1, additive commutative law: Two numbers are added to exchange the position of addend, and the sum is unchanged.

2. Law of additive combination: When three numbers are added, the first two numbers are added first, or the last two numbers are added first, and then the third number is added, and the sum remains unchanged.

3. Multiplication and exchange law: when two numbers are multiplied, the position of the exchange factor remains unchanged.

4. Multiplication and association law: When three numbers are multiplied, the first two numbers are multiplied, or the last two numbers are multiplied first and then the third number, and their products are unchanged.

5. Multiplication and distribution law: When two numbers are multiplied by the same number, you can multiply the two addends by this number respectively, and then add the two products, and the result remains unchanged. Such as: (2+4) × 5 = 2× 5+4× 5

6. Nature of division: In division, the dividend and divisor are expanded (or reduced) by the same multiple at the same time, and the quotient remains unchanged. Divide by any number that is not.

Simple multiplication: multiplication of multiplicand and multiplier with O at the end. You can multiply 1 before o first, and zero does not participate in the operation, and add a few zeros at the end of the product.

7. What is an equation? An equation in which the value on the left of the equal sign is equal to the value on the right of the equal sign is called an equation.

Basic properties of the equation: When both sides of the equation are multiplied (or divided) by the same number at the same time, the equation is still valid.

8. What is an equation? A: Equations with unknowns are called equations.

9. What is a linear equation with one variable? A: An equation with an unknown number of degree 1 is called a linear equation with one variable.

Example method and calculation of learning linear equation of one variable. That is, an example is given to illustrate that the formula is replaced by χ and calculated.

10, fraction: divide the unit "1" into several parts on average, and the number representing such a part or points is called a fraction.

1 1, addition and subtraction of fractions: addition and subtraction of fractions with denominator, only numerator addition and subtraction, denominator unchanged. Fractions of different denominators are added and subtracted, first divided, then added and subtracted.

12. Comparison of fractional sizes: Compared with the denominator, the numerator is large and the numerator is small. Compare the scores of different denominators, divide them first and then compare them; If the numerator is the same, the denominator is big and small.

13, the fraction is multiplied by the integer, and the product of the multiplication of the fraction and the integer is the numerator, and the denominator remains unchanged.

14. Fractions are multiplied by fractions, the product of numerator multiplication is numerator, and the product of denominator multiplication is denominator.

15, the fraction divided by an integer (except 0) is equal to the fraction multiplied by the reciprocal of this integer.

16, true fraction: the fraction with numerator less than denominator is called true fraction.

17. False fraction: the fraction with numerator greater than denominator or numerator equal to denominator is called false fraction. False score is greater than or equal to 1.

18, with fraction: write the false fraction as an integer, and the true fraction is called with fraction.

19, the basic nature of the fraction: the numerator and denominator of the fraction are multiplied or divided by the same number at the same time.

(except 0), the score size remains unchanged.

20. A number divided by a fraction is equal to the number multiplied by the reciprocal of the fraction.

2 1, the number A divided by the number B (except 0) is equal to the reciprocal of the number A multiplied by the number B. ..

Law of fractional addition and subtraction: Fractions with the same denominator are added and subtracted, only the numerator is added and subtracted, and the denominator remains the same. Fractions of different denominators are added and subtracted, first divided, then added and subtracted.

The multiplication of fractions is: use the product of molecules as numerator and the product of denominator as denominator.

22. What is the ratio? The division of two numbers is called the ratio of two numbers. Such as: 2÷5 or 3:6 or 1/3.

The first and second items of the ratio are multiplied or divided by the same number at the same time (except 0), and the ratio remains unchanged.

23. What is proportion? Two expressions with equal ratios are called proportions. For example, 3: 6 = 9: 18

24. The basic nature of proportion: In proportion, the product of two external terms is equal to the product of two internal terms.

25. Solution ratio: Finding the unknown item in the ratio is called solution ratio. Such as 3: χ = 9: 18.

26. Proportion: two related quantities, one of which changes and the other changes. If the corresponding ratio (i.e. quotient k) of these two quantities is certain, these two quantities are called proportional quantities, and the relationship between them is called proportional relationship. For example: y/x=k( k must be) or kx = y.

