Volume and surface area

Area of triangle = base × height ÷2. The formula S= a×h÷2.

Area of square = side length × side length formula S= a2

Area of rectangle = length× width Formula S= a×b

Area of parallelogram = base× height Formula S= a×h

Trapezoidal area = (upper bottom+lower bottom) × height ÷2 Formula S=(a+b)h÷2

Sum of internal angles: sum of internal angles of triangle = 180 degrees.

The surface area of a cuboid = (length× width+length× height+width× height )× 2 Formula: S=(a×b+a×c+b×c)×2.

Surface area of cube = side length × side length ×6 Formula: S=6a2.

Cuboid volume = length× width× height formula: V = abh

Volume of cuboid (or cube) = bottom area × height formula: V = abh.

Volume of cube = side length × side length × side length formula: V = a3.

Circumference = diameter × π formula: L = π d = 2π r

Area of circle = radius × radius× π formula: s = π R2.

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

Surface area of cylinder: the surface area of 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.

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

Volume of cone = 1/3 bottom× product height. Formula: V= 1/3Sh

arithmetic

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

2. Additive associative law: A+B = B+A.

3. Multiplicative commutative law: a× b = b× a.

4. Multiplicative associative law: a × b × c = a ×(b × c)

5. Multiplicative distribution law: a× b+a× c = a× b+c.

6. The nature of division: a ÷ b ÷ c = a ÷(b × c)

7. 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. O is divided by any number that is not O. Simple multiplication: the multiplicand and the end of the multiplier are multiplied by O. 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.

8. Division with remainder: dividend = quotient × divisor+remainder

Equations, Algebras and Equality

Equation: An equation in which the value on the left of the equal sign equals 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.

Equation: An equation with an unknown number is called an equation.

One-dimensional linear equation: An equation with an unknown number of degree 1 is called a one-dimensional linear equation. 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.

Algebra: Algebra means replacing numbers with letters.

Algebraic expression: Expressions expressed by letters are called algebraic expressions. For example 3x = AB+C.

mark

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

Comparison of fraction size: Compared with the fraction of 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.

Addition and subtraction of fractions: add and subtract fractions with the same denominator, only add and subtract numerators, and the denominator remains the same. Fractions of different denominators are added and subtracted, first divided, then added and subtracted.

Fraction multiplied by integer, numerator is the product of fractional and integer multiplication, denominator remains unchanged.

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

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 concept of reciprocal: 1 If the product of two numbers is 1, we call one of them the reciprocal of the other. These two numbers are reciprocal. The reciprocal of 1 is 1, and 0 has no reciprocal.

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

The basic properties of a fraction: the numerator and denominator of a fraction are multiplied or divided by the same number (except 0), and the size of the fraction.

The law of division of fractions: dividing by a number (except 0) is equal to multiplying the reciprocal of this number.

True fraction: The fraction with numerator less than denominator is called true fraction.

False fraction: Fractions with numerator greater than denominator or numerator equal to denominator are called false fractions. False score is greater than or equal to 1.

With a score: write a false score as an integer, and a true score is called with a score.

The basic nature of the fraction: the numerator and denominator of the fraction are multiplied or divided by the same number (except 0) at the same time, and the size of the fraction remains unchanged.

Calculation formula of quantitative relationship

Unit price × quantity = total price 2, single output × quantity = total output

Speed × time = distance 4, work efficiency × time = total workload.

Appendix+Appendix = and one addend = and+another addend.

Negative-negative = differential negative = negative-differential negative = negative+difference.

Factor × factor = product One factor = product ÷ another factor.

Frequency divider/frequency divider = frequency divider = frequency divider/frequency divider = quotient × frequency divider

Length unit:

1 km = 1 km 1 km = 1000 m

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

Area unit:

1 km2 = 1 00ha1hectare =10000m2

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

1 mu = 666.666 square meters.

volume unit

1 m3 = 1000 cubic decimeter

1 cm3 = 1000 cm3

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

Unit right

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

compare

What is the ratio? When two numbers are divided, it is called the ratio of two numbers. For example, the first and second terms of the ratio of 2÷5 or 3:6 or 1/3 are multiplied or divided by the same number at the same time (except 0), and the ratio remains unchanged.

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

The basic property of proportion: in proportion, the product of two external terms is equal to the product of two internal terms.

Solution ratio: the unknown term in the proportion is called solution ratio. Such as 3: χ = 9: 18.

