(1) natural number: When we count objects, 0, 1, 2, 3, ... are all called natural numbers. 1 is the numeration unit of natural numbers. Natural numbers can represent both the number (cardinal number) and the order (ordinal number) of things. For example, "7" in "7 days a week" stands for cardinal number, and "5" and "3" in "May 3" stand for ordinal number. There is no object, so it is represented by 0. 0 is the smallest natural number.

(2) Integers and natural numbers: natural numbers are all integers, but they are only part of integers (integers also include negative integers). The smallest number is 1 instead of 0.

The function of 0: ① it occupies a position in the number, indicating that there is no unit in this position; ② indicates the starting point; ③ indicates the boundary. Such as a thermometer, 0 on the number axis indicates the dividing line between positive and negative numbers.

(3) Score: Divide the unit "1" into several shares on average, and the number representing such one or several shares is called a score. The number representing one of them is a fractional unit.

The relationship between fraction and division: fraction is a number and division is an operation. They are two different concepts, but they are also closely related. For example:

(4) Decimal: Divide the integer "1" into 10, 100, 1000 ... These parts are tenths, hundredths and thousandths ... which can be expressed in decimal.

Classification of decimals:

(5) Number of digits, number of digits and counting unit: the position occupied by each counting unit is called number of digits. How many digits a natural number contains is called digits. Integers and decimals are numbers written in decimal notation, where 1, 10, 100,110,100 are counting units.

(6) Order table of integers and decimal places:

(7) Percentage, percentage and discount:

(1) percentage: A number indicating that one number is a percentage of another number is called a percentage. Percentage is also called percentage or percentage.

② Inheritance: a term commonly used in agriculture. A few percent is a few tenths.

Discount: a common term in business. A few folds is a few tenths.

Note: Percentage, percentage and discount only represent the multiple relationship between two numbers, and the score can be a specific quantity other than the multiple relationship.

2. How to read and write numbers

The reading method of (1) integer: from high to low, read step by step, the zero at the end of each stage is not read, and only one zero is read continuously for other digits.

(2) Writing of integers: from high to low, writing step by step. If there is no unit on any number, write 0 on that number.

(3) Decimal reading and writing: the integer part is read (written) as an integer, the decimal part is read as a point, and the decimal part reads (writes) the number on each bit in turn.

3. Number of rewrites

Rewriting and ellipsis of (1) multi-digits: In order to facilitate reading and writing, we often write a large multi-digit as a number with the unit of "10000" or "1000000", and first find10000 or/kloc-0. Then put a decimal point in the lower right corner of the number 10000 or 1000000, and write it as "1000000 ellipsis", and the result is "≈";

(2) Conversion of Fractions, Decimals and Percentages:

(3) A simplest fraction, if the denominator contains prime factors other than 2 and 5, then this fraction cannot be reduced to a finite decimal.

4. Comparison of figures

Comparison of (1) integers: look at the number of digits first, and the number with more digits is larger; The digits are the same. Starting from the highest digit, the number with the largest number in the same digit is the largest.

(2) Comparison of decimals: First, compare the integer parts of two numbers, and the one with the larger integer part is larger; The same is true for integer parts, and then look at their decimal parts. Starting from the high position, according to the number comparison, the number with the largest number on the same number is the largest.

(3) Score comparison: the score with the same denominator is greater than the score with larger numerator; Scores with the same numerator are larger with smaller denominator. Fractions with different denominators should be divided first and then compared.

Divisibility in the second quarter and the basic properties of fractions and decimals

Key points of knowledge

1, divisible

Meaning of (1) divisibility: When primary school talks about "divisibility of numbers", numbers generally refer to non-zero natural numbers.

When the number A is divided by the number B, the quotient is exactly an integer with no remainder, so we say that A is divisible by B, or B is divisible by A. ..

(2) divisor and multiple: If A is divisible by B, A is called a multiple of B, and B is called a divisor of A. ..

The divisor of a number is finite, in which the smallest divisor is 1 and the largest divisor is itself.