27. Inverse proportion: two related quantities, one of which changes and the other changes accordingly. If the product of the corresponding two numbers in these two quantities is certain, these two quantities are called inverse proportional quantities, and their relationship is called inverse proportional relationship. For example: x×y = k( k must be) or k/x = y.

28. Percentage: The number that indicates that one number is the percentage of another number is called percentage. Percentages are also called percentages or percentages.

29. To convert decimals into percentages, just move the decimal point two places to the right and add hundreds of semicolons at the back. In fact, to convert a decimal into a percentage, just multiply this decimal by 100%.

30. To convert percentages into decimals, just remove the percent sign and move the decimal point two places to the left.

3 1, the fraction is converted into a percentage, usually converted into a decimal (except infinity, three decimal places are usually reserved), and then converted into a percentage. In fact, to turn a fraction into a percentage, you must first turn the fraction into a decimal and then multiply it by 100%.

32, the percentage of the number of components, first rewrite the percentage of the number of components, can be turned into the simplest score.

33. Learn how to divide scores into scores and how to divide scores into scores.

34. greatest common divisor: several numbers can be divisible by the same number at the same time, and this number is called the greatest common divisor of these numbers. (or the common divisor of several numbers is called the common divisor of these numbers. The largest one is called the greatest common divisor. )

35. Prime number: The common divisor is only 1, which is called prime number.

36. Least common multiple: the common multiple of several numbers is called the common multiple of these numbers, and the smallest is called the least common multiple of these numbers.

37. Comprehensive score: Divide scores with different denominators into scores with the same denominator equal to the original score, which is called comprehensive score. (Common divisor is the least common multiple)

38. Approximate fraction: It is called approximate fraction to change a fraction into a fraction that is equal to it, but the numerator and denominator are relatively small. (The greatest common divisor is used for divisor)

39. simplest fraction: The numerator and denominator are fractions of prime numbers, which are called simplest fraction.

40. At the end of the score calculation, the score must be converted into the simplest score.

4 1, numbers with 0, 2, 4, 6 and 8 in the unit can be divisible by 2, that is, can be binary.

42. About integration. A number with a bit of 0 or 5 can be divisible by 5, that is, it can be subtracted by 5. Pay attention to the use of contracts.

43. Even and odd numbers: Numbers divisible by 2 are called even numbers. Numbers that are not divisible by 2 are called odd numbers.

44. Prime number (prime number): If a number only has 1 and its two divisors, it is called a prime number (or prime number).

45. Composite number: a number. If there are other divisors besides 1 and itself, such numbers are called composite numbers. 1 is neither prime nor composite.

46. Interest = principal × interest rate × time (time is generally in years or months, which should correspond to the unit of interest rate).

47. Interest rate: The ratio of interest to principal is called interest rate. The ratio of interest to principal for one year is called annual interest rate. The ratio of interest to principal in January is called monthly interest rate.

48. Natural number: An integer used to represent the number of objects is called a natural number. 0 is also a natural number.

49. Cyclic decimal: a decimal, starting from a certain place in the decimal part, and one or several numbers are repeated in turn. Such decimals are called cyclic decimals. Like 3. 14 14 14.

50. Acyclic decimals: Decimals that start from the decimal part and do not have one or several numbers repeated. Such a decimal is called an acyclic decimal. For example, pi: 3. 14 1592654.

5 1, infinite acyclic decimal: a decimal, from the decimal part to the infinite digits, is called infinite acyclic decimal without one or several numbers repeating in turn. Such as 3. 14 1592654. ...

52. What is algebra? Algebra is to replace numbers with letters.

53. What is algebraic expression? Expressions expressed in letters are called algebraic expressions. For example 3x = AB+C.

relational expression

1, number of copies × number of copies = total number of copies/number of copies = total number of copies/number of copies = number of copies.

2. 1 multiple × multiple = multiple ×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××

4. Unit price × quantity = total price ÷ unit price = total quantity ÷ quantity = unit price

5. Work efficiency × working hours = total workload ÷ work efficiency = working hours ÷ total workload ÷ working hours = work efficiency.