Proportion: two related quantities, one of which changes and the other changes. If the ratio (i.e. quotient k) corresponding to these two quantities is constant, 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.

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.

per cent

Percentage: a number that indicates that one number is a percentage of another number, which is called percentage. Percentages are also called percentages or percentages.

To convert decimals into percentages, just move the decimal point two places to the right and add hundreds of semicolons at the end. In fact, to convert a decimal into a percentage, just multiply this decimal by 100%. To convert percentages to decimals, simply remove the percent sign and move the decimal point two places to the left.

When a fraction is converted into a percentage, the fraction is generally converted into a decimal (three decimal places are generally reserved when it is not used up), and then the decimal is 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%.

Divide the percentage into components, and rewrite the percentage into components first, so that the quotation that can be lowered can be made into the simplest score.

We should learn to decompose fractions into components and fractions into decimals.

Multiples and divisors

Maximum common divisor: The common divisor of several numbers is called the common divisor of these numbers. There is a finite common factor. The largest one is called the greatest common divisor of these numbers.

Least common multiple: The common multiple of several numbers is called the common multiple of these numbers. There are infinite common multiples. The smallest one is called the least common multiple of these numbers.

Prime number: the common divisor has only 1 two numbers, which is called prime number. Two adjacent numbers must be prime numbers. Two consecutive odd numbers must be coprime. 1 and any number coprime.

Comprehensive score: the difference between scores of different denominators is changed into the same denominator score equal to the original score, which is called comprehensive score. (Common divisor is the least common multiple)

Decrement: divide the numerator and denominator of a fraction by the common divisor at the same time, and the fraction value remains unchanged. This process is called dropping points.

Simplest fraction: The numerator and denominator are fractions of prime numbers, which are called simplest fraction. At the end of the score calculation, the score must be converted into the simplest score.

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

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.

Prime factor: If a prime number is a factor of a certain number, then this prime number is the prime factor of this number.

Prime factor decomposition: A composite number is represented by the complementary way of prime factors, which is called prime factor decomposition.

Multiple characteristics:

Characteristics of multiples of 2: You are 0, 2, 4, 6, 8.

Characteristics of multiples of 3 (or 9): The sum of the numbers on each digit is multiples of 3 (or 9).

Characteristics of multiples of 5: You are 0, 5.

Characteristics of multiples of 4 (or 25): The last two digits are multiples of 4 (or 25).

Characteristics of multiples of 8 (or 125): the last three digits are multiples of 8 (or 125).

Characteristics of multiples of 7 (1 1 or 13): the difference (big-small) between the last three digits and other digits is a multiple of 7 (1 1 3).

Characteristics of multiples of 17 (or 59): the difference (big-small) between the last three digits and the rest digits is a multiple of 17 (or 59).

Characteristics of multiples of 19 (or 53): the difference (big-small) between the last three digits and other seven digits is a multiple of 19 (or 53).

Characteristics of multiples of 23 (or 29): The difference (big-small) between the last four digits and the other five digits is multiples of 23 (or 29).

Of the two numbers in the multiple relation, the greatest common divisor is smaller and the smallest common multiple is larger.

The coprime relation between two numbers, the greatest common divisor is 1, and the least common multiple is the product.

When two numbers are divided by their greatest common divisor, the quotient is coprime.

The product of two numbers and the least common multiple is equal to the product of these two numbers.

The common divisor of two numbers must be the greatest common divisor of these two numbers.

1 is neither prime nor composite.

A prime number greater than 3 divided by 6 must get 1 or 5.

Odd and even numbers

Even numbers: Numbers are numbers of 0, 2, 4, 6 and 8.

Odd number: The number is not 0, 2, 4, 6 or 8.

Even even = even Qiqi = Qiqi.

Even numbers add up to even numbers, and odd numbers add up to odd numbers.

Even × even = even × odd = odd × even = even.

The sum of two adjacent natural numbers is odd, and the product of adjacent natural numbers is even.

If one number in the multiplication is even, then the product must be even.

Odd ≠ even number

separable

If c | a, c | b, then c | (a b)

If, then b | a, c | a

If b | a, c | a and (b, c)= 1, then BC | a.

If c | b, b | a, then c | a

decimal

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

Pure Decimal: Decimal in units of 0.

With Decimal: Decimal with more than 0 digits.

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

Acyclic decimal: a decimal, starting from the decimal part, without one number or several numbers appearing repeatedly. Such a decimal is called acyclic decimal. Like 3. 14 1592654.

