Induction of mathematical knowledge points in the last semester of senior two.
fractional equation
I. Understanding the definition
1. Fractional equation: an equation with a fraction and an unknown number in the denominator-fractional equation.
2, the idea of solving the fractional equation is:
(1) Multiplies the simplest common denominator on both sides of the equation, removes the denominator, and becomes an integral equation.
(2) Solve the whole equation.
(3) Bring the root of the whole equation into the simplest common denominator to see if the result is zero, so that the root of the simplest common denominator is the additional root of the original equation and must be discarded.
(4) Write the root of the original equation.
"Four summaries of one transformation, two solutions and three experiments"
3. Root addition: The root addition of fractional equation must meet two conditions:
(1) Finding the root is the simplest, and the common denominator is 0; (2) Increasing root is the root of integral equation formed by fractional equation.
4, the solution of fractional equation:
(1) Simplification before simplification (2) Multiply both sides of the equation by the simplest common denominator and turn it into an integral equation;
(3) solving the integral equation; (4) Root inspection;
Note: When solving the fractional equation, when both sides of the equation are multiplied by the simplest common denominator, the simplest common denominator may be 0, which increases the root, so the fractional equation must be tested.
Test method of fractional equation: bring the solution of the whole equation into the simplest common denominator. If the value of the simplest common denominator is not 0, the solution of the whole equation is the solution of the original fractional equation; Otherwise, this solution is not the solution of the original fractional equation.
5. Fractional equation solves practical problems.
Steps: Examining questions-setting unknowns-listing equations-solving equations-testing-writing answers. Pay attention to the test equation itself and practical problems when testing.
Second, the axisymmetric graphics:
A figure is folded in half along a straight line, and the parts on both sides of the straight line can completely overlap. This straight line is called the axis of symmetry. Points that coincide with each other are called corresponding points.
1, axisymmetric:
Two figures are folded in half along a straight line, and one of them can completely coincide with the other. This straight line is called the axis of symmetry. Points that coincide with each other are called corresponding points.
2, the difference and connection between axisymmetric graphics and axisymmetric:
(1) difference. Axisymmetric graphics discuss "the symmetrical relationship between graphics and straight lines"; Axisymmetry discusses "the symmetrical relationship between two figures and a straight line".
(2) contact. Axisymmetric figures are defined as "the parts on both sides of the axis of symmetry are regarded as two figures". Axisymmetric "two figures as a whole" is an axisymmetric figure.
3, the essence of axial symmetry:
(1) Two symmetric graphs are congruent.
(2) The symmetry axis is perpendicular to the line segment connecting the corresponding points.
(3) The distances from the corresponding points to the symmetry axis are equal.
(4) The connecting lines of the corresponding points are parallel to each other.
Third, use coordinates to represent the axis symmetry.
1, and the coordinates of the point (x, y) which is symmetrical about x axis are (x,-y);
2. The coordinates of the point (x, y) about the Y axis symmetry are (-x, y);
3. The coordinates of the point (x, y) symmetrical about the origin are (-x, -y).
Fourthly, about the symmetry of the bisector of the coordinate axis.
The point P(x, y) is symmetrical about the bisector y=x of the first and third quadrant coordinate axes, and the coordinate of this point is (y, x).
The point P(x, y) is symmetrical about the bisector y=-x of the second and fourth quadrant coordinate axes, and the coordinate of this point is (-y, -x).
Eight-grade mathematics knowledge points
1, the corresponding edge of congruent triangles is equal to the corresponding angle.
2. Angular Axiom (SAS) has two sides and two triangles with equal included angles.
3. Angle and Angle Axiom (ASA) has 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.
5. The side-by-side axiom (SSS) has two triangles with equal sides.
6. Axiom of hypotenuse and right-angled edge (HL) Two right-angled triangles with hypotenuse and a right-angled edge are congruent.
7. Theorem 1 The distance between a point on the bisector of an angle and both sides of the angle is equal.
8. Theorem 2 The point where two sides of an angle are equidistant is on the bisector of this angle.
9. The bisector of an angle is the set of all points with equal distance to both sides of the angle.
10, the property theorem of isosceles triangle, the two base angles of isosceles triangle are equal (that is, equilateral angles)
1 1, it is inferred that the bisector of the vertices of 1 isosceles triangle bisects the base and is perpendicular to the base.
12. 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.
13, inference 3 All angles of an equilateral triangle are equal, and each angle is equal to 60.
14, the judgment theorem of 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).
15, inference 1 A triangle with three equal angles is an equilateral triangle.
16, inference 2 An isosceles triangle with an angle equal to 60 is an equilateral triangle.
17. In a right-angled triangle, if an acute angle is equal to 30, the right-angled side it faces is equal to half of the hypotenuse.