6. Appendix+Appendix = sum, and-one addend = another addend.

7. Minus-Minus = Minus-Minus = Minus+Minus = Minus

8. Factor × factor = product ÷ one factor = another factor.

9. Dividend = quotient dividend = divisor quotient × divisor = dividend

Total number ÷ Total number of copies = average value

Formula of sum and difference problem

(sum+difference) ÷ 2 = large number

(sum and difference) ÷ 2 = decimal

And folding problems.

Sum \ (multiple-1) = decimal

Decimal × multiple = large number

(or sum-decimal = large number)

Difference problem

Difference ÷ (multiple-1) = decimal

Decimal × multiple = large number

(or decimal+difference = large number)

Tree planting problem

1 The problem of planting trees on unclosed lines can be divided into the following three situations:

(1) If trees are planted at both ends of the non-closed line, then:

Number of plants = number of nodes+1 = total length-1.

Total length = plant spacing × (number of plants-1)

Plant spacing = total length ÷ (number of plants-1)

2 If you want to plant trees at one end of the unclosed line and not at the other end, then:

Number of plants = number of segments = total length ÷ plant spacing

Total length = plant spacing × number of plants

Plant spacing = total length/number of plants

(3) If no trees are planted at both ends of the non-closed line, then:

Number of plants = number of nodes-1 = total length-1.

Total length = plant spacing × (number of plants+1)

Plant spacing = total length ÷ (number of plants+1)

The quantitative relationship of planting trees on the closed line is as follows

Number of plants = number of segments = total length ÷ plant spacing

Total length = plant spacing × number of plants

Plant spacing = total length/number of plants

The question of profit and loss

(Profit+Loss) ÷ Difference between two distributions = number of shares participating in distribution.

(Big profit-small profit) ÷ Difference between two distributions = number of shares participating in distribution.

(big loss-small loss) ÷ The difference between two distributions = the number of shares participating in the distribution.

encounter a problem

Meeting distance = speed × meeting time

Meeting time = meeting distance/speed and

Speed Sum = Meeting Distance/Meeting Time

Catch up with the problem

Catch-up distance = speed difference× catch-up time

Catch-up time = catch-up distance ÷ speed difference

Speed difference = catching distance ÷ catching time

Tap water problem

Downstream velocity = still water velocity+current velocity

Countercurrent velocity = still water velocity-current velocity

Still water velocity = (downstream velocity+countercurrent velocity) ÷2

Water velocity = (downstream velocity-countercurrent velocity) ÷2

Concentration problem

Solute weight+solvent weight = solution weight.

The weight of solute/solution × 100% = concentration.

Solution weight × concentration = solute weight

Solute weight-concentration = solution weight.

Profit and discount problem

Profit = selling price-cost

Profit rate = profit/cost × 100% = (selling price/cost-1) × 100%.

Up and down amount = principal × up and down percentage

Discount = actual selling price ÷ original selling price× 1 00% (discount <1)

Interest = principal × interest rate× time

After-tax interest = principal × interest rate × time × (1-20%)

Exchange rate between units

1 km = 1 km 1 km = 1000 m

1 m = 10 decimeter 1 decimeter =10 cm1cm =10 mm.

1 m2 = 100 square decimeter 1 square decimeter = 100 square centimeter 1 square centimeter = 100 square millimeter

1 m3 = 1000 cubic decimeter 1 cubic decimeter = 1000 cubic centimeter 1 cubic centimeter = 1000 cubic millimeter.

1 ton = 1 000kg1kg = 1 000g = 1 kg =1kg.

1 hectare = 1 10,000 square meters. 1 mu = 666.666 square meters.

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

Area unit conversion

1 km2 = 100 hectare

1 ha = 1 10,000 m2

1 m2 = 100 square decimeter

1 square decimeter = 100 square centimeter

1 cm2 = 100 mm2

Volume (volume) unit conversion

1 m3 = 1000 cubic decimeter

1 cubic decimeter = 1000 cubic centimeter

1 cubic decimeter = 1 liter

1 cm3 = 1 ml

1 m3 = 1000 liter

Weight unit conversion

1 ton = 1000 kg

1 kg =1000g

1 kg = 1 kg

Rmb unit conversion

1 yuan = 10 angle.

1 angle = 10 point

1 yuan = 100 integral.

Time unit conversion

1 century = 100 1 year =65438+ February.

The big month (3 1 day) includes:1\ 3 \ 5 \ 7 \ 8 \10 \ 65438+February.

