General comment on the twelve volumes of mathematics
1. Numbers and numerical operations
Target requirements:
1. Make students further understand the meanings of natural numbers, integers, decimals and fractions, and be able to read and write integers, decimals and rewrite numbers correctly and skillfully.
2. Make students systematically master the related concepts of divisibility, further understand the meanings of divisibility, multiples, divisors, prime numbers, composite numbers, common divisors, common multiples and coprime numbers, understand and master the basic properties of fractions and decimals, and find the greatest common divisor and the smallest common divisor correctly and quickly.
3. Make students further understand the significance and laws of four operations of addition, subtraction, multiplication and division and elementary arithmetic order, flexibly choose reasonable calculation methods, and perform elementary arithmetic operations of integers, decimals and fractions correctly and skillfully.
4. Be able to understand the mathematical terms in the four operations, list comprehensive formulas to solve word problems, and further improve the calculation ability.
Class hours: 6-8 class hours.
teaching process
The Meaning, Reading and Writing of Numbers
First, review the meaning of numbers
1, natural number, integer.
1, 2, 3, … represents the number of objects called natural numbers. Natural numbers have two meanings: one is called cardinal number, which is used to represent the number of things. For example, the "8" in "8 trees" is the cardinal number; The second is the ordinal number used to indicate the order of things. For example, "10" in "Page 10" is the ordinal number.
There is no object, so it is represented by 0, and 0 is also a natural number. Both 0 and natural numbers are integers.
1, fractions and decimals
Divide the unit "1" into several parts on average, and the number representing such 1 or several parts is called a fraction. The number of copies representing 1 is the decimal unit of this fraction.
When people calculate and measure, they often can't get integer results, so they need to be expressed by decimals.
Divide the integer "1" into 10, 100, 1000 ... How many tenths, how many percent, how many thousandths are such a 1 0 ... Such a number can be expressed in decimals. Decimals such as 0. 1, 0.25, 0.001... are actually fractions with denominators of 10, 1000,1000 ... but they are written differently.
Relationship between Fraction and Division
When two natural numbers are divisible and cannot be divisible, their quotient can be expressed by a fraction. The numerator is equivalent to the dividend, the denominator is equivalent to the divisor, and the fractional line is equivalent to the divisor, that is, dividend/divisor =, so the denominator of the fraction cannot be zero.
Fraction is closely related to division, but there are also differences; Division is an operation and fractions are numbers.
Decimals with an integer part of 0 are called pure decimals, such as 0.24, 0.3, 0.2 16. Decimals whose integer part is not 0 are called decimals, such as 3. 14 and 4.2.
The decimal part of the decimal system, in which one or several numbers are repeated in turn from a certain number, is called a cyclic decimal. Cyclic decimals must meet two conditions: ① Infinite digits; (2) One or several numbers appear repeatedly, and the repeated numbers are called cyclic segments.
There are two kinds of cyclic decimals: ① The cyclic segment starting from the first place on the left of the decimal part is called pure cyclic decimal; (2) The cycle segment is not called mixed cycle decimal from the first place in the decimal part. For example, 4.37 is a pure cyclic decimal; 4.037 and 3. 12 are mixed cyclic decimals.
The classification of decimals can be shown in the following figure:
Endless decimal
Decimal infinite acyclic decimal
Infinite decimal pure cyclic decimal
Cyclic decimal mixed cyclic decimal
3. Numbers
(1) Counting unit
Integer and decimal numbers are numbers written in decimal notation. The size of a number is different in different positions. The counting units of integers are: one (one), ten, one hundred, one thousand, ten thousand, one hundred thousand, one million, ten million, ..., and the counting units of decimals are: one tenth, one hundredth, one thousandth, one thousandth, ...
(2) Decimal counting method
The propulsion rate between every two adjacent units is 10. This counting method is called decimal counting method.
(3) When counting numbers, the positions occupied by the numbers are called digits. These figures are arranged in a certain order. (See page 74 of the textbook for details)
(4) Number of digits For an integer, a number containing several digits is the number of digits. For example, 3 is a number, 32 is a number, and 348,070 is a number.
For decimals, a few digits in the decimal part are decimals. For example, 3. 17 is two decimal places, and 320.438+07 is also two decimal places.
