New curriculum standards. General review outline of junior high school mathematics of China Normal University Edition

General review materials of junior high school mathematics

Numbers and algebra

⒈ number sum formula

(1) rational number: finite or infinite cyclic number (irrational number: infinite acyclic decimal)

⑵ Number axis: "three elements"

(3) reciprocal

(4) absolute value: │ a │ = a (a ≥ 0) │ a │ =-a (a

5] mutual

[6] Index

① Zero exponent: = 1(a≠0) ② Negative integer exponent: (a≠0, n is a positive integer)

Once the square formula is completed:

⑻ Square difference formula: (a+b)(a-b)= 1

Levies the operating properties of the power supply:

① ? = ② ÷ = ③ = ④ = ⑤ ⑽ Scientific notation: (1 ≤ A < 10, n is an integer)

⑾ arithmetic square root, square root, cube root,

⑿

2. Equality and inequality

(1) unary quadratic equation

(1) definition and general form:

② Solution:

1. Direct leveling method.

2. Matching method

3. Formula method:

4. Factorization method.

③ Discrimination formula of roots:

> 0, there are two solutions.

< 0, no solution.

= 0, there are 1 solutions.

(4) Vader theorem:

⑤ Common equations:

⑥ Application problems

1. Travel problems: When encountering problems, catch up with them and sail in the water:

2. The growth rate problem: the starting number (1+X)= the ending number.

3. Engineering problem: workload = working efficiency × working time (workload is often considered as "1").

4. Geometric problems

(2) Fractional equation (attention test)

Find the value of the parameter by adding the root:

① Change the original equation into an integral equation.

(2) Bring the added root into the transformed whole equation, and get the value of the parameter.

(3) the essence of inequality

①a & gt； b→a+c & gt； b+c

②a & gt； b→AC & gt； BC (c>0)

③a & gt； b→AC & lt； BC (c<0)

④a & gt； b，b & gtc→a & gt； c

⑤a & gt； b，c & gtd→a+c & gt； b+d。

3. Function

(1) linear function

① definition: y=kx+b(k≠0)

② Image: the intersection of straight line (0, b)- and Y axis and (-b/k, 0, b)- and X axis.

③ Attribute:

K>0, the straight line passes through the first and third quadrants, and y increases with the increase of x.

K<0, the straight line passes through two or four quadrants, and y decreases with the increase of x.

When b>0, the straight line must pass through the first and second quadrants.

When b=0, the straight line passes through the origin.

When b<0, the straight line must pass through three or four quadrants.

(4) Four images:

(2) Proportional letter:

① definition: y=kx(k≠0)

② Image: straight line (passing through the origin)

⑶ Inverse proportional function

① Definition: (k≠0).

② image: hyperbola (two branches)

③ Attribute:

K>0, the two curves are located in the first and third quadrants respectively, and the value of y decreases with the increase of the value of x.

K<0, the two curves are located in the second and fourth quadrants respectively, and the value of Y increases with the increase of X value. ;

④ Two curves are infinitely close to the coordinate axis but can never reach the coordinate axis.

(4) Quadratic function.

① Definition:

② Image: parabola

Vertex:

Vertex: (h, k)

③ Attribute:

(1) when a >; 0, the opening is upward; When a<0, the opening is downward. The larger the |a|, the smaller the opening of the parabola.

(2) When A and B have the same number (ab>0), the symmetry axis is on the left side of the Y axis; When a and b have different numbers (AB

(3) When c>0, the positive semi-axis intersects the Y-axis; When c<0, it intersects the Y axis on the negative semi-axis; When c=0, it intersects the y axis at the origin.

(4) the law of parallel motion:

When h>0, y=ax moves horizontally by h units to the right, and y=a(x-h) is obtained.

When h < 0, it is obtained by moving |h| units in parallel to the left.

When h>0, k>0 and y=ax move in parallel by H units to the right, and then move up by K units, y=a(x-h) +k is obtained.

When h>0, k<0, y=ax are moved in parallel by h units to the right, and then moved down by |k| units, y = a (x-h)+k is obtained.

When h < 0, k >; 0, y=ax moves |h| units to the left in parallel, and then moves up by k units to get y = a (x-h)+k.

When h < 0, k<0, y=ax moves |h| units to the left in parallel, and then moves |k| units down, and y = a (x-h) 2+k is obtained.

Second, space and graphics.

1. triangle

⑴ Area formula: bottom multiplied by height divided by 2.