Abortion (30 days) includes: April \ June \ September \165438+1October.

February 28th in a normal year and February 29th in a leap year.

There are 365 days in a normal year and 366 days in a leap year.

1 day =24 hours 1 hour =60 minutes.

1 minute =60 seconds 1 hour =3600 seconds.

Calculation formula of perimeter, area and volume of mathematical geometry in primary schools

1, the perimeter of the rectangle = (length+width) ×2 C=(a+b)×2.

2. The circumference of a square = side length ×4 C=4a.

3. Area of rectangle = length× width S=ab

4. Square area = side length x side length s = a.a = a.

5. Area of triangle = base × height ÷2 S=ah÷2.

6. parallelogram area = bottom x height S=ah

7. trapezoidal area = (upper bottom+lower bottom) × height ÷ 2s = (a+b) h ÷ 2.

8. Diameter = Radius× 2D = 2r Radius = Diameter ÷2 r= d÷2

9. The circumference of a circle = π× diameter = π× radius× 2c = π d = 2π r.

10, area of circle = π× radius× radius.

1 1, the surface area of a cuboid = (length× width+length× height+width× height )× 2 formula: S=(a×b+a×c+b×c)×2.

12, cuboid volume = length× width× height formula: V = abh.

13, the surface area of the cube = side length× side length× 6 formula: S=6a2.

14, cuboid (or cube) volume = bottom area × height formula: V = abh.

15, volume of cube = side length × side length × side length formula: V = a3.

16, surface (edge) area of the cylinder: the surface (edge) area of the cylinder is equal to the perimeter of the bottom multiplied by the height. Formula: s = ch = π DH = 2π RH.

17, the surface area of the cylinder: the surface area of the cylinder is equal to the perimeter of the bottom multiplied by the height plus the area of the circles at both ends. Formula: S=ch+2s=ch+2πr2.

18, volume of cylinder: the volume of cylinder is equal to the bottom area multiplied by the height. Formula: V=Sh

19, cone volume = 1/3 bottom × product height. Formula: V= 1/3Sh

prime number

A number has no divisor (factor) except 1 and itself. This number is called prime number (also called prime number).

composite number

A number has other divisors besides 1 and itself. This number is called a composite number.

Note: 1 has only one divisor, which is itself. 1 is neither prime nor composite.

The smallest prime number is 2, which is the only even number among prime numbers (see below for the explanation of even numbers), and all other prime numbers are odd numbers (see below for the explanation of odd numbers).

even number

Even numbers are natural numbers (including 0) divisible by 2, also called even numbers. Even numbers are usually represented by "2k".

odd number

Odd numbers are natural numbers that are not divisible by 2, also known as odd numbers. Odd numbers are usually represented by 2k+ 1

Note: Even numbers are composite numbers except 2. Even number: a number divisible by 2. (Also includes 0)

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

Natural number: a number representing the number of objects, and the smallest natural number is "0".

Natural numbers are also integers. 0 is the dividing line between positive and negative integers.

Complex number: Besides "1" and itself, there are other divisors. The smallest composite number is "4".

Prime number: a number with only "1" and its own two divisors. The smallest prime number is "2".

1' is neither a composite number nor a prime number.

Prime number: the common divisor has only two numbers "1".

Common divisor: the common divisor of two numbers.

Common multiple: a multiple shared by two numbers.

Prime factor: decomposing a composite number into several prime numbers is called the prime factor of this composite number.

Prime factor decomposition: the process of decomposing a composite number into several prime numbers is called prime factor decomposition.

Characteristics of numbers divisible by 2: The numbers in the unit are 0, 2, 4, 6, 8.

The characteristic of a number divisible by 3: the sum of the numbers on each bit is a multiple of 3.

Characteristics of numbers divisible by 5: The number in a unit is 0,5.

The characteristic of a number divisible by 9: the sum of the numbers on each bit is a multiple of 9.

Characteristics of numbers divisible by 4 or 25: The number of the last two digits is a multiple of 4 or 25.

Characteristics of numbers divisible by 8 or 125: the last three digits are multiples of 8 or 125.

decimal

Basic properties of decimals: Add "0" or remove "0" at the end of decimals, and the size of decimals remains unchanged.

