1. Representation of position: a (column, row) For example, a (3,4) means that point A is in the third column and the fourth row.
Generally, look at the horizontal figures first, and then look at the vertical figures. Notice that there is a comma in the middle.
2. The meaning of fractional multiplication: a number × a fraction.
Fraction × a number
3. Two numbers whose product is 1 are reciprocal. The reciprocal of 1 is 1 0 without reciprocal.
4. dividing by a number that is not equal to 0 is equal to multiplying the reciprocal of this number.
Division of two numbers is also called the ratio of two numbers. The ratio is usually expressed in fractions, and can also be expressed in fractions or integers.
6. The basic nature of the ratio: the first term and the second term of the ratio are multiplied or divided by the same number at the same time (except 0), and the ratio remains unchanged.
7. The ratio of the circumference to the diameter of a circle is called pi, which is expressed by Wu, and Wu ≈ 3.438+04.
8. About the formula of circle:
C= d = 2 r S = r 2。
d = c÷d = 2 r r = d÷2 r = c÷2
The area of the ring S =σR2-σR2.
9. Original price × discount = current turnover × tax rate = tax payable × principal × interest rate × time = interest.
10. Bar chart: How much data can be seen clearly.
Broken line statistical chart: you can clearly see the increasing and decreasing trend of data.
Department statistical chart: you can clearly see the relationship between each part and the whole.
Knowledge points in the second volume of sixth grade mathematics
I. Proportion
1, the basic property of proportion is that the product of two internal terms is equal to the product of two external terms in proportion.
2.x and Y are used to represent two related quantities, and K is used to represent their ratio (certain), so the positive proportional relationship is expressed as:
Y: x = k (ok)
3.x and Y are used to represent two related quantities, and K is used to represent their product (certain), so the inverse relationship is expressed as:
Xy=k (ok)
Second, number and algebra (review)
1, natural numbers and 0 are integers.
2. Natural numbers: When we count objects, 1, 2, 3 ... are called natural numbers. There is no object, which is represented by 0. 0 is also a natural number.
3. Counting units: one, ten, hundred, thousand, ten thousand, one hundred thousand, one million, ten million, one hundred million ... are all counting units.
The propulsion rate between every two adjacent counting units is 10. This counting method is called decimal counting method.
4. Digits: Counting units are arranged in a certain order, and their positions are called digits.
5. Number divisibility: The integer A is divisible by the integer B (b ≠ 0), and the divisible quotient is an integer with no remainder, so we say that A is divisible by B, or that B is divisible by A. ..
6. Multiplies and factors: If the number A is divisible by the number B (b ≠ 0), then A is called a multiple of B and B is called a factor of A. Multiplication and factors are interdependent. Because 35 is divisible by 7, 35 is a multiple of 7 and 7 is a factor of 35.
7. The number of factors of a number is limited, of which the smallest factor is 1, and the factor of a number is itself. For example, the factors of 10 are 1, 2,5, 10, where the smallest factor is 1 and the factor is 10.
8. The number of multiples of a number is infinite, and the smallest multiple is itself. The multiple of 3 is: 3, 6, 9, … The minimum multiple is 3, and there is no multiple.
9. Numbers divisible by 2 are called even numbers. Numbers that are not divisible by 2 are called odd numbers. 0 is also an even number. Natural numbers can be divided into odd and even numbers according to their divisibility by 2.
10, a number, if only 1 and its own two factors, such a number is called a prime number (or prime number), and the prime numbers within 100 are: 2, 3, 5, 7, 1 1,/kloc.
1 1, a number, if there are other factors besides 1 and itself, such a number is called a composite number, such as 4, 6, 8, 9 and 12.
12 and 1 are neither prime nor composite, and natural numbers except 1 are either prime or composite. If natural numbers are classified according to the number of their factors, they can be divided into prime numbers, composite numbers and 1.
13, each composite number can be written as the product of several prime numbers. Every prime number is a factor of this composite number, which is called the prime factor of this composite number. For example, 15=3×5, and 3 and 5 are called prime factors of 15.
14, the common factor of several numbers is called the common factor of these numbers. One of them is called the common factor of these numbers. For example, the factor of 12 is 1, 2,3,4,6,12; The factors of 18 are 1, 2,3,6,9 and 18. Where 1, 2, 3, 6 is the common factor of 12 and 1 8, and 6 is their common factor.
Two numbers whose common factor is 15 and only 1 are called prime numbers. There are the following situations:
16. If the smaller number is a factor of the larger number, then the smaller number is the common factor of the two numbers.
17. If two numbers are prime numbers, their common factor is 1.
18, the common multiple of several numbers is called the common multiple of these numbers, and the smallest is called the minimum common multiple of these numbers. For example, the multiple of 2 is 2, 4, 6, 8, 10, 12, 14, 16, etc.