(2) "Four Hearts":

① Vertical center: the intersection of three heights of a triangle.

② Inner heart: the intersection of bisectors of three inner angles of a triangle, that is, the center of the inscribed circle.

③ Center of gravity: the intersection of the three midlines of a triangle.

(4) Outer circle center: the intersection of the perpendicular lines of the three sides of a triangle, that is, the center of the circumscribed circle.

(3) the relationship between the sides of the triangle:

The sum of two sides is greater than the third side. (two shorter sides)

The difference between the two sides is smaller than the third side. (longest and smallest sides)

(4) The relationship between the sum of the inner angles of a triangle and the outer and inner angles:

The sum of the internal angles of a triangle is 180 degrees.

The outer angle of a triangle is equal to the sum of two non-adjacent inner angles.

The outer angle of a triangle is greater than any inner angle that is not adjacent to it.

5] prove

Judgment and nature

straight

corner

three

corner

Form (1) In a right triangle, if there is an acute angle equal to 30, then the right side it faces is equal to half of the hypotenuse.

(2) If the midline of one side of a triangle is equal to half of the hypotenuse, then the angle subtended by this side is a right angle.

① The two acute angles of a right triangle are complementary.

② The median line on the hypotenuse of a right triangle is equal to half of the hypotenuse.

③ In a right triangle, the sum of squares of two right-angled sides A and B is equal to the square of hypotenuse C, that is, A2+B2 = C2.

Isosceles

Triangle ① The two base angles of an isosceles triangle are equal. (equilateral and angular)

② The bisector of the top angle of the isosceles triangle, the median line on the bottom edge and the height on the bottom edge coincide. (three in one)

An isosceles triangle with an angle equal to 60 is an equilateral triangle.

mutually

be similar

Triangle ① similar triangles corresponds to the height ratio, and the ratio corresponding to the center line and the ratio corresponding to the angular bisector are all equal to the similarity ratio.

(2) the ratio of similar triangles perimeter is equal to the similarity ratio.

③ The ratio of similar triangles area is equal to the square of similarity ratio.

④ The corresponding angles of similar triangles are equal, and the corresponding sides are proportional.

accomplish

wait for

three

corner

The three sides of shape ① correspond to the coincidence of two equal triangles. (SSS)

② Two sides and their included angles correspond to the congruence of two triangles. (SAS)

(3) The two corners and their clamping edges are congruent with each other. (ASA)

(4) The opposite sides of two angles and one angle correspond to the congruences of two equal triangles. (AAS)

⑤ Two triangles with hypotenuse and a right angle are congruent. (HL)

⑥ The sides and angles corresponding to congruent triangles are equal.

triangle

Midline ① The line segment connecting the midpoints on both sides of the triangle is called the midline of the triangle.

② The midline of the triangle is parallel to the third side, equal to half of it.

3. Special angle:

(1) diagonal

(2) Complementary angle

(3) supplementary angle

3. Line segment

theorem

The point on the midline of the line segment (1) is equal to the distance between the two endpoints of the line segment.

Trapezoidal midline ① The midline of the trapezoid is parallel to the two bottoms and equal to half of the sum of the two bottoms.

Parallel lines ① have equal internal dislocation angles. ② The internal angles on the same side are complementary. ③ congruent angles are equal.

The vertical section ① is the shortest distance from a point to a straight line.

Angle bisector ① The point on the angle bisector is equal to the distance on both sides of this angle.

3. Trigonometric function

(1) acute trigonometric function:

Sine: opposite hypotenuse of sin A=∠A tangent of the adjacent hypotenuse of cosine: cos A =∠A: tangent of opposite hypotenuse of tan a = ∠ a.

(2) trigonometric function of complementary angle:

①sin A=co s(90 -A) cos A=sin(90 -A)

② Tan A = Kurt (90 -A) Kurt A = Tan (90 -A)

(3) The trigonometric function relation of the same acute angle:

sin2A+cos2A= 1 tanA？ CotA = 1 tanA = Sina cosA

(4) trigonometric function value of special angle:

Trigonometric function sinα cosα tanα

30 12

45 22

60 32

three

5] Handling of practical problems:

① Slope: The greater the Sin A value, the steeper the step; The smaller the value of Cos A, the steeper the step.

② Orientation (up north, down south, left west, right east)

(3) prone and elevation:

5. Quadrilateral.

(1) area formula:

① Trapezoid, the sum of the upper bottom and the lower bottom multiplied by the height divided by 2.