The number of multiples of a number is infinite, the smallest is itself, and it has no maximum multiple.

(3) Odd and even numbers: Numbers divisible by 2 are called even numbers, because 0 can also be divisible by 2, so the smallest even number is 0; Numbers that are not divisible by 2 are called odd numbers, and the smallest odd number is 1.

(4) Characteristics of numbers divisible by 2,3,5:

① Numbers divisible by 2: The units are 0, 2, 4, 6, 8.

2 Numbers divisible by 3: The sum of the numbers in each bit is divisible by 3.

③ The number in a unit that can be divisible by 5: 0 or 5.

(5) Prime number and composite number: A number is called a prime number if it only has 1 and its own two divisors; A number is called a composite number if there are other divisors besides 1 and itself. 1 is neither prime nor composite. The smallest prime number is 2 and the smallest composite number is 4.

(6) prime factor decomposition: each composite number can be written as the multiplication of several prime numbers, which is called the prime factor of this composite number. A composite number is expressed by multiplying several prime factors, which is called prime factor decomposition. Usually we use short division to decompose prime factors.

(7) Common divisor and greatest common divisor: The common divisor of several numbers is called the common divisor of these numbers. The largest one is called the greatest common divisor of these numbers.

(8) Prime number: The common divisor is only 1, which is called a prime number.

(9) Common multiple and minimum common multiple: The common multiple of several numbers is called the common multiple of these numbers. The smallest one is called the least common multiple of these numbers.

(10) The method of finding the greatest common divisor and the least common multiple: short division is generally adopted. If a large number of two numbers is a multiple of a decimal and a decimal is a divisor of a large number, then this large number is their least common multiple and this decimal is their greatest common divisor. If two numbers are prime numbers, their greatest common divisor is 1, and their least common multiple is the product of the multiplication of two numbers.

2. The basic properties of fractions and decimals

The basic property of (1) fraction: the numerator and denominator of the fraction are multiplied or divided by the same number (except zero) at the same time, and the size of the fraction remains unchanged.

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

(3) The size of decimal point changes with the movement of decimal point position: the decimal point moves one, two and three places to the right ... the original number is enlarged 10 times, 1000 times,1000 times ... on the contrary, the decimal point moves one, two and three places to the left ... the original number is reduced/kloc-0.

Operation of numbering in the third quarter

Key points of knowledge

Significance and law of 1 and four operations

The significance of (1) four operations:

digital

classify

Integer decimal part of operation name

The operation of combining two numbers into one number. It means the same as integer addition. It means the same as integer addition.

Subtraction is the operation of finding the sum of two numbers and one addend and the other addend. It has the same meaning as integer subtraction. It has the same meaning as integer subtraction.

A simple operation to find the sum of several identical addends by multiplication. Decimal multiplication by integer has the same meaning as integer multiplication.

Multiplying a number by a decimal is to find a few tenths, a few percent and a few thousandths of this number. Fractional multiplication of integers has the same meaning as integer multiplication.

Multiplying a number by a fraction is to find a fraction of this number.

The operation of finding another factor by knowing the product of two factors and one of them. It means the same as integer division. It means the same as integer division.

(2) Four algorithms:

1) addition and subtraction:

Add and subtract in the same unit, the unit remains the same, and the number of units is added and subtracted.

Integer fraction

1. The same number is aligned;

2. Start counting from the low position;

3. If you add dozens, you will advance to one place; When subtraction is not enough, retreat from the previous one, several times to dozens of times. 1. Same digit alignment (decimal point alignment);

2. Start counting from the low position;

3. Calculate by integer addition and subtraction;

4. The decimal point in the result is aligned with the decimal point in the addition and subtraction. 1. Fractions with the same denominator are added and subtracted, the denominator remains the same, and the numerator is added and subtracted.

2. Addition and subtraction of scores with different denominators, division first, and then calculation.

3. The bidding score that can be reduced as a result is that the false score is converted into a score.

(2) Multiplication and division:

Diverse

Standard integer fraction

1. Starting from the unit, multiply the first factor by the number in each bit of the second factor in turn.

2. Multiply the number on which bit of the second factor, and the last bit of the number will be aligned with which bit of the second factor.