18. The midline of the hypotenuse of a right triangle is equal to half of the hypotenuse.
19, it is proved that the distance between the point on the middle vertical line of a line segment and the two endpoints of the line segment is equal.
20. The inverse theorem and the point where the two endpoints of a line segment are equidistant are on the vertical line of this line segment.
2 1, the middle vertical line of a line segment can be regarded as the set of all points with equal distance at both ends of the line segment.
22. Theorem 1 Two graphs symmetric about a straight line are conformal.
23. Theorem 2 If two figures are symmetrical about a straight line, then the symmetry axis is the perpendicular line connecting the corresponding points.
24. Theorem 3 Two graphs are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, then the intersection point is on the axis of symmetry.
25. Inverse Theorem If the straight line connecting the corresponding points of two graphs is vertically bisected by the same straight line, then the two graphs are symmetrical about this straight line.
26. Pythagorean Theorem The sum of squares of two right angles A and B of a right triangle is equal to the square of the hypotenuse C, that is, A 2+B 2 = C 2.
27. The Inverse Theorem of Pythagorean Theorem If the three sides of a triangle are related A 2+B 2 = C 2, then the triangle is a right triangle.
Ten Skills of Math Learning Methods in Grade Two of Junior High School
1, matching method
The so-called formula is to change some items of an analytical formula into the sum of positive integer powers of one or more polynomials by using the method of constant deformation. The method of solving mathematical problems with formulas is called matching method. Among them, the most common method is to make it completely flat. Matching method is an important method of constant deformation in mathematics. It is widely used in factorization, simplifying roots, solving equations, proving equality and inequality, finding extreme values of functions and analytical expressions.
2, factorization method
Factorization is to transform a polynomial into the product of several algebraic expressions. Factorization is the basis of identity deformation. As a powerful mathematical tool and method, it plays an important role in solving algebra, geometry and trigonometry problems. There are many methods of factorization, such as extracting common factors, formulas, grouping decomposition, cross multiplication and so on. Middle school textbooks also introduce the use of decomposition and addition, root decomposition, exchange elements, undetermined coefficients and so on.
3. Alternative methods
Method of substitution is a very important and widely used method to solve problems in mathematics. We usually call unknowns or variables variables. The so-called substitution method is in a complex 4. Discriminant method and Vieta theorem.
The root discrimination of unary quadratic equation ax2+bx+c=0(a, B, C belongs to R, a≠0) and△ = B2-4ac is not only used to judge the properties of roots, but also widely used in algebraic deformation, solving equations (groups), solving inequalities, studying functions and even geometric and trigonometric operations as a problem-solving method.
Vieta's theorem not only knows one root of a quadratic equation, but also finds another root. Knowing the sum and product of two numbers, we can find the symmetric function of the root, calculate the sign of the root of quadratic equation, solve the symmetric equation and solve some problems about quadratic curve. , has a very wide range of applications.
5, undetermined coefficient method
When solving mathematical problems, it is first judged that the obtained results have a certain form, which contains some undetermined coefficients, then the equations about undetermined coefficients are listed according to the problem setting conditions, and finally the values of these undetermined coefficients or some relationship between these undetermined coefficients are found. This method is called undetermined coefficient method to solve mathematical problems. It is one of the commonly used methods in middle school mathematics.
6. Construction method
When solving problems, we often use this method to construct auxiliary elements by analyzing conditions and conclusions, which can be a figure, an equation (group), an equation, a function, an equivalent proposition and so on. And establish a bridge connecting conditions and conclusions, so that the problem can be solved. This mathematical method of solving problems is called construction method. Using construction method to solve problems can make algebra, trigonometry, geometry and other mathematical knowledge permeate each other, which is beneficial to solving problems.
7. reduce to absurdity
Reduction to absurdity is an indirect proof method. First, a hypothesis contrary to the conclusion of the proposition is put forward, and then from this hypothesis, through correct reasoning, contradictions are led out, thus denying the opposite hypothesis and affirming the correctness of the original proposition. The reduction to absurdity can be divided into reduction to absurdity (with only one opposite conclusion) and exhaustive reduction to absurdity (with more than one opposite conclusion). The steps to prove a proposition by reduction to absurdity can be roughly divided into: (1) reverse design; (2) return to absurdity; (3) conclusion.
Anti-design is the basis of reduction to absurdity. In order to make correct anti-design, we need to master some commonly used negative expressions, such as: yes/no; Existence/non-existence; Parallel/non-parallel; Vertical/not vertical; Equal to/unequal to; Large (small) inch/small (small) inch; Both/not all; At least one/none; At least n/ at most (n-1); At most one/at least two; /At least two.
