1, addition, subtraction, multiplication and division are called four operations.

2. In the formula without brackets, if there is only addition, subtraction or multiplication and division, it should be calculated from left to right.

3. There are multiplication, division and addition and subtraction in the formula without brackets, and multiplication and division must be calculated first, then addition and subtraction.

4. If there are brackets in the formula, count the inner side of brackets first, and then the outer side of brackets; The calculation order of the formulas in brackets follows the above calculation order.

5. Addition, subtraction, multiplication and division are called four operations.

6. Multiply first, then divide, then add and subtract, with brackets, and calculate in advance.

On the operation of "0"

1 and "0" are inseparable; Letter means: 0 error.

2. Add 0 to a number to get the original number; The letter means: A+0 = A.

3. Subtract 0 from a number to get the original number; The letter means: a-0 = a.

4. The minuend is equal to the minuend, and the difference is 0; Letter: A-A = 0

5, a number multiplied by 0, still get 0; Letter: a×0= 0

6. Divide 0 by any number other than 0 to get 0; The letter means: 0÷a(a≠0)= 0.

7,0 ÷ 0 can't get the fixed quotient; 5÷0 can't get the business.

Location and direction:

1, according to the direction and distance to determine or draw the specific position of the object. (Drawing and measuring scale and angle)

Note: 1, scale 2, due north direction 3, draw a corner.

2. Relativity between positions. The mutual positional relationship between two objects will be described. (Determination of observation points)

3. Draw a simple road map.

4. Three elements of a map: legend, direction and scale.

5. When determining the direction: A, first determine the observation point.

(1) From there, it is the observation point.

(2) The observation point follows the word "zhong".

Look at the direction at the observation point.

For example: ① East-south 25 (the corner marked with 25 is near the east)

② 35 north of the west (the angle marked with 35 is close to the west).

6. When describing the route and drawing the road map: there is only one line, and the lines made are end to end.

7. Eight common directions: east, south, west, north, southeast, northeast, southwest and northwest.

Regular operation and simple operation;

First, the law of addition:

1, additive commutative law: Two numbers are added, the addend positions are exchanged, and the sum is unchanged. a+b=b+a

2, the law of addition and association: three numbers are added, you can add the first two numbers first, and then add the third number; Or add the last two numbers first, and then add the first number, and the sum remains the same. (a+b)+c=a+(b+c)

These two laws of addition are often used together.

For example:165+93+35 = 93+(165+35) What is the basis?

3. The essence of continuous subtraction: one number subtracts two numbers continuously, which is equal to the sum of this number MINUS those two numbers. a-b-c=a-(b+c)

Second, the law of multiplication:

1, multiplication method of substitution: When two numbers are multiplied, the exchange factor position remains unchanged. a×b=b×a

2. Multiplication and association law: When three numbers are multiplied, you can multiply the first two numbers and then the third number, or you can multiply the last two numbers and then the first number, and the product remains unchanged.

(a×b )× c = a× (b×c)

These two multiplication laws are often used in combination. Such as: 125×78×8.

3. Multiplication and distribution law: the sum of two numbers is multiplied by one number. You can multiply these two numbers with these two numbers first, and then add up the products. (a+b)×c=a×c+b×c (a-b)×c=a×c-b×c

Application of multiplication and distribution laws;

① type 1: (a+b) × c (a-b )× c

= a×c+b×c = a×c-b×c

② Type Ⅱ: A× C+B× C A× C-B× C

=(a+b)×c =(a-b)×c

③ Type Ⅲ: A× 99+A× B-A.

= a×(99+ 1) = a×(b- 1)

④ Type 4: a×99 a× 102.

= a×( 100- 1)= a×( 100+2)

= a× 100-a× 1 = a× 100+a×2

Third, simple calculation.

1. Simple calculation of continuous addition:

(1) Use the law of additive combination (combine sum with integer ten, integer hundred, integer thousand and sum).

② Positions: 1 and 9,2 and 8,3 and 7,4 and 6,5 and 5, combined.

③ Decimal digits: 0 and 9, 1 and 8, 2 and 7, 3 and 6, 4 and 5, combined.

