1, a rational number can be expressed as (A and B are integers), so it is not a rational number.

2. The integer part of 1 .2 is1and the decimal part is 0.2, while the integer part of-1.2 is -2 and the decimal part is 0.8.

3. Points on the number axis correspond to real numbers one by one. When representing points on the number axis, we should pay attention to three elements: positive direction, origin and unit length.

4, pay attention to the conditions:

5. Pay attention to the positive and negative of a when removing the absolute value. Similarly, when simplifying the radical, we should also pay attention to the sign of the radical. For example:

6. Pay attention to the positive and negative square roots of positive numbers, such as

Negative numbers have no square root.

7. To find the middle term C in the ratio of A to B, it should be noted that if C is a number, it can be positive or negative, and if C is a line segment, it can only be positive.

8. To judge whether it is a similar item, it must be simplified before judging.

9. Find the value of algebraic expression: (1) Simplify first and then substitute.

(2) Pay attention to the format (when x=…, original formula = original value substitution = simplification = answer)

10, factorization: (1) Pay attention to the range of decomposition, which is generally within the real number range.

(2) No matter which decomposition method is adopted, the common factor is extracted first.

(3) Use letters accurately.

For example, the topic is for you. Do not decompose into (x-3)(x+ 1).

(4) cross multiplication, to split accurately, don't take it for granted.

For example:

(5) When using the root formula, please pay attention to the following points:

(1) One is not missing, and two are preceded by minus signs.

②

Don't miss y

1 1, the exponential power should be transformed into a radical. For example,

12, expressed by algebraic expression, simple should be changed, complex should not be changed.

Equation (group)

1, first observe the equation, then choose the appropriate method, don't just pick up the questions and do it. Pay more attention to whether the topic specifies a method. If the title says "Use method of substitution", you can't use "method of substitution".

2. See clearly whether the topic is integral equation, fractional equation or irrational number equation? You don't need to test the whole equation, but don't forget to test the score and irrational number equation. It is suggested to substitute the original equation for checking calculation, and the solution equation should be 100% correct.

3, the conclusion should be correct, depending on whether it is an equation or a system of equations.

4, a quadratic equation:

(1) If the equation has two real solutions, it must be.

② For the relationship between root and coefficient, pay attention to whether the quadratic coefficient A is 1.

(3) If the topic uses the relationship between root and coefficient, then the calculated value should be replaced by delta check.

④ The quadratic equation with one variable has multiple roots, but the equation has no multiple roots.

⑤

⑥

⑦ If there are no two problems, set two equations as follows

5. If there is a letter coefficient, it is necessary to discuss whether it is a linear equation or a quadratic equation.

6. The solution of the equation is: it should be said that the equation has no real number solution, not that X has no solution.

7. Application questions:

(1) test questions

(2) The setting and answer should be complete, and the unit should be written.

(3) The units in the questions should be unified, and the answers should not be omitted.

④ Pay attention to the range of values. For example, if several people are involved, an integer is required.

⑤ Indirect setting must be required.

6. Pay attention to the standard quantity if it is more or less.

⑦ Common types: growth rate problem (compared with the origin), handshake problem, telephone problem ...

⑧ The irrational equation of fractional order equation, whether it is practical problem or geometric calculation, must be tested. Check whether it is the solution of the original equation, but also whether it meets the meaning of the question.

8、

9. When method of substitution transforms the fractional equation, it depends on whether the topic is to transform you simply or to transform you into an integral equation.

Inequality (group)

1, both sides of the inequality are multiplied (or divided) by the same negative number at the same time, and the inequality needs to be signed.

Such as-2x >; six

2, pay attention to the topic is to let you find the solution of the inequality group or integer solution.

function

1, function increase and decrease (positive and negative ratio, linear function, sine, cosine, tangent, cotangent) can be combined with images.

2, find the coordinates of the point, abscissa, ordinate don't make a mistake.

If the point in the topic is on the coordinate axis, it should be considered as being on the X axis or the Y axis.

4. If y=2x+m does not pass through the second quadrant, then m 0

5. the coordinate of point a is X 1, the coordinate of point b is X2, AB=, if X2 >；; Then AB= X2 -X 1.

6, quadratic function:

① Vertex coordinates are clearly recited.