4. The meaning and percentage of.
A number indicating that one number is a percentage of another number is called a percentage. Also called percentage or percentage.
Score is a common noun in industry, agriculture and daily life. In fact, it refers to the fraction whose denominator is 10, and a few percent is a few tenths. For example, 40% is four tenths, and if it is rewritten as a percentage, it is 40%.
5. What is the connection and difference between percentage and score?
Fractional percentage
Meaning can not only represent specific quantity, but also
To represent the multiple relationship between two numbers. It only represents the multiple relation of two quantities,
Does not mean the specific quantity.
Fractions can be followed by units of measurement,
There can also be no unit of measurement. Don't write the unit of measurement after the percentage.
General method of writing scores
Scores are generally simplified.
Fraction is not decimal. There is a special way to write them.
There is no need to simplify.
Molecules can be decimals.
Second, review the reading and writing methods of numbers
How to read (1) integers (see page 73 of the textbook)
(2) Writing of integers (see page 73 of the textbook)
(3) Decimal reading method: first read the integer part according to the integer reading method, and then directly read each bit of the decimal part.
(4) Decimal writing: first write the integer part according to the integer writing, then put the decimal point behind the integer part, and then write the number of the decimal part.
1, read the numbers below.
106000800 52000803 100 40030050080 1 200000005
0.00 16 80. 105 206.723
2. Write down the following figures.
902.503 billion 0.2305.208
40,800.36 20.005 163 75.24 1 1.
Rewriting numbers and divisors
(1) Rewrite the number to "10,000" or "100 million".
For a relatively large integer, in order to facilitate reading and writing, it can often be rewritten as a number in units of "10,000" or "100 million". The specific method is:
(1) Rewrite a number into a number in units of "ten thousand". Move the decimal point of the number four places to the left, and then add the word "10 thousand" after it. For example, 43000 = 43000.
(2) Rewrite a number into a number with "100 million" as the unit. Move the decimal point of the number eight places to the left, and then add the word "100 million" after it. Such as 576 million = 576 million. Note: Rewriting should get an accurate value, so use the equal sign.
False fractions and band fractions or integers can also be rewritten with each other.
For example, 2 =-, =( 30), =(25)
(2) Several methods of approximate value:
(1) rounding method: Look at the number after the one to be reserved. If the number of digits of this number is greater than or equal to 5, remove this number and all the digits after it, and then move forward 1 to get the required divisor; If the number after the number to be retained is less than or equal to 4, the number and all the numbers after it are removed to obtain the required divisor.
Example: Find the divisor of the following figures.
3.54963≈3.5 (retained to the tenth place) 3.54963≈3.55 (retained to the hundredth place)
3.54963≈3.550 (reserved to thousands) Note, why can't the 0 at the end of 3.550 be removed?
(2) Tailing method
As needed, no matter how many digits are left, remove them. This method of taking divisor is called "tail-cutting method"
(3) one-step method
According to the actual needs, no matter how many places are reserved, we should move forward one place. This approximate method is called one-step method.
(3) Conversion between decimals, fractions and percentages
Examples of mutual assistance methods
The number of decimal parts used to have several decimal places.
Just write a few zeros after 1.
As the denominator, the original decimal
Remove the decimal point as a molecule. Can be reduced to the simplest part of the quotation. 0. 19=
3.24=3 =3
Decimals are converted into percentages. The decimal point is shifted to the right by two places (the number of digits is not enough to be filled with 0), followed by hundreds of semicolons. 1.365= 136.5%.
0.4=40%
2=20%
The percentage is converted into decimal, the percent sign is removed, and the decimal point is moved to the left by two places (the number of digits is not enough to make up 0), 1% = 0.0 1.
150%= 1.5
Scores are converted into percentages. Fractions are converted to decimals first (in the case of infinite division, three decimal places are generally needed), and then converted to percentages. 1 ≈ 1.667.
= 166.7%
The percentage component number is rewritten as a fraction with the mother 100, which can simplify the quotation; If it is a false fraction or, it must be converted into a fraction or an integer. 80%=
125%=
A simplest fraction, if the denominator does not contain other prime factors except 2 and 5, this fraction can be reduced to a finite decimal; You can also enlarge or reduce the numerator and denominator of this fraction by the same multiple at the same time, and divide them into fractions with letters 10, 100,1000 ... and then write them directly as decimals.