3. Add up the multiplied numbers again. 1. First, integrate according to the integer multiplication rule.

2. Look at a factor * * *, how many decimal places there are, and count the decimal points from the right side of the product. 1. Fractions are multiplied by fractions, the product of numerator multiplication is numerator, and the product of denominator multiplication is denominator.

2. If there is an integer, treat it as a false fraction with the denominator of 1.

3. If there is a score, generally turn the score into a fake score first.

Except for ...

Dividing by law is the division of integers: from the high order of the dividend, look at the first few digits of the dividend first, and if it is not enough, look at one digit. Except which one, which one do you want to write the business on? The decimal point of quotient is aligned with the decimal point of dividend. Divider is the division of decimals: first move the decimal point of the divisor to make it an integer. The decimal point of the divisor is shifted to the right by several digits, and the decimal point of the dividend is also shifted to the right by the same digits ("the digits are not enough to add 0"), and then the calculation is carried out according to the division in which the divisor is an integer. The number A divided by the number B (except 0) is equal to the reciprocal of the number A multiplied by the number B.

(3) the relationship between the four operating parts:

2, operating rules and simple operation

(1) algorithm:

① additive commutative law a+b = b+a.

② additive associative law (a+b)+c=a+(b+c)

③ Multiplicative commutative law a×b=b×a

④ Multiplicative combination rate a×b×c=a×(b×c)

⑤ Multiplication distribution law a×(b+c)=a×b+a×c

(2) Business nature:

① the operational nature of subtraction a-(b+c) = a-b-ca-(b-c) = a-b+c.

② The nature of division operation A ÷ (b× c) = A ÷ B ÷ C A ÷ (b ÷ C) = A ÷ B× C c.

(a+b)÷c=a÷c+b÷c (a-b)÷c=a÷b-b÷c

3, the order of the four operations

These four operations are divided into two levels. Addition and subtraction are called primary operations, and multiplication and division are called secondary operations. Operation order: in an expression without brackets, if it only contains the same level of operation, it should be calculated from left to right in turn; If there are two levels of operation, do the second level operation first, and then do the first level operation.

In the formula with brackets, the contents in brackets are calculated first, and then the contents outside brackets are calculated.

The second chapter is the basic knowledge of algebra.

The first section simple equation

Key points of knowledge

1, alphanumeric

The letters (1) can represent natural numbers, integers, decimals and percentages. ...

(2) Mathematical concepts, arithmetic rules and mathematical calculation formulas can be concisely expressed by formulas containing letters. The quantitative relationship can also be expressed concisely.

Note: (1) When multiplying with letters, you can omit the multiplication sign or "?" Express delivery. For example, a×x is written as ax or a 10. When multiplying numbers, you cannot omit the multiplication sign.

(2) When numbers and letters are multiplied, they can be simplified to numbers and put in front. For example, a×4×b is written as 4ab.

(3) When 1 is multiplied by letters,1is omitted. For example, a× 1 is written as a.

2. Simple equation

(1) equation: The expression of an equation is called an equation.

(2) Equation: An equation containing unknowns is called an equation.

(3) Solution of the equation: the value of the unknown quantity that makes the left and right sides of the equation equal is called the solution of the equation.

(4) Solving equations: The process of solving equations is called solving equations.

(5) Steps for solving simple equations: ① For equations with only one step operation, the reciprocal relationship of addition, subtraction, multiplication and division can be used to solve them. For the equation with two or three operations, the order of operations is determined according to the equation, and then the solution of the equation is obtained according to the reciprocal relationship of the four operations.

(2) Substitute the obtained unknown values into both sides of the original equation for calculation (that is, find the value of the formula containing letters). If both sides of the equal sign of the original equation are equal, the unknown number obtained is the solution of the original equation.

Section 2 Proportion and Specific Gravity

Key points of knowledge

1, and ratio

Proportional ratio

The division of two numbers in meaning is also called the ratio of two numbers. Two expressions with equal ratios are called proportions.