2. Simple calculation of continuous reduction:

Subtracting several numbers in a row is equal to subtracting the sum of these numbers. Such as:106-26-74 =106-(26+74)

② Subtracting the sum of several numbers is equivalent to subtracting these numbers continuously. Such as:106-(26+74) =106-26-74.

3. Simple calculation of addition and subtraction:

The position of the first number remains the same, and other addends and subtractions can be interchanged (you can add first and then subtract).

For example:123+38-23 =123-23+38146-78+54 =146+54-78.

4. Simple calculation of multiplication:

Apply the law of multiplicative association: combine two common numbers, 25 and 4; 125 and 8; 125 and 80, etc. Find 4 when you see 25, and find 8 when you see 125;

5. Simple calculation of division:

(1) divided by a series of numbers is equal to dividing by the product of these numbers.

(2) The product divided by several numbers is equal to the continuous division by these numbers.

6. Simple calculation of multiplication and division:

The position of the first number remains unchanged, and the remaining factors and divisors can be interchanged. (Multiply first and then divide) For example: 27× 13 ÷ 9 = 27× 13.

Fourth, the nature of continuous division: a number divided by two numbers equals the product of these two numbers. a÷b÷c = a÷(b×c)

1, common multiplication operations:

25×4= 100 125×8= 1000

2. additive commutative law simple calculation example: 3. Simple calculation example of additive associative law;

50+98+50 488+40+60

=50+50+98 =488+(40+60)

= 100+98 =488+ 100

= 198 =588

4. Simple example of multiplicative commutative law: 5. Simple example of multiplicative associative law:

25×56×4 99× 125×8

=25×4×56 =99×( 125×8)

= 100×56 =99× 1000

=5600 =99000

6. Simple calculation with additive commutative law and the law of association:

65+28+35+72

=(65+35)+(28+72)

= 100+ 100

=200

7. Simple calculation with multiplicative commutative law and associative law:

25× 125×4×8

=(25×4)×( 125×8)

= 100× 1000

= 100000

Simple examples of multiplication and distribution laws;

1, decomposition formula 2, combination formula

25×(40+4) 135× 12— 135×2

=25×40+25×4 = 135×( 12—2)

= 1000+ 100 = 135× 10

= 1 100 = 1350

3. Special 1 4, special 2

99×256+256 45× 102

=99×256+256× 1 =45×( 100+2)

=256×(99+ 1) =45× 100+45×2

=256× 100 =4500+90

=25600 =4590

5, special 3 6, special 4

99×26 35×8+35×6—4×35

=( 100— 1)×26 =35×(8+6—4)

= 100×26— 1×26 =35× 10

=2600—26 =350

=2574

A simple example of continuous subtraction;

528—65—35 528—89— 128 528—( 150+ 128)

=528—(65+35) =528— 128—89 =528— 128— 150

=528— 100 =400—89 =400— 150

=428 =3 1 1 =250

Second, the simple operation example of continuous division:

3200÷25÷4

=3200÷(25×4)

=3200÷ 100

=32

Third, other simple examples:

256—58+44 250÷8×4

=256+44—58 =250×4÷8

=300—58 = 1000÷8

=242 = 125

Five, the simple calculation of expansion:

102×38-38×2 125×25×32 125×88

37×96+37×3+37

Error-prone situation: 38×99+99

The meaning and essence of decimals;

1. Generation of decimals: When measuring and calculating, it is often impossible to get the exact result of integers, so decimals are often used to express them.

2. Fractions with denominators of 10, 100, 1000 ... can be expressed in decimals.

3.Decimal is another form of decimal.

4. Decimals are counted in tenths, hundredths and thousandths ... Write 0. 1, 0.0 1, 0.00 1 ...

5. The propulsion rate between every two adjacent counting units is 10.

6. Decimal numbers are decimals, percentiles, thousandths ... The highest digit is decimals. The lowest bit of the integer part is one bit. The propulsion rate of unit and decile is 10.

7. Decimal digit sequence table

Integer part decimal part decimal part

Digital ...10,000,100,10? Ten percentage points, thousands, tens of thousands ...

Counting unit ... one thousandth, one thousandth ...

(1) The counting unit of 6.378 is 0.00 1. (The counting unit of the lowest bit is the counting unit of an integer)

(2) There are six ones in 6.378, three tenths (0. 1) and seven percent (0.0 1).