For example: ÷25=0.28 or
A simplest fraction, if the denominator contains prime factors other than 2 and 5, can't be converted into a finite decimal, but can only be converted into an infinite circulating decimal, or an approximate value can be taken as needed.
For example: 4÷ 15=0.26≈0.267 (three decimal places are reserved).
Remember the following commonly used data, which is good for improving the operation speed.
=0.5 =0.25 =0.75 =0.2 =0.4 =0.6
=0.8 =0. 125 =0.375 =0.625 =0.875
=0.05
Comparison of figures
(1) integer size comparison
An integer with more digits is greater than an integer with less digits. If seven digits are greater than six digits.
(2) The number of digits is the same. Comparing from high to low, the number with the highest digit is larger.
Big; If the highest digits are the same, compare the second digit from the left, the second digit is larger, and so on.
(2) Decimal size comparison
Look at the integer part first (compared by integer size), and the decimal part with large integer part is larger; If the integer parts are the same, look at decimals, and the decimals with larger decimals are larger.
(3) Comparison of scores (see page 77 for details)
practise
fill (up) a vacancy
1, 530,456,070 Write ()
Rounding to ten thousand places is () ten thousand.
2. A number consists of 8 1, 6 0 1 and 7 0 0 1. This number is
(), rounded to ten places, is roughly equal to ().
3, 0.303, 0.33 and 0.3 are arranged from small to large.
()& lt()& lt()& lt( ).
4, 6.752 billion writing (), rounded off.
Billions of records ().
5, 0.245, 0.245, 0.245, 0.25, which is the largest?
(), the smallest number is ().
6, three thousand seven hundred and fifty-six writing (), rounded.
Ten thousand bits is about ().
7,809.205 million (), rewritten as
The number of ten thousand units is ().
8, 0.3, 0.33, 3.3% away from the big column. ()
9, 5.907 accurate to the percentile is ().
10, the smallest natural number is (), and the smallest integer is ().
1 1、36028=3×( )+6×( )+2×( )+8×( ).
12, the unit of natural number is (), and 48 is composed of such units as (). The two natural numbers adjacent to the largest two digits are () and () respectively.
13 and 0.027 are () one thousandth.
14 and 1 contain () 0. 1 and () 1%.
15, in 0.8, 30.9, 0, 1 00.0/,1,0.6, 6.362, 8.906, () is an integer, () is a cyclic decimal, and () is a pure cyclic decimal.
16, a number consists of 45000, 30 1 and 26 percent. This number is ().
17,210760000 The mantissa after the ellipsis billion is ().
18,90.3006 Writing ().
Six thousand one point zero zero two writing ().
19, where hundreds of millions are 1, tens of thousands are 8, hundreds are 6, and everyone else is 0. This number is (), pronounced ().
20. 4.206 consists of () 1, () 1/10 and 6.
().
2 1, the simple notation for writing down the business cycle decimal 1 1÷6 is (), and the three decimal places are reserved for about ().
22. If a two-digit decimal is rounded to approximate, you will get 0.2. The maximum value of this decimal is () and the minimum value is ().
23. The decimal number consisting of 10 decimals, 8 decimals, 9 decimals and 7 percentiles is (), rounded to the tenth place is ().
24, with 1, 0, 4, 8 can form the largest three digits is (), the smallest three digits is ().
25. Fill in 1.42, 1, 1.4 and 142% in the following brackets as required.
()& gt()& gt()& gt( )
26. Divide 3 kilograms of apples into 8 parts, each part belongs to this pile of apples, and each part weighs () kilograms.
27, a wire length 15 meters, after cutting, there are () meters left.
28, hours = () minutes.
29、 1 12÷( ) ≈( )%
30. The decimal unit of (m is a natural number) is (), and it has () such decimal units.
3 1, and the meter can be regarded as 5 meters-; It can also be regarded as 1 m-.
32. Among the three fractions, the one that cannot be converted into a finite fraction is (). If it is converted into a circular decimal, it can be simply recorded as () and three decimal places can be reserved as ().