② The distance between two intersections on the X axis =

(4) If the topic does not specify the coordinates of the intersection of the function and the X axis, then it should be set.

⑤ Whenever the relationship between root and coefficient is used, it must be replaced by △.

⑥ If the function value is always greater than 0, then a>0 and △ < 0.

⑦ Movement of functional images (left+right-)

⑧ Finding the maximum value of a function depends on whether the vertex is within the allowable range. If the vertex is within the allowable value range, it is the maximum value; If the vertex is not within the allowable range, the two endpoints of the range are the maximum.

Pet-name ruby function image vertex is on the x axis, then δ= 0.

7, trigonometric function:

① Recite special values.

②

③

(4) Sina Times cosA times A, which should be expressed as asinAcosA.

⑤ When trigonometric function is used, it should be explained which angle is right angle.

⑥ the slope is expressed as I =1:m.

⑦ Find the elevation angle and depression angle.

What should I pay attention to when I encounter the slope ratio problem? Is to find the horizontal distance? Vertical distance? Or slope?

For example, if the flying height of the plane is m kilometers and the depression angle of the ground control point is a when viewed from the plane, the distance between the plane and the ground control point is

Pet-name ruby objects under sunlight, shadows may all shine on the ground, may also shine on the wall.

statistics

1, how to judge whether the data given to you can represent the whole as a sample?

(1) depends on whether the sample is randomly selected from the population (2) depends on whether the randomly selected sample is within the range required by the population.

2、

3. Find out the difference between frequency distribution histogram and frequency distribution histogram:

The ordinate of the histogram of frequency distribution is: frequency

The ordinate of the histogram of frequency distribution is: (the sum of the areas of all small rectangles is equal to the sum of all group frequencies, which is equal to 1).

4. Examination: See if the topic requires you to complete the full-frequency (frequency) distribution map.

Don't forget to divide the standard deviation and variance by the number of items.

6. Distinguish between variance and standard deviation

7. Analysis stability: the average value should be combined with variance.

triangle

1, "Four Hearts" and Its Properties

2. The sum of two sides of a triangle is greater than the third side.

For example, the triangle a=2, b=3 and c=5 does not exist.

3. When using Pythagorean theorem, we should first explain which angle is right angle.

If one side of a triangle is equal to half of the other side, it can't be said that its right angle must be 30 degrees.

5. In a right triangle, there is an angle of 60 degrees, so it cannot be directly deduced that there is a doubling relationship between the two sides.

6. When meeting the height of a triangle, pay attention to whether the triangle is an acute triangle, a right triangle or an obtuse triangle.

For example, it is known that the waist of an isosceles triangle is higher than half of the waist, so the vertex angle of this isosceles triangle is 30 or 150 degrees.

7. If two sides or two angles of an isosceles triangle are known, they should be classified and discussed.

For example, suppose that both sides of an isosceles triangle are 2 and 5, and find the perimeter.

Analysis: the lengths of three sides are divided into 2, 2, 5 and 2, 5, 5, and the former is not valid.

8. The area of a triangle should be multiplied by half.

9. See the right triangle and the midpoint of the hypotenuse, often connecting the midline of the hypotenuse; Seeing the midpoint of a general triangle may double the length of the midline, and the midline attribute may be used.

10, knowing the height on the hypotenuse of a right triangle, I think of the projective theorem, but I need to prove it.

quadrilateral

1, clarifying the judgment theorem of each special quadrilateral. (See whether the conditions given in the title are quadrilateral or parallelogram)

Rectangle: A parallelogram with equal diagonals is a rectangle.

A parallelogram with right angles is a rectangle.

There are three angles that are right angles, quadrangles or rectangles.

Diamond: A set of parallelograms with equal adjacent sides is a diamond.

Parallelograms with diagonal lines perpendicular to each other are diamonds.

A quadrilateral with four equilateral sides is a diamond.

Square: A set of rectangles with equal adjacent sides is a square.

Diamonds with right angles are squares.

2, isosceles trapezoid

The nature of isosceles trapezoid is: (1) isosceles (2) with the same bottom and two corners equal (3) with the same diagonal.

Determination of isosceles trapezoid: (1) An isosceles trapezoid is an isosceles trapezoid.

(2) A trapezoid with the same base and equal angles is an isosceles trapezoid.