The multiple of 3 is 3,6,9, 12, 15, 18 ... where 6, 12, 18 ... are the common multiples of 2 and 3, and 6 is their least common multiple. .
19. If the larger number is a multiple of the smaller number, the larger number is the least common multiple of the two numbers. If two numbers are prime numbers, then the product of these two numbers is their least common multiple.
20. The number of common factors of several numbers is limited, while the number of common multiples of several numbers is infinite.
(2) Decimals
1, meaning of decimals: Decimals, percentiles and thousandths obtained by dividing the integer 1 into 10, 100, 1000 ... can be expressed in decimals.
One decimal place indicates a few tenths, two decimal places indicate a few percent, and three decimal places indicate a few thousandths. ...
2. Decimal system consists of integer part, decimal part and decimal part. The point in a number is called a decimal point, the number to the left of the decimal point is an integer part, and the number to the right of the decimal point is called a decimal part.
3. In decimals, the series between every two adjacent counting units is 10. The propulsion rate between the decimal unit "one tenth" of the decimal part and the lowest unit "one" of the integer part is also 10.
(3) scores
1, meaning of score: divide the unit "1" into several parts on average, and the number representing such one or several parts is called a score. In the score, the middle horizontal line is called the dividing line; The number below the fractional line is called the denominator, indicating how many copies the unit "1" is divided into on average; The number below the fractional line is called the numerator, indicating how many copies there are.
2. Divide the unit "1" into several parts on average, and the number representing one part is called fractional unit.
3. Classification of scores
True fraction: The fraction with numerator less than denominator is called true fraction. The true score is less than 1. 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 fraction: False fraction can be written as a number consisting of integer and true fraction, which is usually called with fraction.
4. Approximation: It is called approximation to change a fraction into a fraction equal to it, but with smaller numerator and denominator.
5. The fraction whose numerator and denominator are prime numbers is called simplest fraction.
6. Changing scores of different denominators into scores of the same denominator is equal to the original score, which is called the total score.
(4) Approximate points and general points
1, reduction method: divide the denominator by the common factor of the denominator (except 1); Usually, we have to separate it until we get the simplest score.
2. General fractional method: first find the least common multiple of the denominator of the original fraction, and then turn each fraction into a fraction with this least common multiple as the denominator.
Three properties and laws
1, Quotient Invariance Law: Quotient Invariance Law: In division, the dividend and divisor expand or shrink by the same multiple at the same time, and the quotient remains unchanged.
2. Properties of decimals: Add or remove zero at the end of decimals, and the size of decimals remains unchanged.
3. The movement of decimal position causes the change of decimal size.
(1) If the decimal point moves one place to the right, the original number will be expanded by 10 times; If the decimal point is moved two places to the right, the original number will be expanded by 100 times; If the decimal point is moved three places to the right, the original number will be enlarged by 1000 times. ...
(2) The decimal point is shifted to the left by one place, and the original number is reduced by 10 times; If the decimal point is moved two places to the left, the original number will be reduced by 100 times; If the decimal point is moved three places to the left, the original number will be reduced by 1000 times. ...
(3) When the decimal point moves left or right, the number should be supplemented with "0".
(5) the basic nature of the score
The basic nature of a fraction: both the numerator and denominator of the fraction are multiplied or divided by the same number (except zero), and the size of the fraction remains unchanged.
(6) the relationship between fraction and division
1. divider/divider = divider/divider
2. Because zero can't be divisible, the denominator of the fraction can't be zero.
3. Divider is equivalent to numerator and divisor is equivalent to denominator.
The significance of four operations
Integer operation
Appendix+Appendix = Sum
One addend = and-the other addend.
Negative-negative = difference
Negative = negative+difference
Subtraction = minuend-difference
Coefficient × coefficient = product
One factor = product ÷ another factor
Dividend = quotient
Divider = Divider
Dividend = quotient × divisor
(2) Operation law
1. additive commutative law: When two numbers are added, the positions of addends are exchanged, and the sum is unchanged, that is, A+B = B+A.
2. The law of addition and association: when three numbers are added, the first two numbers are added first, and then the third number is added; Or add the last two numbers first, and then add the first number, and their sum remains the same, that is, (a+b)+c=a+(b+c).
3. Multiplicative commutative law;
When two numbers are multiplied, the position of the exchange factor remains unchanged, that is, a× b = b× a.
4. Multiplication and association law: multiply three numbers, first multiply the first two numbers and then multiply the third number; Or multiply the last two numbers first, and then multiply them with the first number, and their products are unchanged, that is, (a×b)×c=a×(b×c).