33, meters long rope, the average is divided into three sections, each section () meters long, each section is full length.
34, the decimal unit of 1 is (), and the decimal unit like () is 2.
35.A number is 50, B number is 40, and B number is less than A number ()%.
36. Of the four numbers 1 .87, 187.6%,1and 1.87, the smallest is () and the largest is ().
37. The smallest prime number is smaller than the smallest composite number ()%, and the greatest common divisor of 4 and 5 is their least common multiple ()%.
The reciprocal of17 is (), and the reciprocal of 5 is ().
39, a simplest fraction, its numerator is expanded by 3 times, and the denominator is reduced by 2 times, which is equal to 4. The original score is ().
40. The fractional unit is the sum of all the simplest true fractions of ().
Second, right or wrong (tick right, tick wrong ×)
1. If the decimal point is 0.45, the number obtained is 100 times the original number. ()
2.0 is the smallest natural number. ()
3. All decimals are smaller than integers. ()
Xiaoming won the fourth place in the long jump. The number 4 here is not a natural number. ()
5.6.131313 is a cyclic decimal. ()
6. Integers less than 5 are only 1, 2, 3, 4. ()
7. Add 0 or delete 0 after the decimal point. The size of the decimal point remains the same. ()
8.π is a cyclic decimal. ()
9.2. 19 and 2. 19 are equal. ()
10.2.999 is rounded to two decimal places, and the approximate value is 3.00. ()
1 1. Divide the unit "1" into several parts, and the number representing such one or several parts is called a fraction. ()
12. The denominator of the false fraction is less than the numerator. ()
13. When the numerator and denominator are two adjacent natural numbers, the fraction is the simplest fraction. ()
14. There are countless scores greater than and less than. ()
/kloc-the counting unit of 0/5. 6.4 and 6.40 are the same. ()
16. Decimal is smaller than integer. ()
17. The percentages are all less than 1. ()
18. There are only one or two decimal places greater than 0.63 and less than 0.65. I don't know ().
19. The mantissa of an integer after omitting ten thousand digits is about 200000, and the maximum number is 199999. ()
20. 1% is equal to 10 per thousand. ()
2 1. If it is a false fraction, then the numerator must be greater than the denominator. ()
Third, multiple choice questions
1. The number "7" with a decimal of 2.507 is in the () position.
A. Decades C unit D
2. Move the decimal point one place to the right and then two places to the left. This number ().
A. expansion 100 times B. expansion 10 times C. reduction 10 times D. unchanged
3. Among the following figures, the one with the same size after removing 0 is ().
A.0.045 B.3.20 C.4.03 D
4. 1.59 The two decimal places reserved are ().
a . 2.00 b . 1.6 c . 1.60d . 1.59
5. The following figures are not equal to 0.75 ().
75% BC to 75% BC
6. With three 1 and three zeros, the number that reads two zeros is ().
a . 1 1 1000 b . 10 100 1 c . 1000 1 1d . 10 10 1
7. In the following figures, the first number is the second number and the divisor is ()
A.0.2 and 0.4 B. 0.3 and 0.6 C. 3 and 6 D. 10 and 5
8. Rounding 0.789 to one thousandth is ()
0.789 B. 0.780 C.0.7890 D. 0.790
9.7.131313 ... Yes ()
A. Pure cyclic decimal B. Mixed cyclic decimal C. Infinite cyclic decimal D. Finite decimal
10. Decimals greater than 3.7 and less than 3.75 are limited ()
A.5b.4c. Countless D. 10.
1 1. Among the three numbers of 0.57 1 and 57. 1%, the largest is ().
A.b.0.571c.57.1%d.cannot be determined.
12. Among the following three scores, the simplest score greater than and less than is ().
A. BC
13. Among the following fractions, finite () cannot be converted into finite decimals.
A.B. C. D。
Add 4 to the numerator of 14. and the denominator should be () to keep the score unchanged.