② Diamond, diagonal multiplied by diagonal divided by 2.

(3) parallelogram line, the base multiplied by the height.

⑵

Judgment property

flat

line

four

edge

Shape ① Two groups of opposite sides are parallel respectively.

② The opposite sides of the two groups are equal.

③ The two diagonal groups are equal.

(4) Two diagonal lines are equally divided.

⑤ A group of opposite sides are parallel and equal.

6. A set of diagonal lines are equal, and a set of opposite sides are parallel. (1) diagonally equal.

② Two groups of opposite sides are parallel and equal.

③ The two diagonals are equally divided.

water caltrop

shape

① There is a group of parallelograms with equal adjacent sides.

② A parallelogram with two diagonal lines perpendicular to each other.

(3) A quadrilateral with four equilateral sides. ① It has all the properties of a parallelogram.

② All four sides are equal.

③ Diagonal lines are perpendicular to each other, and each diagonal line bisects a set of diagonal lines.

④ It is not only an axisymmetric figure, but also a centrally symmetric figure.

A square/rule/moment

Form ① A parallelogram with a right angle.

② Parallelogram with equal diagonals.

(3) A quadrilateral with three right angles.

① It has all the properties of a parallelogram.

② All four corners are right angles.

③ Diagonal lines are equal.

④ It is not only an axisymmetric figure, but also an axisymmetric figure.

Square ① has a set of rectangles with equal adjacent sides.

② A diamond with a right angle.

③ There is a group of parallelograms with equal adjacent sides and a right angle.

④ Quadrilateral whose diagonals are perpendicular to each other and equally divided. ① It has all the properties of parallelogram, rectangle and diamond.

② Diagonal lines are perpendicular to each other, equally divided and equal.

③ It is not only an axisymmetric figure, but also a centrally symmetric figure.

wait for

waist

ladder

Form ① One set of opposite sides is parallel and the other set is equal.

② Two trapezoids with equal bottom angles on the same bottom. ① The two waists are equal.

② Diagonal lines are equal.

(3) The figure obtained by connecting the midpoints of each side in turn:

① Connect the midpoint of each side of the quadrilateral with equal diagonal lines in turn to get a diamond.

(2) Connect the midpoints of the sides of the quadrilateral in turn with diagonal lines perpendicular to each other to obtain a rectangle.

③ Connect the midpoints of the sides of the quadrilateral with equal diagonals in turn to get a square.

(4) parallelogram connecting the midpoints of each side of the quadrilateral in turn.

[6] Circle

(1) vertical diameter theorem:

Passing through the center of the circle, perpendicular to the chord, bisecting the chord and bisecting the upper and lower arcs of the chord. (Know two and push three)

(2) Angle related to the circle:

central angle

Defines the angle of the vertex at the center of the circle and the angle of the vertex on the circumference.

nature

quality

The degree of the central angle is equal to its radian. The circumferential angle of the diameter is 90 degrees.

In the same circle or in the same circle, the isocentric angle (circumferential angle) has equal arc, chord and chord center distance.

The angle of the circle opposite to the arc is equal to half the angle of the center of the circle opposite to it.

(3) the positional relationship between circles: (the distance between the centers of circles is d, and the radii are R r and R >； respectively; r)

Exterior: d & gtR+r circumscribed: d=R+r intersected: R-R.

(4) the positional relationship between the straight line and the circle: (the radius is R, and the distance from the center of the circle O to the straight line L is D)

Separation: d>R Tangency: d=R Intersection: D

5] The positional relationship between the point and the circle: (the radius is R, and the distance from the point to the center of the circle is D)

Outside the circle: d>r point is in the circle: D.

[6] Calculation formula:

① perimeter formula:

② Formula of circular area:

③ Sector area formula:

④ Arc length formula:

(7) Concept: chord and diameter; Arc, equal arc, upper arc, lower arc, semicircle; Distance from chord to center; Equal circle, same circle, concentric circle.

Once ruler drawing requirements

(1) makes a line segment equal to a known line segment.

(2) Make an angle equal to the known angle.

(3) the bisector of the angle

(4) perpendicular bisector as a line segment.

5] Make a triangle

① It is known that all three sides are triangles.

② Two sides and their included angles are called triangles.

③ It is known that two angles and their sides are triangles.

④ Known base and isosceles triangle on the base.

[6] Make a circle by one point, two points and three points that are not on the same straight line.

⒏' s views and predictions