5. Multiplicative distribution law:
When the sum of two numbers is multiplied by a number, you can multiply the two addends by this number, and then add the two products, that is, (a+b) × c = a× c+b× c.
6. The essence of subtraction:
If you subtract several numbers from a number continuously, you can subtract the sum of all subtractions from this number, and the difference remains unchanged, that is, a-b-c=a-(b+c).
(3) Algorithm
1. integer addition calculation rules:
The same numbers are aligned from the low order. When the numbers add up to ten, they will advance to the previous number.
2. The integer subtraction calculation rules:
The same numbers are aligned from the lower number. If the number of digits in the digit is not enough, step back from the previous digit to make ten, combine it with the number of digits in the standard, and then subtract it.
3. The integer multiplication calculation rules:
First, multiply the number on each bit of one factor by the number on each bit of another factor, then multiply the number on which bit of the factor, align the end of the multiplied number with which bit, and then add the multiplied numbers.
4. The integer division calculation rules:
Divide from the high order of the dividend, which is a few digits, depending on the first few digits of the dividend; If the division is not enough, look at another place and the quotient is written on the dividend. If any number is not quotient 1, a "0" placeholder should be added. The remainder of each division should be less than the divisor.
5. Decimal multiplication rules:
First, calculate the product according to the calculation rules of integer multiplication, and then look at the factor * * *, how many decimal places there are, just count a few from the right side of the product and point to the decimal point; If the number of digits is not enough, make up with "0".
6. The divisor is an integer fractional division calculation rule:
First of all, according to the law of integer division, the decimal point of quotient should be aligned with the decimal point of dividend; If there is a remainder at the end of the dividend, add "0" after the remainder to continue the division.
7. The divisor is a decimal division calculation rule:
First, move the decimal point of the divisor to make it an integer, then move the decimal point of the divisor to the right by several digits (the number of digits is not enough, make up "0"), and then calculate according to the division rule that the divisor is an integer.
8. The same denominator fraction addition and subtraction calculation method:
Add and subtract fractions with denominator, only add and subtract numerators, and the denominator remains the same.
9. Different denominator fractions addition and subtraction calculation method:
Divide first, and then calculate according to the addition and subtraction law of fractions with the same denominator.
10. Calculation method of fractional addition and subtraction: add and subtract the integer part and the decimal part respectively, and then combine the obtained numbers.
Correction/wholeness
(A) the significance and rules of decimal multiplication and division
1. Importance of decimal multiplication:
Decimal multiplication has the same meaning as integer multiplication, and it is a simple operation to find the sum of several identical addends. Example: 3.5×4 represents the sum of four 3.5. Or 4 times 3.5.
The meaning of multiplying a number by a decimal number is different from that of integer multiplication. Is to find a few tenths, a few percent, and a few thousandths of this number. Example: 25×0. 17, which means what is 17% of 25.
2. The significance of fractional division
Fractional division has the same meaning as integer division. It is an operation to find the other factor by knowing the product of two factors and one of them. Example: It means that the product of two known factors is 0.75, and one factor is 0.5, so what is the other factor? Or 0.75 is a multiple of 0.5.
(2) Calculation rules of decimal multiplication and division method
1. decimal multiplication rule:
(1) is first calculated according to the law of integer multiplication;
(2) Look at a factor * * *, how many decimal places there are, just count the decimal places from the right side of the product and point to the decimal point.
2. The law of fractional division:
(1) is first removed according to the law of integer division;
(2) The decimal point of quotient is aligned with the decimal point of dividend;
(3) There is a remainder at the end of the dividend, and 0 is added after the remainder to continue the division.
Second, weights and measures
Length unit conversion
1 km = 1 000m1m = 10 decimeter.
1 decimeter =10cm1m =10cm.
1 cm = 10/0mm
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.
Basic knowledge of algebra
First, use letters to represent numbers.
1 Use letters to indicate the meaning and function of numbers.
Use letters to express common quantitative relations, operation rules and properties, and calculation formulas of geometric shapes.
(1) Common quantitative relations
The distance is represented by s and the speed by t, and the relationship between them is as follows:
t=s/v
The total price is represented by A, the unit price is represented by B, and the quantity is represented by C. The relationship between them is as follows:
a=bc b=a/c c=a/b
(2) Operating rules and characteristics
Additive commutative law: A+B = B+A.
Additive associative law: (a+b)+c=a+(b+c)
Multiplicative commutative law: ab=ba
Law of multiplicative association: (ab)c=a(bc)
Multiplication and distribution law: (a+b)c=ac+bc
The essence of subtraction: A-(B+C) = A-B-C.
(3) Formulas for expressing geometric shapes with letters.