The first and second terms of the basic property ratio are multiplied or divided by the same number at the same time (except zero), and the ratio remains unchanged. In proportion, the product of two internal terms is equal to the product of two external terms.

2. The relationship between ratio, fraction and division

The ratio of the former to the latter.

Fraction (fractional line) numerator denominator decimal value

Division quotient of divisor

3. The difference and connection between comparison and simplification.

Conventional method results

Find the ratio according to the meaning of the ratio, and divide the former by the latter. Is a quotient, which can be an integer, decimal or fraction.

Simplified ratio According to the basic nature of the ratio, the first and second terms of the ratio are multiplied or divided by the same number (except zero). It is a ratio, and its first and last terms are integers.

4. Scale

The ratio of the distance on the picture to the actual distance is called the scale of this picture. That is, the distance on the map: actual distance = scale. Usually, the scale is written as the ratio of the previous term (or the latter term) to 1.

5. The difference and connection between positive proportion and inverse proportion.

similarities and differences

characteristic relation

In direct proportion, there are two related quantities, a change and a change. The ratio of two corresponding numbers in two quantities is certain. Yx = k (ok)

The product of two corresponding numbers in two quantities of inverse proportional relation is certain. X×y=k (sure)

Chapter III Application Problems

Section 1 General composite application problems

Key points of knowledge

1, composite application problem

Two-step or more application problems are usually called compound application problems. Compound application problems are composed of several related simple application problems. Compound application problems without specific structural characteristics and problem-solving rules are called general compound application problems.

2. Solution of general composite application problems

Generally, there is no certain rule to solve the problem of compound application. It can be decomposed into several simple single-step application problems, and indirect problems can be solved separately, and then the results can be obtained. In the concrete analysis and solution, analytical method, synthesis method or analytical synthesis method are generally used. For more complex problems, graphic method, hypothesis method and transformation method can be used to help analyze.

(1) Analysis method: Start with the problem and analyze the known conditions in the problem step by step.

(2) synthesis method: that is, from the known conditions of the application problem, it is gradually pushed to the unknown until the solution is found.

(3) Analytical synthesis method: it is a method that combines analytical method and comprehensive method for alternate use. When there is an obvious calculation process in the known conditions, the comprehensive method is used to push forward, and when there is difficulty, the problem raised in the original problem is turned to the analytical method to help, and the problem is solved by pushing back a few steps.

3. Steps to solve general composite application problems

To solve common composite application problems, please follow the following steps:

(1) Examine the meaning of the question and find out the known conditions and problems;

(2) analyze the quantitative relationship in the topic, so as to determine what to calculate first, then what to calculate ... finally, what is important;

(3) List formulas and calculate the number;

(4) Test and write the answer.

Typical application problems in the second quarter

Key points of knowledge

1, typical application problems

An application problem that is solved by two or more operations and has certain solving rules is called a typical application problem. For example, average application problem, chance encounter problem and normalized application problem. We should pay special attention to the characteristics of various application problems and master the law of solving them.

2. Average problem

(1) The characteristic of the average problem is to combine all the "partial quantities" into a "total quantity", and then average according to the "total number of copies" to find out what one of them is.

(2) The rule of solving the average problem: The key to solving this kind of problem is to first find out the "total amount" and "total number of copies", and then use the total amount ÷ total number of copies = average.

(3) Some complex average problems are solved by the method of "shifting more to make up less" according to the essence that the average is to get an equal number by removing a decimal from the redundant part of a large number.

3. The problem of standardization

The characteristic of (1) normalization problem is to find a "single quantity" from the known conditions, and then calculate the required quantity according to this "single quantity". Normalization problems are usually divided into positive normalization and negative normalization.

(2) The rule of solving the normalization problem: In the process of solving the problem, first get a unit quantity, and then take this unit quantity as the standard, and according to the requirements of the topic, calculate what the unit quantity is by multiplication, which is the rule of solving the normalization problem. Or calculate how many "units" the sum contains by division, which is the law of anti-normalization. The normalization problem can also be solved by solving the multiple ratio problem.

