chapter V Data Collection and Processing
First, the time to do housework every week
1. The whole object to be investigated is called the whole. ※:
Call every survey object that constitutes the whole population an individual;
Some individuals extracted from the population are called samples of the population.
2. A comprehensive survey of all disciplines for a specific purpose is called a general survey. ※:
A survey of certain objects for a specific purpose is called a sampling survey.
Two. data collection
1. The characteristics of sampling survey are: small survey scope, saving time, manpower and material resources. However, it is not as accurate as the survey results obtained by the census, and only the estimated value is obtained. ※ 。
Whether the estimated value is close to the actual situation depends on whether the sample is representative.
Chapter VI Proof (1)
Two. Definitions and propositions
1. Generally speaking, a sentence that can clearly point out the meaning or characteristics of a concept is called a definition. ※ 。
The definition must be strict. In general, vague terms such as "some", "possible" and "almost" should be avoided.
2. A sentence that can judge right or wrong is called a proposition. ※ 。
Correct propositions are called true propositions, and false propositions are called false propositions.
3. The correctness of some propositions in mathematics is summed up by people in long-term practice, and they serve as the original basis for judging the truth value of other propositions. Such a true proposition is called axiom. ※ 。
4. Some propositions can be judged to be correct by axioms or other true propositions through logical reasoning, and can be further used as the basis for judging the truth value of other propositions. Such a true proposition is called a theorem. ※ 。
5. According to topics, definitions, axioms, theorems, etc. Logic judges whether a proposition is correct. This reasoning process is called proof.
Three. Why are they parallel?
1. axiom of parallel judgment: the same angle is equal and two straight lines are parallel. ※
2. Parallel judgment theorem: the same side is complementary internally, and the two straight lines are parallel. ※ 。
3. Parallel judgment theorem: the same angle is equal and two straight lines are parallel. ※ 。
4. If two straight lines are parallel.
1. Axiom of two parallel lines: two parallel lines with the same angle; ※:
2. The property theorem of two parallel lines: two parallel lines with equal internal angles. ※:
3. Parallel theorem of two straight lines: two straight lines are parallel and the internal angles on the same side are complementary. ※ 。
Proof of triangle sum theorem of verb (abbreviation of verb)
1. Theorem of the sum of interior angles of a triangle: the sum of three interior angles of a triangle is equal to 180. ※
2. A triangle has at most one right angle.
3. A triangle has at most one obtuse angle.
4. A triangle has at least two acute angles.
6. Pay attention to the outer corner of the triangle
1. Two inferences about the theorem of sum of interior angles of triangles. ※:
Inference 1: one outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it;
Inference 2: An outer angle of a triangle is larger than any inner angle that is not adjacent to it.
(Note: ※ indicates the key part; (1644) indicates the understanding part; ◎ Represents the part for reference only; )