Reduction to absurdity is the key to reduction to absurdity. There is no fixed model in the process of derivation of contradiction, but it must be based on reverse design, otherwise the derivation will become passive water and trees without roots. Reasoning must be rigorous. There are the following types of contradictions: contradictions with known conditions; Contradicting with known axioms, definitions, theorems and formulas; There are dual contradictions; Contradictions
8. Find the area method
The area formula in plane geometry and the property theorems related to area calculation derived from the area formula can be used not only to calculate the area, but also to prove that plane geometry problems sometimes get twice the result with half the effort. The method of proving or calculating plane geometric problems by using area relation is called area method, which is commonly used in geometry.
The difficulty in proving plane geometry problems by induction or analysis lies in adding auxiliary lines. The characteristic of area method is to connect the known quantity with the unknown quantity by area formula, and achieve the verification result through operation. Therefore, using the area method to solve geometric problems, the relationship between geometric elements becomes the relationship between quantities, and only calculation is needed. Sometimes there may be no auxiliary lines, even if auxiliary lines are needed, it is easy to consider.
9, geometric transformation method
In the study of mathematical problems, the transformation method is often used to transform complex problems into simple problems and solve them. The so-called transformation is a one-to-one mapping between any element of a set and the elements of the same set. The transformation involved in middle school mathematics is mainly elementary transformation. There are some exercises that seem difficult or even impossible to start with. We can use geometric transformation to simplify the complex and turn the difficult into the easy. On the other hand, the transformed point of view can also penetrate into middle school mathematics teaching. It is helpful to understand the essence of graphics by combining the research of graphics under isostatic conditions with the research of motion.
Geometric transformation includes: (1) translation; (2) rotation; (3) symmetry.
10, objective problem solving method
Multiple-choice questions are questions that give conditions and conclusions and require finding the correct answer according to a certain relationship. Multiple-choice questions are ingenious in conception and flexible in form, which can comprehensively examine students' basic knowledge and skills, thus increasing the capacity and knowledge coverage of test papers.
Fill-in-the-blank question is one of the important questions in standardized examination. Like multiple-choice questions, it has the advantages of clear test objectives, wide knowledge coverage, accurate and fast marking, and is conducive to examining students' analytical judgment and calculation ability. The difference is that the fill-in-the-blank question does not give an answer, which can prevent students from guessing the answer.
In order to solve multiple-choice questions and fill-in-the-blank questions quickly and correctly, in addition to accurate calculation and strict reasoning, there are also methods and skills to solve multiple-choice questions and fill-in-the-blank questions. The following examples introduce common methods.
(1) Direct deduction method: Starting directly from the conditions given by the proposition, using concepts, formulas, theorems, etc. Carry out reasoning or operation, draw a conclusion and choose the correct answer. This is the traditional method of solving problems, which is called direct deduction.
(2) Verification method: find out the appropriate verification conditions from the questions, and then find out the correct answer through verification, or substitute alternative answers into the conditions for verification to find out the correct answer. This method is called verification method (also called substitution method). This method is often used when encountering quantitative propositions.
(3) Special element method: substitute appropriate special elements (such as figures or numbers) into the conditions or conclusions of the topic, so as to get the solution. This method is called the special element method.
(4) Exclusion and screening method: for multiple-choice questions with only one correct answer, according to mathematical knowledge or reasoning and calculus, the incorrect conclusion is excluded and the remaining conclusions are screened, so that the solution to make the correct conclusion is called exclusion and screening method.
(5) Graphic method: According to the nature and characteristics of the graphics or images that meet the conditions of the topic, make a judgment and make a correct choice, which is called graphic method. Graphic method is one of the common methods to solve multiple-choice questions.
(6) Analysis method: directly through the conditions and conclusions of multiple-choice questions, make detailed analysis, induction and judgment, so as to select the correct result, which is called analysis method.
Articles related to eighth grade mathematics knowledge points published by People's Education Publishing House;
★ Summary of knowledge points in the first volume of eighth grade mathematics published by People's Education Press
★ Summary of knowledge points in the first volume of eighth grade mathematics published by People's Education Press
★ People's Education Edition Grade 8 Mathematics Book 1 Knowledge Point Arrangement
★ Sort out and summarize the knowledge points of eighth grade mathematics.
★ Arrangement of Mathematics Knowledge Points in Grade Eight
★ Summary of knowledge points in the first volume of the eighth grade mathematics textbook published by People's Education Press.
★ The first volume of mathematics knowledge points induction of the second grade teaching edition.
★ Knowledge points of the first volume of eighth grade mathematics published by People's Education Edition
★ Summary of Mathematics Knowledge Points in the First Volume of Grade 8 of People's Education Edition
★ Knowledge points of the first volume of eighth grade mathematics published by New People's Education Edition