Step 4 encounter problems

(1) Features: A. Two moving objects; B. the movement direction is opposite; C. synchronous movement time.

(2) Law of solving problems: speed sum × meeting time = distance/speed sum = meeting time.

Distance ÷ Meeting time = speed and

Section 3 Application of Fractions and Percentages

Key points of knowledge

1, fractional multiplication application problem

Given a number, find its fraction (percentage) and multiply it.

That is, "a number times a fraction (percentage)".

The relationship between the three quantities is expressed by an equation: the quantity of unit "1" × corresponding fraction = corresponding quantity.

2. The application of fractional division.

(1) What is the fraction (percentage) of a number? Find this number and divide it by it. That is, "how much is a score."

The relationship between the three quantities is expressed by an equation: corresponding quantity ÷ corresponding fraction = quantity with the unit of "1".

(2) Find the fraction (percentage) of one number to another and divide it. That is, "one number ÷ another number"

The relationship between the three quantities is expressed by an equation: the quantity corresponding to the unit "1" = the corresponding fraction.

3, the application of engineering problems

The total amount of work is expressed as "1", and the work efficiency is expressed as "a fraction" of the total amount of work done in unit time. According to the total amount of work and work efficiency, find out the time for cooperation to complete the work.

The relationship between the three quantities: working efficiency × working time = total work.

Total workload ÷ working time = working efficiency

Total amount of work ÷ work efficiency = working hours

Section 4 Setting Equations to Solve Application Problems

Key points of knowledge

1, solving application problems by equations.

Solving application problems with column equations means replacing the unknowns in application problems with letters, and column and solve equations according to the equal relationship between quantity and quantity.

2. General steps to solve application problems with column equations

(1) Find out the meaning of the problem, and find out that the unknown is represented by X;

(2) Find out the equal relationship between the quantities in the application problem and make an equation;

(3) solving the equation;

(4) check or check, and write the answer.

Section 5 Application of Ratio and Proportion

Key points of knowledge

The application problems of ratio and proportion include: scale, proportion distribution and positive and negative proportion application.

In the application of (1) scale, the relationship among distance, actual distance and scale on the map is as follows: distance on the map: time distance = scale. If we know any two of the three related quantities, we can find another quantity according to the relationship. In the calculation, it should be noted that the units of various quantities in the formula must be unified.

(2) The application of proportional distribution: a quantity is divided into several parts according to a certain proportion. Assigning application problems in proportion is based on the meaning of proportion and the relationship between proportion and score. The key is to determine the relationship between the number of each part and the total according to the proportion of each part, that is, how many parts each part accounts for. Then answer the question "What is the score of a number (in this case, the amount allocated)?" .

(3) There is a positive proportional relationship between the related quantities in the direct proportional application problem, the relationship is yx = k (certain), and there is an inverse proportional relationship between the related quantities in the inverse proportional application problem, the relationship is X? Y= k (certain). The basic steps to solve the problem of positive-negative ratio application are:

(1) Analyze the quantitative relationship, and judge their proportions according to the quantitative relationship between related quantities;

(2) According to the relation, list the equivalence relation;

(3) setting an unknown number and setting an equation according to the equivalence relation;

④ Solving the equation; ⑤ Test and write the answers.

Chapter IV Quantity Calculation

Key points of knowledge

1, meaning of quantity, measurement and unit of measurement

Quantity, length, weight, speed, etc. Among things, the measurable characteristics of these objective things are called quantities. Comparing the measured quantity with the standard quantity is called measurement. The quantity used as the standard of measurement is called the unit of measurement.

2, commonly used units of measurement and their ratios

(1) Length, area, plot, volume, volume, weight unit and its propulsion rate:

Length 1 km = 1 000m1m = 10 decimeter =100cm.

1 decimeter = 1 0cm1cm =10mm

Area1km2 =1million m2.

1 m2 = 100 square decimeter

1 square decimeter = 100 square centimeter

1 cm2 = 1 00mm2 plot1km2 =100ha.