Eight thousandths (0.00 1).

(3)6.378 is (6,378) thousandth (0.005438+0).

(4) 4 in 9.426 means four tenths (0. 1) [10th 4]

8. Decimal reading: first read the integer part (according to the original reading method), then read the decimal point, and then read the decimal part. Read the decimal part, the decimal part should read each number in turn, and there are several zeros to read.

9. Decimal writing: first write the integer part (according to the original writing method), then write the decimal point, and then write the decimal part: write the decimal part, and write each number in turn in the decimal part, and write several zeros if there are several zeros.

10, the nature of decimal: add "0" or remove "0" at the end of decimal, and the size of decimal remains unchanged. Note: the "0" in the middle of the decimal cannot be removed, and some "0" at the end cannot be removed when approximating. Functions can be reduced to decimals, etc.

1 1, decimal size comparison: (1) Compare the integer parts first; (2) If the integer parts are the same, compare decimals; (3) When the deciles are the same, compare the percentiles; (4) and so on until the size is compared.

12, movement of decimal point

Decimal point moves to the right:

Move one place, and the decimal will be expanded to 10 times the original number;

Move two places, and the decimal will be expanded to 100 times of the original number;

Move three places, and the decimal will be expanded to 10 00 times of the original number; ……

Decimal point moves to the left:

If you move one place, the decimal will be reduced by 10 times, that is, the decimal will be reduced to the original number;

Move two places, the decimal will be reduced by 100 times, that is, the decimal will be reduced to the original number;

If you move three digits, the decimal will be reduced by 1000 times, that is, the decimal will be reduced to the original number; ……

13, commonly used units in life:

Mass:1t =1000 kg; 1 kg =1000g

Length:1km =1000m1decimeter =10cm1cm =10mm.

1 decimeter = 100 mm 1 m = 10 decimeter = 100 cm = 1000 mm.

Area: 1 m2 = 100 square decimeter 1 square decimeter = 100 square centimeter.

1 km2 = 1 00ha1hectare =10000m2

Rmb: 1 yuan = 10 angle 1 angle = 10 minute 1 yuan = 100 minute.

Unit of length: kilometer —————————————— decimeter ——————— centimeter.

Area unit: square kilometer-hectare-square meter-square decimeter-square centimeter.

Mass unit: ton-kilogram-gram

Unit conversion:

(1) High-level units are converted into low-level units = = = = times the forward speed, and the decimal point is moved to the right.

(2) The low-level unit is converted into the high-level unit = = = = divided by the forward speed, and the decimal point is moved to the left.

14. Approximate decimal places (rounded):

(1) Keep an integer, indicating that it is accurate to one place, that is, omit the decimal part, depending on the number of decimal places. If the number of decimal places is greater than or equal to 5, advance one place. If it's less than five, give it up.

(2) Keep one decimal place, which means accurate to ten decimal places. It is necessary to omit everything after the first decimal place. At this time, it depends on the second place after the decimal point. If the second decimal place is less than 5, it is completely discarded. On the contrary, we should move forward one by one.

(3) Keep two decimal places, which means it is accurate to 1%, so the part after the second decimal place should be omitted. At this time, look at the third place after the decimal point. If the third decimal place is less than 5, it is completely discarded. On the contrary, we should move forward one by one.

(4) In order to facilitate reading and writing, numbers that are not whole tens of thousands or billions are often rewritten as numbers with units of "tens of thousands" or "hundreds of millions". When the number is rewritten as "10,000", it means that the decimal point is moved four places to the left, that is, the decimal point is placed on the right side of the 10,000 digits, and the word "10,000" is added after the number. When the number is rewritten as "1 100 million", it means that the decimal point is shifted to the left by 8 places, that is, the decimal point is placed to the right of 1 100 million, and the word "1 100 million" is added after the number. Note: bring your unit. Then remove the zeros after the decimal point according to the nature of the decimal point.

(5) When representing the divisor, the "0" after the decimal point cannot be removed.

Triangle:

1. Definition of triangle: A figure surrounded by three line segments (the endpoints of every two adjacent line segments are connected or overlapped) is called a triangle.

Draw a vertical line from the vertex of the triangle to its opposite side. The line segment between the vertex and the vertical foot is called the height of the triangle, and the opposite side is called the bottom of the triangle. This triangle is only three stories high. Key point: the drawing method of triangle height.