(3) A trapezoid with equal diagonals is an isosceles trapezoid.

3. Can calculate and distinguish the internal angle, external angle and central angle of a regular polygon.

Axisymmetry and rotation

1, see clearly the rotation center, rotation direction and rotation angle.

2, will distinguish between axisymmetric graphics and graphics about linear symmetry graphics.

3, will distinguish between the central symmetrical figure and the symmetrical figure of the figure about a certain point.

4. Axisymmetric graphics: isosceles triangle, equilateral triangle, isosceles trapezoid, regular polygon, rectangle, diamond, square, line segment, straight line, angle (including angle) and circle.

5. Centrally symmetric figures: parallelogram, rectangle, diamond, square, regular N-polygon (n is an even number), line segment, straight line and circle.

6. Write a conclusion after drawing.

similar triangles

1, pay attention to the corresponding relationship of similar triangles. If the topic gives you "△ABC is similar to △DEF", discuss it in categories; If the topic is "△ABC∽△DEF", the corresponding relationship has been determined and need not be discussed.

2. The area ratio is equal to the square of the similarity ratio, and the corresponding ratio of height, angle bisector and median line is equal to the similarity ratio.

3. See clearly whether the topic is comparison or comparison.

For example:

4. Prove that triangles are similar. If AA cannot be proved, consider SAS.

5. The condition of "applying the proportion theorem of parallel lines" is that "three straight lines are parallel to each other" (AB//CD//EF).

6. There is no inverse theorem for the proportion theorem of parallel lines and line segments, that is, "two straight lines are cut by three straight lines, and the cut line segments are proportional, so these three line segments are parallel to each other" is a false proposition.

Golden section:

1. There are two golden sections on a line segment.

2.p is the golden section of line segment AB (AP >；; BP), then,

circle

1, the basic properties of circles

(1) Determination of a circle: Three points that are not on the same straight line determine a circle.

(2) Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the arc opposite to the chord.

Vertical Diameter Theorem and its Inference: ① Passing through the center of the circle; ② perpendicular to the chord; ③ bisecting the chord; ④ bisecting the lower arc of the chord; ⑤ bisecting the upper arc of the chord.

If the above two conditions are met, the remaining three conclusions can be deduced.

But pay attention to the condition of "bisecting the diameter (non-diameter) of a chord perpendicular to the chord and bisecting the arc opposite to the chord".

(3) In the same circle, if the central angles are equal, the chords are equal, the distances between chords are equal, and the arcs are equal.

(4) Reciting the formula of arc length; Area formulas of sector, arc length and bow

(5) It should be proved that "four-point * * * circle" and "the angle of the circle opposite to the diameter is a right angle".

2. The positional relationship between a straight line and a circle

(1) The nature of the tangent of the circle: the tangent of the circle is perpendicular to the radius of the tangent point.

A straight line passing through the center (tangent point) and perpendicular to the tangent line must pass through the tangent point (center).

Two tangents drawn from a point outside the circle are equal in length, and the connecting line between the center of the circle and the point bisects the included angle of the two tangents.

(2) Theorem for determining the tangent of a circle:

(1) Know the tangent point, connect the center of the circle and the tangent point, and prove the verticality (using the judgment theorem of tangent-verticality+outer end of radius).

(2) The tangent point is unknown, and the center of the circle is taken as the vertical line, which proves that d = r.

(3) The intersection chord, secant and secant theorems need to be proved.

(4) To judge the position relationship between a circle and a straight line, we only need to study the size relationship between D and R..

3. The positional relationship between circles.

(1) Two circles have different radii, and there are five positional relationships.

The radii of two circles are equal, and there are only three positional relationships.

(2) The positional relationship between two circles cannot be said to be tangent or separated. Because tangency includes internal cutting and external cutting; Separation includes tolerance and alienation. If the topic says that two circles are tangent, there are two situations-inscribed and circumscribed.

(3) When two circles intersect, it should be noted that the centers of the two circles may be on the same side of the chord or on different sides of the chord.

For example, if the radii of two intersecting circles are 5 and 4 respectively and the chord length is 6, the distance between the centers of the two circles is.

(5) To calculate the distance between two chords, we should also consider the same side and different side of the center of the circle.