Infinite cycle decimal: a decimal, from the decimal part to the infinite digits, and one or several numbers are repeated in turn. Such decimals are called infinite cyclic decimals. For example, 3. 14 14 14 ...

Infinite acyclic decimal: a decimal, from decimal part to infinite digits, is called infinite acyclic decimal without one number or several numbers appearing repeatedly. Such as 3. 14 1592654. ...

profit

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

Interest rate: The ratio of interest to principal is called interest rate. The ratio of one year's interest to principal is called the annual exam description: this question 15, full mark 120, and the answer time is 90 minutes.

First, fill in the blanks:

1, a two-digit number, divided by 58+2, 73+3, 85+ 1, this two-digit number is 14.

2. 173□ is a four-digit number. The math teacher said, "I filled in three numbers one by one in this □, and the three four-digit numbers I got were divisible by 9, 1 1 and 6 in turn." Then the sum of the three numbers filled in in turn is 19.

3. A in the score is a natural number. In order to make this score reducible, the minimum value of a is 1 1.

A five-digit number plus 200,000, after tripling, the result is exactly the same as that of the right-hand side of the five-digit number plus 2. This five-digit number is _ _ _ _ _ _ _.

5. A worker put the parts into two kinds of boxes, each big box contains 12 parts, and each small box contains 5 parts, which is just finished. If the number of parts is 99 and the number of boxes is greater than 10, it is 8 and 5 boxes respectively.

6. When the goods with the unit price of 40 yuan are sold in 50 yuan, the profit per piece is 10 yuan, but only 500 pieces can be sold. It is known that the sales volume of this kind of goods will decrease by 1 0 for every price increase of1yuan. In order to make the biggest profit, the price should be set at RMB.

7. Two cups are filled with 40% and 10% salt solution respectively, and the concentration is 30% after being poured together. If you add 300 grams of 20% salt solution, the concentration will become 25%. So the original 40% salt solution is grams.

8. At a certain moment, the hour hand of the clock is between 10 and 1 1. Six minutes later, the minute hand is exactly opposite to the hour hand three minutes ago and is in a straight line. Then the time represented by this clock is.

9. Grandpa is six times older than Xiaoming this year. In a few years, my grandfather will be five times as old as Xiaoming. In a few years, my grandfather will be four times as old as Xiaoming. Q: How old is my grandfather this year?

10, the film crew had a one-day trip from city a to city b, and planned to walk more in the morning than in the afternoon 100 km to go to city c for lunch. Because of the traffic jam, they arrived at a small town at noon, only one third of the original plan was opened. After crossing the town, the car traveled 400 kilometers and then stopped to have a rest in the evening. The driver said, from C City to here, we have to walk half way.

Second, answer questions:

1 1. As shown in the right figure, AD, BE and CF divide △ABC into six small triangles, and the areas of the four small triangles have been marked on the figure. Try to find the area of △ABC. (Unit: square centimeter)

12. My mother gave Honghong some money to buy New Year cards. There are three kinds of New Year cards: A, B and C. The first card is 0.50 yuan each, and the third card is 1.20 yuan. Using this money to buy a card is more than 8 cards for B card, and 6 cards for B card. How much did mom give Hong Hong? How much is the second card?

13. The escalator runs from bottom to top at a constant speed. Two impatient children thought that the escalator was running too slowly, so on the escalator, the boy walked up 1 step every second and the girl walked 2 steps every 3 seconds. As a result, boys go upstairs in 50 seconds and girls go upstairs in 60 seconds. How many steps does this escalator have?

14. A workshop plans to finish the task of processing a batch of parts in five days. 120 parts were processed on the first day, the remaining 150 parts on the second day, the remaining 80 parts on the third day, the remaining 20 parts on the fourth day and the last 1800 parts on the fifth day.

15, two cars, A and B, drive in opposite directions from two cities for 6 hours respectively, and they can meet somewhere on the way. A car failed to track on the way, and it took 2.5 hours to continue driving. So it took 7.5 hours from departure to meeting. So, how many hours did car A travel from one city to another?

Interest rate. The ratio of interest to principal in January is called monthly interest rate.

Mid-term examination questions of sixth grade mathematics in primary school

Class: Name: Grade:

Fill in the blanks.

1、0.7÷5 = 7:( ) = =( )%。

2.5A=4B(A and B are not equal to 0). A: B = (): ().