3. Characteristics of triangle: 1. Physical characteristics: stable. Such as: tripod for bicycles, tripod on telephone poles.

4. Characteristics of sides: the sum of any two sides is greater than the third side.

5. For the convenience of expression, letters A, B and C are used to represent the three vertices of a triangle, which can be represented as triangle ABC.

6, the classification of triangle:

By angle: acute triangle, right triangle, obtuse triangle.

Divided by side length: triangle with unequal sides and isosceles triangle (equilateral triangle or regular triangle is a special isosceles triangle).

The three sides of an equilateral triangle are equal and each angle is 60 degrees. (The concepts of top angle, bottom angle, waist and bottom)

7. A triangle with three acute angles is called an acute triangle.

8. A triangle with a right angle is called a right triangle.

9. A triangle with an obtuse angle is called an obtuse triangle.

10, each triangle has at least two acute angles; Each triangle has at most 1 right angles; Each triangle has at most 1 obtuse angles.

A triangle with 1 1 equal sides is called an isosceles triangle.

12. A triangle with three equilateral sides is called an equilateral triangle, also called a regular triangle.

13, an equilateral triangle is a special isosceles triangle.

14, and the sum of the internal angles of the triangle is equal to 180 degrees. The sum of the internal angles of the quadrilateral is 360, and the calculation and format of the correlation degree.

15, combination of figures: two identical triangles can definitely be combined into a parallelogram.

16. Two identical triangles can be used to form a parallelogram.

17. Two identical right triangles can be combined into a parallelogram, a rectangle and a big triangle.

18. Two identical isosceles right triangles can be combined into a parallelogram and a square. Large isosceles right triangle.

19, Dense Shop: The figures that can be Dense Shop are rectangle, square, triangle and regular hexagon.

Addition and subtraction of decimals:

1, calculation rule: align with the same digit (decimal point alignment). According to the integer calculation method, the decimal point of the obtained number should be aligned with the decimal point on the horizontal line. The result is that decimals should be simplified according to the nature of decimals.

2. Vertical calculation and check calculation. Pay attention to the horizontal answer, not the result of checking calculation.

3. The four operation sequences and algorithms of integers are also applicable to decimals. (Simplified calculation)

Statistics:

Advantages of 1. Bar chart: It directly reflects the quantity.

2. Advantages of broken-line statistical chart: It can reflect both quantity and increase or decrease of quantity.

3. In the dotted line statistical chart, the change trend refers to: rising or falling.

4. Broken line statistical chart: a unit length is used to represent a certain quantity, points are drawn according to the quantity, and then the points are connected in turn by line segments.

5. Advantages: You can not only see the quantity, but also see the change of quantity, predict the future trend, and provide guidance and help for future production and life.

Mathematics wide angle: planting trees

Planting trees:

1, both ends should be planted: interval number = total length ÷ spacing; Total length = spacing × number of intervals; Tree number = interval number+1; Number of intervals = number of trees-1

2. No planting at both ends: number of intervals = total length ÷ spacing; Total length = spacing × number of intervals; Number of trees = number of intervals-1; Number of intervals = number of trees+1

Number of intervals = total length ÷ interval length

Situation classification: 1, planting at both ends: number of plants = number of intervals+1.

2. Planting one head and not planting the other: number of plants = number of intervals.

3. No planting at both ends: number of plants = number of intervals-1.

4. Close: number of trees = number of intervals.

(2) Sawing problem: number of segments = times+1; Times = number of segments-1

Total time = x times each time

(3) Square matrix problem: The outermost layer is: side length × 4-4 or (side length-1) × 4.

The total number of the whole square matrix is: side length × side length.

(4) Closed figure (if surrounded by circle or ellipse): total length ÷ spacing = number of intervals; Number of trees = Number of intervals

(5) Number of chessboards:

1. Number of outermost chessboards: number of chessmen per side × number of sides-number of sides.

2. The total number of chessboards: the number of chessmen in each row × the number of chessmen in each column.

3. Number of people on the outermost layer of the phalanx: number of people on each side × 4-4.

4. Place flowerpots on polygons: the number of flowerpots placed on each side × the number of sides-the number of sides.