(6) When two circles are inscribed, multiple solutions should also be considered.

For example, two circles are inscribed, the center distance is 3, and the radius of one circle is 5. Find the radius of the other circle.

Solution: /x-5/=3, x=8 or x=2.

(7) The common tangent of two circles should consider the internal common tangent and the external common tangent.

(8) When studying the positional relationship between circles, the formula can be applied as long as the relationship between D and R+r and R-r is considered. You don't have to draw.

(9) When the radii of two circles are unknown, the absolute value should be added to the radius subtraction.

Intersection:

Internal cutting:

Include:

(10) The intersection of the common tangent of two circles and the center of two circles and the three-point * * * line need to be proved.

Matters needing attention

1, calculating

(1) The calculation result should be simplified as follows: ① There is no common factor between numerator and denominator.

② Radical is the simplest radical.

(2) When removing the brackets, multiply by the coefficient outside the brackets and pay attention to the symbols.

(3) The topic requires similarity and needs to be kept at the end.

2. The trajectory should be clearly explained.

For example, the equidistant trajectory from point A to point B is the midline of AB.

Another example is the trajectory of a point 3cm away from point A: a circle with point A as the center and 3cm as the radius.

3. Draw with a ruler, keep traces of drawing, and write conclusions.

4. Fill in the blanks and don't miss the unit

5. Mathematical thought: reduction, combination of numbers and shapes, classified discussion, conjecture induction, analogy association, letter substitution, analysis and synthesis, and equation thought.

6. Theorems that need to be proved in the senior high school entrance examination: ① Angular bisector theorem ② Projective theorem ③ Four-point * * circle ④ The circumference angle corresponding to the diameter is 90 degrees ⑤ Some three-point * * lines.

7, easy to skip (proof questions must not be skipped)

(1) don't write vertical, directly get 90 (or don't write vertical after 90). ② In a right triangle, the multiplication relation of line segments is directly derived from the angle of 60 degrees. ③ Quadrilateral+condition, and directly deduce the square to solve equations and inequalities.

8. Multiple solutions:

① Know the two sides of a right triangle, and find the height of the third side ② Know the triangle and divide it into acute angle, right angle and obtuse angle ③ Know the corresponding relationship between two sides or two angles of an isosceles triangle ④ similar triangles uncertainty ⑤ The circle is tangent to or separated from the circle ⑤ Two circles are inscribed, and the radius and center distance of one circle are known, while the radius of the other circle ⑦ Two circles intersect, and the radius of two circles is known, and the center distance ⑧ Find the distance between two chords.

9, classified discussion

Classification discussion caused by uncertainty of congruence or similar correspondence of (1) graphs.

(2) Classification discussion caused by point uncertainty.

(3) Classification discussion caused by the change of position relationship between graphs caused by graph movement.

10, test questions

(1) Learn to find out the implicit conditions in the topic and pay attention to the relationship between each small topic.

(2) See clearly whether the moving range of the point is on a line segment, a ray or a straight line?

(3) dynamic topic, when thinking, draw more pictures of various states (general/special) on the draft paper, so that you can see the position change of points and the trend of graphic change.

(4) After completing the question, review it again to see if it meets the meaning of the question.

For example, y =-x2+bx+c (c >; 0), the vertex is on the straight line AB, Pa: Pb = 1: 3, and find the analytical formula of parabola.

Analysis: To solve y=-x2+4x or y=-x2+2x+6, the former does not meet the requirement of c>0 and needs to be abandoned.

1 1, domain:

(1) Algebra Application Problem: In real life problems, most values are greater than 0. Sometimes you have to consider restrictions, such as a few cars, and you have to take an integer.

(2) Graphic exercises: ①x is meaningful and Y is meaningful; ② Take the limit state.

Note: ① Conforms to the meaning of the question (whether two points can coincide, whether the points are on a straight line (or a ray or a straight line) ② The figure exists.

12. If the topic asks "When the value of x, ... and prove your conclusion", you need to prove it in reverse. If the topic asks "Is there ... which makes ...", it is generally assumed that the conclusion is established before solving it, and there is no need for reverse proof.

13, do not exceed the dotted line.

14. Bring good tools: pencil, ruler, a set of triangular rulers, compasses and protractors.

You can use tools to do exercises.