A. be good at using 3 b. be good at dividing 4 C by 4 d. score
15. Any _ _ _ _ _ number has a reciprocal. ()
A. The natural number is unclear B. Integer C. Decimal D. Fraction
16. Among the following figures, the largest one is ().
A.B.0.84 C.84%
17. A natural number divided by a true fraction, quotient _ _ _ _ dividend.
A. greater than B. less than C. equal to
Divisibility of numbers
1. concept
(1) is divisible (see page 80 of the textbook)
(2) Division: the number A is divided by the number B, the quotient of division is an integer or a finite decimal, and the remainder is 0, so we say that the number A can be divided by the number B. For example, 10÷4=2.5, that is to say, 10 can be evenly divided by 4.
Division can be divided into two situations according to the result: division and division. Division is a special case of division, which requires that two numbers must be natural numbers, the divisor cannot be 0, and the result must just get an integer. Division must be divisible, division must be divisible.
(3) divisor and multiple: Generally speaking, if A and B are natural numbers, and b≠0, A can be divisible by B, then A is a multiple of B, and B is the divisor of A. 。
The divisor of a number is finite, in which the smallest divisor is 1 and the largest divisor is itself. For example, the divisors of 12 are 1, 2, 3, 4, 6, 12, and the divisors often appear in pairs. Find the divisor of a number and put this.
The number of multiples of a number is infinite, and the smallest multiple is itself. For example, the multiple of 5 is 5, 10, 15, 20 ... The minimum multiple is 5.
(4) Common divisor and greatest common divisor
The common divisor of several numbers is called the common divisor of these numbers, and the largest is called the greatest common divisor of these numbers. For example, the common divisor of 12 and 18 is 1, 2,3,6, and the greatest common divisor is 6. The common divisor of all natural numbers is 1.
(5) common multiple, minimum 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. For example, the common multiples of 6 and 8 are 24, 48, 72, 96, ... and the least common multiple is 24. The number of common multiples of several numbers is infinite.
(6) Prime numbers and composite numbers
If a number has only 1 and its two divisors, it is called a prime number. If a number has other divisors besides 1 and itself, it is called a composite number. 1 is neither prime nor composite.
(7) Prime factor and decomposition prime factor: Each composite number can be written in the form of several prime numbers, which are called prime factors of these composite numbers.
For example, 24=2×2×2×3, and the prime factors of 2 and 3 are all 24.
Representing a composite number in the form of a good prime factor is called prime factor decomposition. Prime factors are usually decomposed by short division, and the divisor must be a prime number (usually starting from the smallest) until the final quotient is a prime number, and then the composite number is written in the form of a good prime number. For example, decompose 84 into prime factors.
2 84
2 42
3 2 1
seven
84=2×2×3×7
(8) Two numbers whose common divisor is only 1 are called prime numbers. For example, 4 and 5 are prime numbers and 8 and 9 are prime numbers.
Two prime numbers are not necessarily prime numbers, but can be a prime number and a composite number, or two composite numbers, or of course two prime numbers.
(9) Odd and even numbers that are divisible by 2 are called even numbers, and numbers that are not divisible by 2 are called odd numbers. For example, 2, 4, 6, 24, 324, … are all odd numbers, and 3, 5, 7, 9, 2 1, 532 1, … are all odd numbers.
2. The method of finding the greatest common divisor and the least common multiple
(1) To find the greatest common divisor and the least common multiple of two numbers, there are three basic situations. The differences are as follows:
Greatest common divisor least common multiple
The product of two numbers (7 and 9) 1.
7×9=63
Multiple relations
(6 and 18) decimal 6 big 8
It is neither a prime number nor a multiple relationship (12 and 18). Decomposition of prime factors by short division.
Connect all the divisors 2×3=6 and connect all the divisors and quotients.
2×3×2×3=36
2. Separability of numbers
(1) The characteristic of a number divisible by 2 is that there are 0, 2, 4, 6 and 8 divisible by 2. For example: 3160,248,964, 10726, … is divisible by 2.
(2) Numbers with characteristic bits of 0 or 5 can be divisible by 5. For example: 3160,450,75, ... can be divisible by 5.
(3) Characteristics of Numbers Divisible by 3 The sum of numbers on each digit can be divisible by 3, and this number can also be divisible by 3.
practise
Fill in the blanks
1, integers include () and (), and the smallest natural number is ().
The divisor of 2 and 24 is (), the largest of which is
(), the smallest is ().
3. Among the natural numbers of1~ 20, the largest odd number is () and the smallest even number is (); In odd numbers () is a composite number, and in even numbers () is a divisor.