3. The ratio of: to the simplest integer is ().

4. If =, then A and B are proportional ().

5. The volume of a cylinder with a base diameter and a height of 6 decimeters is ().

6. The radius of the cylinder bottom is 5m, the volume is157m3, and the height is () m..

7. In a proportion, two internal terms are reciprocal, one external term is and the other external term is ().

8. A rectangular land with a length of 75 meters and a width of 30 meters was drawn on the drawing, and the scale was () long and () wide.

9. The radius of the bottom of the cylinder is 2 cm and the height is 2 cm. Its side development is ()-shaped. The circumference of this figure is () cm and the area is () cm 2.

The ratio of 10 to: 8 is (). If you use it to write a ratio, this ratio can be ().

1 1. It is known that the relationship between A, B and C is A ÷ B = C, and if A is certain, B is directly proportional to C (). If c is certain, a and b are proportional ().

12, the number of students who passed the sixth-grade math contest accounted for the number of students who failed, and the passing rate of sixth-grade students in this competition was ().

13, the sum of the minuend, subtraction and difference is 40, the ratio of subtraction and difference is 3:2, the minuend is (), and the subtraction is ().

14, a kind of brine, is made by mixing salt and water at the ratio of 1: 100. Now, to prepare 8008 grams of this brine, 1 kg salt is needed.

15, the two external terms of a ratio are 5 and 6 respectively, and their ratio is 3, which is ().

16. On the map of China with the scale of 1: 400000, the distance between the two places is 30 cm, and the actual distance is () km.

17, () The statistical chart can not only show the quantity, but also clearly show the change of the quantity.

18, a certain length of iron wire is divided into several sections on average, and the length of each section is proportional to the number of sections.

Second, choose.

1, the following ratio, can be proportional to the ().

①3:4 ②4:3 ③ : ④ :3

2. Put 1 g salt into 100 g water, and the ratio of salt to brine is ().

① 100: 10 1 ② 1: 10 1 ③ 1:99 ④ 1: 100

3. In proportion, the product of two external terms is certain, and the two internal terms become ().

① Positive proportion ② Inverse proportion ③ Out of proportion ④ It is impossible to judge.

4. The existing three numbers 9, 3 and () can form a proportion.

① ② ④4 ④2

5. Solution ratio =2: 1, χ = ().

①6 ② 1.5 ③0.7 ④9

6, two reciprocal, they must be ().

① Positive proportion ② Inverse proportion ③ Out of proportion ④ It is impossible to judge.

7. Xiao Wang's height and weight are ().

① Positive proportion ② Inverse proportion ③ Out of proportion ④ It is impossible to judge.

8. The radius of the small circle is 2 cm, the radius of the big circle is 3 cm, and the ratio of the area of the small circle to the big circle is ().

①2:3 ②3:2 ③4:9 ④9:4

9, a project, the ratio of completed to this project is 3:5, the rest () for this project.

①60% ②40% ③20% ④ 166.6%

10, the distance on the drawing is 3cm, the actual distance is 1.5mm, and the scale is ().

① 1:20 ② 1:2 ③ 1:200 ④20: 1

1 1, the class size is certain, and the number of attendees and absentees is ().

① Positive proportion ② Inverse proportion ③ Out of proportion ④ It is impossible to judge.

12, a cylinder, if the height remains the same, the bottom area will be expanded by 3 times, and its volume will be expanded ().

①3 times, ②6 times, ③9 times, ④27 times.

Third, the judge.

1. The total amount of money subscribed to Little Torch is directly proportional to the number of copies ordered. ( )

2. Legends should be used to make composite bar charts. ( )

3. The scale is 1 cm on the map, which means the actual distance is 20km. ( )

There are 55 students in the class, the ratio of male to female is 5:6, so there are 30 boys in this class. ( )。

5. The lateral area of two cylinders is equal, and their bottom perimeters are equal. ( )

6. In the proportion, the product of two external terms is 10, one internal term is 5, and the other internal term is also 5. ( )

7 =B, then a and b are inversely proportional. ( )

8. The circumference and diameter of a circle must be in direct proportion. ( )

Fourth, the solution ratio.

= : = :χ =

40:χ=2.5: 15 :χ=5: 16 : =20:χ

Five, according to the following data, calculate the pass rate of each class to participate in the competition, and then make statistics.

Six (1) classes 16 students, 12 students passed; 6 (2) class 15 students, 13 students passed;

Class 6 (3) 1 1 person, 8 people passed.