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).
Eighth grade mathematics knowledge point book 1
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 the same 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. Inverse Theorem of Pythagorean Theorem If the three sides of a triangle are related to A 2+B 2 = C 2, then the triangle is a right triangle.
Mathematics learning methods and skills
1, intensive training, there are many problems calculated and proved this semester, so we should strengthen this training in review. Especially for linear functions, we should practice by types in the review process, pay attention to the correct choice of problem-solving methods, and let students develop the habit of checking the calculation results. There are also geometric proof questions, and we should try our best to make fewer points through targeted exercises to achieve concise and rigorous proof.
2. Strengthen strict management requirements. According to the strict requirements of each student's own situation and learning level, the contents of the corresponding knowledge meeting are repeatedly explained and practiced. It is necessary to learn a little, learn a little. Students with poor acceptance ability should strengthen counseling after class, correct their mistakes in time, and check more at ordinary times. For students with strong ability, we should guide them to do more extracurricular exercises and appropriately improve the difficulty of doing them.
3. Strengthen the training of proof questions. Through the recent study, I found that students are not sure about the proof question, can't find a suitable analysis method, and some students can't understand the meaning of the question and have no ideas. In the future review, I am going to spend some time practicing the proof questions to guide students how to understand the meaning, how to analyze and how to write the proof process. Try to let students do all kinds of questions and grasp their own characteristics.
4. Strengthen the counseling of students with unsatisfactory grades, make detailed review plans, praise and encourage them more, mobilize their enthusiasm for learning, and use their spare time for counseling. Be patient and calm when tutoring. Speak the knowledge you don't know several times, and don't be afraid of trouble until you understand it.
Relevant articles on knowledge points in the second volume of mathematics in the second grade of Hunan Education Press;
★ Catalogue of eighth grade mathematics textbooks of Hunan Education Press.
★ Review materials of junior high school mathematics by Hunan Education Press.
★ Guidance of eighth grade learning methods
★ Hunan Education Edition Eighth Grade Mathematics Volume II Teaching Plan
★ Hunan Education Edition Senior One Mathematics Knowledge Points
★ Hunan Education Press Eighth Grade Mathematics Volume II Teaching Plan (2)
★ Hunan Education Press eighth grade mathematics volume II teaching plan.
★ Final examination paper of eighth grade mathematics of Hunan Education Publishing House