4. The minimum composite number is () and the minimum prime number is ().
5. 16 and 15 are (), and their greatest common divisor is ().
6. The least common multiple of three prime numbers is 42. These three prime numbers are (), () and () respectively.
7. Fill in () in the box of 74. This number is divisible by 2 and 3. Fill in () in the box of 969. This number is divisible by 5 and 3.
8. The prime factor of decomposition 30 is 30= ()
9, a true score, its denominator is the product of the smallest odd number and the smallest composite number, the largest true score is ().
The minimum common multiple of10,32,36 is (), and the maximum common divisor is ().
1 1, and the smallest three digits divisible by 2, 3 and 5 at the same time is ().
12, the highest digit of a nine-digit number is the smallest composite number, the smallest prime number on 10 million digits, the smallest odd number on 100 digits, and all other digits are 0. This number is written as () and rewritten as a number in tens of thousands.
13, the greatest common divisor of three prime numbers is 1, and the least common multiple is 105. These three numbers are ().
14, if 33, 27 and 2 1 are divisible by the same number and the remainder is 3, then the divisor is at most ().
15, the largest four-digit number consisting of 0, 1, 5 and 3 that can be divisible by 2, 5 and 3 at the same time is ().
The greatest common divisor of 16, 12, 18 and 24 is ().
17, write a number () divisible by 3 but not by 3.
18, number A =2×2×3×5, number B =2×3×7, the greatest common divisor of number a and number b is (), and the least common multiple is ().
19. 1, 2, 4, 5, 9 Among these numbers, odd numbers have (), even numbers have (), prime numbers have (), and composite numbers have ().
20. Prime numbers have only () divisors, and composite numbers have at least () divisors.
2 1, the sum of three consecutive odd numbers is 33, and these three consecutive odd numbers are () ().
The least common multiple of 22, 12 and 24 is (), and the prime factor to decompose this number is ().
23. The largest five digits divisible by 2 () and the smallest five digits divisible by 3 are ().
24, can be divisible by 2, 3, 5 at the same time the largest three digits is ().
Second, the judgment question
1, 12÷4=3, 12 is a multiple, and 4 is a divisor. ( )
2. All numbers divisible by 7 are composite numbers. ( )
3. All prime numbers except 2 are odd numbers. ( )
4. Two adjacent natural numbers must be prime numbers. ( )
5. Prime numbers are odd numbers and even numbers are composite numbers. ( )
6. Natural numbers are either prime numbers or composite numbers. ( )
7. Because 4.8÷0.8=6, 4.8 can be divisible by 0.8. ()
8. 10 is divisible by 4. ( )
9. The sum of all prime numbers within10 is 17. ( )
10, because 2 and 5 are prime numbers, so 2 and 5 have no common divisor. ( )
Third, multiple choice questions
The divisor of 1 and 30 is ()
A 5 B 7 C 6 D 8
2. In the following three groups of numbers,-is a prime number. ( )
A 15 and 30 B 13 and 52 C 29 and 30 D 4 and 10.
3. The prime factor of decomposition 24 is ()
A 24= 1×2×2×3×2 B 24=3×8
C 24=2×2×2×3 D 24= 12×2
4,6 is divisible by A, so the minimum value of A is ()
A 12 B 6 C 1 D 2
5, with 0, 3, 4, 5, four digits can be evenly divided by-().
A 2 B 3 C 5 D 9
6.x is a natural number, and the following three statements are incorrect ().
A x must be an integer, B x is either odd or even, and C x is either prime or composite.
7. The sum of prime factors of natural number 23 1 is ()
A 20 B 2 1 C 22 D 40
8, the following statement is correct ()
Even numbers are composite numbers. B 200 1 is a leap year.
C, month, day and the prime factors of a number are all prime numbers, and D and odd numbers are all prime numbers.
9. If the least common multiple of A and B is ab, then A and B are ().
A prime number b composite number c prime number d multiple
Fourth, the greatest common divisor and the least common multiple of the following numbers
(1) 16 and 48 (2) 13 and 52 (3) 5 and 13.
8, 16 and 24 (5)2, 3 and 4 (6)30, 36 and 48
Basic properties of fractions and decimals