Finite decimal: the number of digits in the decimal part is limited.

Infinite decimal: The number of decimal parts is infinite. Infinitely circulating decimal: The decimal part has regular digits.

Infinitely cyclic decimal: the decimal part is irregular (also called irrational number)

Pure Cyclic Decimal: Cycle from the first digit in the decimal part.

Mixed cycle decimal: does not start with the first decimal place.

Cycle segment: starting from somewhere in the decimal part, it means repeating one or several numbers in turn. These numbers are called cyclic parts.

mark

Significance of fraction: Divide the unit "1" into several parts, and the number of one or several parts is called fraction. The basic nature of a fraction: the numerator and denominator of a fraction are multiplied or divided by a number at the same time (except 0). The size of the score remains the same.

True score < 1. False score ≥ 1.

Dividing the numerator and denominator of a fraction by their greatest common factor at the same time is called divisor, and the obtained fraction is called simplest fraction.

Simplest fraction: When the denominator and the numerator are coprime, this score is called simplest fraction.

Using the basic properties of fractions, several fractions with different denominators are transformed into fractions with the same denominator. This process is called general score. In the comparison of scores, general scores will be widely encountered.

Geometric knowledge

A closed figure surrounds him with 1 circle, and the length of this circle is his circumference.

The size of a plane occupied by an object is called its area.

The space occupied by an object is called its volume.

The volume that an object can hold other objects is called the volume of this object.

The surface area of an object is called the surface area.

The sum of the internal angles of a triangle is 180 degrees. The sum of the internal angles of a quadrilateral is 360 degrees. The sum of the internal angles of n sides is (side length -2)× 180 degrees.

Exterior Angle: The angle between the reverse extension line of 1 side and an adjacent side is called the exterior angle. The outer angle of a triangle is the sum of two non-adjacent inner angles.

The sum of the external angles of any closed figure is 360 degrees.

Line:

Straight line: no end point, no length and infinite extension.

Ray: There is an end point, no length and infinite extension.

Line segment: It has two endpoints and a length.

Two rays drawn from a point, the part sandwiched between these two rays is called an angle, and that point is called a vertex. Angles are divided into several angles: acute angle (greater than 0 degrees and less than 90 degrees), right angle (equal to 90 degrees), obtuse angle (greater than 90 degrees and less than 180 degrees), right angle (equal to 180 degrees) and rounded corner (equal to 360 degrees).

Make a vertical line from the point 1, and this point is called vertical foot.

When two straight lines never intersect, it means that they are parallel to each other.

Plane graphics:

Triangle:

If the largest angle in a triangle is an obtuse angle, this triangle is called an obtuse triangle.

If the largest angle in a triangle is a right angle, this triangle is called a right triangle.

If the largest angle in a triangle is an acute angle, this triangle is called an acute triangle.

Draw a vertical line from the vertex to the other side. The length of this vertical line is called the height of this triangle. 1 Triangle has three heights.

When the two sides of a triangle are equal, it is called an isosceles triangle. The two sides of an isosceles triangle are equal in length, and the rest is called the base. When three sides of a triangle are equal, it is called an equilateral triangle, which is a special isosceles triangle. Its three angles are all 60 degrees.

Quadrilateral:

All four corners of a quadrilateral are right angles. When any two nonadjacent sides are parallel to each other, a quadrilateral is called a rectangle. When all four sides are equal and each angle is 90 degrees, it is a square. A square is a special rectangle.

When any two sides of a quadrilateral are parallel to each other, this figure is a parallelogram (a rectangle is a special parallelogram). A parallelogram has countless heights. When the four sides are equal in length, this figure is called a diamond (a diamond is a special parallelogram).

When only one set of opposite sides are parallel to each other, this figure is called trapezoid. The upper side of the trapezoid is called the upper bottom, the lower side is called the lower bottom, and the left and right sides of the trapezoid are called the waist of the trapezoid.

When the left and right sides are equal in length, this trapezoid is called an isosceles trapezoid.

The ratio of the circumference to the diameter of a circle is always colonial. People call it pi, which is generally expressed by the letter π≈3. 14.

Three-dimensional graphics:

Cuboids and cubes have 6 faces, 12 diamonds and 8 vertices.