The length of a rectangle is represented by a, the width by b, the circumference by c and the area by s. c=2(a+b) s=ab
The side length a of a square is denoted by, the perimeter is denoted by c, the area is denoted by s, and c=4a s=a2.
The base A of the parallelogram is represented by, the height is represented by H, the area is represented by S, and s=ah.
The base of a triangle is represented by a, the height by h and the area by s.
s=ah/2
The upper base of the trapezoid is represented by a, the lower base by h, the height by h, and s=(a+b)h/2.
Calculation formula of mathematical graphics in primary schools
1, square c perimeter s area a side length perimeter = side length× 4c = 4a area = side length× side length s = a× a.
2. Cube V: volume A: side surface area = side length × side length× 6s table =a×a×6 volume = side length× side length× side length V = a× a× a.
3. rectangular
Perimeter area side length
Circumference = (length+width) ×2
C=2(a+b)
Area = length × width
S=ab
4. Cuboid
V: volume s: area a: length b: width h: height.
(1) Surface area (L× W+L× H+W× H) ×2
S=2(ab+ah+bh)
(2) Volume = length × width × height
V=abh
5 triangle
S area a bottom h height
Area = bottom × height ÷2
s=ah÷2
Height of triangle = area ×2÷ base.
Triangle base = area ×2÷ height
6 parallelogram
S area a bottom h height
Area = bottom × height
S = ah
7 trapezoid
Height of upper bottom b and lower bottom h in s area a
Area = (upper bottom+lower bottom) × height ÷2
s=(a+b)× h÷2
8 laps
Area c perimeter d= diameter r= radius
(1) circumference = diameter ×∏=2×∏× radius
c =∏d = 2r
(2) area = radius × radius×∈
Cylinder 9
V: volume h: height s; Bottom area r: bottom radius c: bottom perimeter
(1) lateral area = bottom circumference × height.
(2) Surface area = lateral area+bottom area ×2
(3) Volume = bottom area × height
(4) Volume = lateral area ÷2× radius.
10 cone
V: volume h: height s; Bottom area r: bottom radius
Volume = bottom area × height ÷3
1 1, diameter = radius× 2d = 2r radius = diameter ÷2 r= d÷2.
12, circumference = π× diameter = π× radius× 2c = π d = 2π r.
13, area of circle = π× radius× radius.
(b) Application of scores and percentages
1, fractional addition and subtraction application problem: the application problem of fractional addition and subtraction is basically the same as the application problem of integer addition and subtraction in structure, quantitative relationship and problem solving method, except that there are scores in known numbers or unknowns.
2. Fractional multiplication application problem: refers to the application problem of finding the fraction of a given number.
Features: The quantity and fraction of the unit "1" are known, and the actual quantity corresponding to the fraction is found.
The key to solving the problem is to accurately judge the number of units "1". Find the score corresponding to the required question, and then formulate it correctly according to the meaning of multiplying a number by a score.
3. Application of fractional division:
(1) Find a number that is a fraction (or percentage) of another number.
Features: Knowing one number and another, find the fraction or percentage of one number. "One number" is a comparative quantity, and "another number" is a standard quantity. Find a fraction or percentage, that is, find their multiple relationship.
The key to solving the problem: start with the problem and find out who is regarded as the standard number, that is, who is regarded as "unit one" and who is the bonus compared with the number of unit one.
A is the fraction (percentage) of B: A is the comparative quantity and B is the standard quantity. Divide a by b ..
How much is A more (or less) than B (a few percent): A minus B is more (or less) or (a few percent) than B ... Relationship: (A minus B)/B or (A minus B)/A. ..
(2) Given the fraction (or percentage) of a number, find this number.
Features: Knowing an actual quantity and its corresponding fraction, find the quantity with the unit of "1".
The key to solving the problem: according to the meaning equation of fractional multiplication, or according to the meaning equation of fractional division, but we must find the known actual quantity corresponding to the fraction.
4. Percentage:
Germination rate = number of germinated seeds/number of experimental seeds × 100%
Wheat flour yield = flour weight/wheat weight × 100%.
Product qualification rate = number of qualified products/total number of products × 100%.
Employee attendance = actual attendance/attendance × 100%
5. Engineering problem: it is a special case of fractional application problem, which is closely related to the work of integers. It is an applied problem to explore the relationship among total workload, work efficiency and working hours.
The key to solving the problem: regard the total amount of work as the unit "1", and the work efficiency is the reciprocal of the working time, and then use the formula flexibly according to the specific situation of the topic.
Quantitative relationship: total work = work efficiency × work time.
Work efficiency = total workload ÷ working hours
Working hours = total amount of work ÷ working efficiency
Total workload ÷ work efficiency = cooperation time