Axisymmetric: If a figure is folded in half along a straight line and the two parts completely overlap, such a figure is called an axisymmetric figure, and this straight line is called an axis of symmetry. (Square, rectangle, triangle, parallelogram, circle)

Rotation: in a plane, a figure rotates around a vertex by a certain angle to get the change of another figure. The fixed point o is called the rotation center, and the rotation angle is called the rotation angle. After rotation, another point on the original graph becomes the corresponding point.

The essence of rotation: the rotation of a figure is the positional movement of each point on the figure rotating around a fixed point at a fixed angle on the plane; Wherein the distances from the corresponding points to the rotation center are equal; The size and shape of the figure have not changed before and after rotation; The angles formed by the connecting line of two groups of corresponding points and the rotation center are equal, which are both equal to the rotation angle; The center of rotation is a fixed point.

Knowledge point connection: the difference between translation, axial symmetry and rotation

Two factors and multiples

1, divisible: divisor, divisor and quotient are natural numbers with no remainder.

When a large number is divisible by a decimal, it is a multiple of this decimal, and this decimal is a factor of this large number.

How to find the factor:

The number of factors of a number is limited, of which the smallest factor is 1, and the factor of a number is itself.

The number of multiples of a number is infinite, and the smallest multiple is itself.

Factor and multiple exist relatively and cannot be separated: 2 is a factor of 4 and 4 is a multiple of 2.

Factors and multiples usually refer to integers, not decimals. 2.4×5= 12, so 5 is the factor (×) of 12.

2. Natural numbers can be divided into odd and even numbers according to whether they can be divisible by 2.

Odd number: a number that is not divisible by 2.

Even number: a number divisible by 2.

The smallest odd number is 1 and the smallest even number is 0.

Numbers in units of 0, 2, 4, 6 and 8 are all multiples of 2.

A number with 0 or 5 is a multiple of 5.

The sum of the numbers on each digit of a number is a multiple of 3, and this number is a multiple of 3.

The two digits divisible by 2, 3 and 5 at the same time are 90, and the smallest three digits are 120.

3. Natural number divided by the number of factors: prime number, composite number, 1.

Prime number: There are only two factors, 1 and itself.

Composite number: There are at least three factors, 1, itself and other factors.

1: There are only 1 factors. 1' is neither a prime number nor a composite number.

The smallest prime number is 2 and the smallest composite number is 4.

Prime numbers within 20: 8 (2, 3, 5, 7, 1 1 3, 17, 19).

4. Prime factorization

Decomposition of prime factors by short division (a composite number is written in the form of multiplication of several prime numbers)

5. Common factor, common factor

The common factor of several numbers is called the common factor of these numbers. One of them is called their common factor.

Find the common factor of two or three numbers by short division (multiply all the divisors until they are coprime)

The common factor of several numbers is only 1, so they are coprime.

Special case of coprime of two numbers;

(1) 1 is coprime with any natural number; (2) Two adjacent natural numbers are coprime; (3) Two prime numbers must be coprime;

(4) 2 and all odd coprime; 5] prime number is coprime with a composite number smaller than it;

If two numbers are multiples, then the smaller number is their common factor.

If two numbers are coprime, then 1 is their common factor.

0、 1、2、3、4

6, common multiple, minimum common multiple

The common multiple of several numbers is called the common multiple of these numbers. The smallest one is called their least common multiple.

Find the least common multiple of two numbers by short division (multiply all divisors and quotients until they are coprime)

Find the least common multiple of three numbers by short division (multiply all divisors and quotients until they are pairwise coprime)

If two numbers are multiples, then the larger number is their least common multiple.

If two numbers are coprime, then their product is their least common multiple.

1. lap problem

2. Bus problem

3. Common factor

Cuboid and cube

concept

1, a three-dimensional figure surrounded by six rectangles (two opposite faces are squares in special cases) is called a cuboid. In a cuboid, the opposite faces are exactly the same, and the opposite sides are equal in length.

2. The edge where two faces intersect is called an edge. The point where three sides intersect is called a vertex. The length of three sides intersecting at a vertex is called the length, width and height of a cuboid.

A three-dimensional figure surrounded by six identical squares is called a cube (also called a cube). A cube has 12 sides, and all sides have the same length and the same face.

4. The number of faces, edges and vertices of cuboids and cubes is the same, but the edges of cubes are all equal. A cube can be said to be a cuboid with equal length, width and height. It is a special cuboid.

5. A cuboid has 6 faces, 8 vertices and 12 sides. The areas of the opposite faces are equal, and the lengths of the opposite sides are equal. A cuboid has at most 6 rectangular faces, at least 4 rectangular faces and at most 2 square faces. A cube has six faces, each face is square, each face has the same area, and there are 12 sides, and each side has the same length.

The sum of the sides of a cuboid = (length+width+height) ×4 L=(a+b+h)×4.

Length = sum of side lengths ÷4- width-height a=L÷4-b-h

Width = sum of side lengths ÷4- length-height b=L÷4-a-h

Height = sum of sides ÷4- length, width and h=L÷4-a-b

Sum of cube sides = side length × 12 L=a× 12.

Side length of a cube = sum of side lengths ÷ 12 a=L÷ 12.

6. The six faces and the total area of a cuboid or cube are called its surface area.

The surface area of a cuboid = (length× width+length× height+width× height )× 2s = 2 (AB+AH+BH)

The surface area of a bottomless (or uncovered) cuboid = length× width+(length× height+width× height )× 2.

S=2(ab+ah+bh)-ab S=2(ah+bh)+ab

The surface area of a cuboid without a bottom and a cover = (length× height+width× height )× 2s = 2 (ah+BH)

Surface area of cube = side length × side length× 6 s = a× a× 6.

6. The size of the space occupied by an object is called its volume.

Volume of cuboid = length× width× height V=abh

Length = volume ÷ width ÷ height a=V÷b÷h

Width = volume ÷ length ÷ height b=V÷a÷h

Height = volume ÷ length ÷ width H = V ÷ A ÷ B.

Volume of cube = side length × side length × side length v = a× a× a.

7. The volume of objects that can be accommodated in boxes, oil drums, warehouses, etc. Often referred to as their volume.

The commonly used unit of volume are liters and milliliters, which can also be written as L and ml.

1 l = 1 cubic decimeter 1 ml = 1 cubic centimeter 1 l = 1000 ml.

8.a3 is pronounced as "the cube of A", which means that three A's are multiplied (that is, a a a).

Unit of volume converted high-level units into low-level units.

Low-level unit and high-level unit

Propulsion rate: 1 m3 = 1000 cubic decimeter = 1000000 cubic centimeter.

1 cubic decimeter = 1000 cubic centimeter = 1 liter = 1000 ml.

1 cm3 = 1 ml

1 m2 = 100 square decimeter = 10000 square centimeter

1 km2 = 100 hectares =1100 million m2.

Weight unit rate, time unit rate and length unit rate

The emphases and difficulties of this chapter:

1, the problem of finding the side length:

2. The problem of finding the area: the occupied area, the area of irregular figure and the change of the surface area of block three-dimensional figure.

3. Solve the volume problem: segmentation problem, irregular figure volume and drainage method. (plus one method, tail removal method)