1 ha = 1 10,000 m2

Volume 1 m3 = 1000 cubic decimeter.

1 cubic decimeter = 1000 cubic centimeter

1 cm3 = 1000 cubic millimeter volume 1 liter = 1000 ml.

1 cubic decimeter = 1 liter

1 cm3 = 1 ml

Weight1t =1000kg1kg =1000mg.

(2) Common time units and their relationships:

The relationship between (1) dates can be illustrated by the following table:

There are 12 months in a year, 365 days in a normal year and 366 days in a leap year. According to the size of the month, 65438+ 10 month, March, May, July, August, 65438+ 10 month and 65438+February are big months, with 3 1 day per month.

April, June, September,165438+1October are abortion, with 30 days per month.

February is neither a big moon nor a small moon. The average year is February 28th, and the leap year is February 29th.

According to four quarters, 65438+ 10, February and March belong to the first quarter.

April, May and June are the second quarter.

July, August and September are the third quarter.

65438+1October,165438+1October and 65438+February belong to the fourth quarter.

② Every month is divided into early, middle and late days, with 65,438+00 days in the early and middle days, and the last time should be determined according to the month, including 65,438+065,438+0 days in the last month, 65,438+00 days in the second half of a normal year, 8 days in the second half of a leap year and 9 days in the second half of a leap year.

③ 1 week =7 days 1 day =24 hours 1 hour =60 minutes 1 minute =60 seconds.

(4) According to the Gregorian calendar year, the method of judging whether the year is a flat year or a leap year is as follows:

The whole thousand years are divisible by 400, and other years are leap years, and vice versa.

3. Summary between similar units of measurement

(1) transformation method: the method of changing the singular number and composite number of high-level units into the singular number of low-level units is called transformation method. Multiply the number of advanced units by the corresponding advanced rate.

(2) Aggregation method: The method of changing the single number of a subordinate company into the single number or multiple number of a superior company is called aggregation method. In the process of polymerization, the relevant amount should be removed at a corresponding rate.

(3) the relationship between chemical method and polymerization method:

Chapter V Preliminary knowledge of geometry

Section 1 Understanding and Calculation of Plane Graphics

Key points of knowledge

1, line

2. Angle

(1) Angle: The figure formed by two rays from a point is called an angle.

(2) Classification of angles:

3, plane graphics

(1) triangle

① Definition of triangle: A figure surrounded by three end-to-end line segments is called a triangle.

② Classification of triangles:

(2) quadrilateral

① Definition of quadrilateral: A closed figure surrounded by four sequentially connected line segments is called quadrilateral.

(2) the classification of quadrilateral:

(3) Calculation formula of characteristics, perimeter and area:

Chapter VI Statistical Chart

Key points of knowledge

1, statistics

(1) Statistical table: collate the collected data, make a table, analyze the situation and reflect the problems. This kind of table is called statistical table, which is generally divided into simple statistical table, composite statistical table and percentage statistical table.

(2) Making statistical tables: When making statistical tables, we should first collect and sort out the data, and then determine the format and items of the table according to the data and tabulation requirements. General statistical tables include general titles (table names), vertical titles (column titles), horizontal titles (column titles), data columns, etc. In addition, the quantity unit and the tabulation date shall be indicated, and the tabulator shall be indicated if necessary.

2, statistical chart

(1) statistical chart: a graph using points, lines, faces, etc. A chart showing the quantitative relationship between related quantities is called a statistical chart. There are three common statistical charts: bar chart, line chart and fan chart.

(2) Bar chart:

① A bar chart represents a certain quantity with unit length, draws straight bars with different lengths according to the quantity, and then arranges these straight bars in a certain order. It is easy to see various numbers from the bar chart.

(2) The drawing method of bar graph:

A. collating data; B. Draw a vertical axis and a horizontal axis, and express a certain quantity in units of length; C. Draw straight lines with the same width and different lengths according to the quantity, and arrange them in a certain order; D write the name and drawing date of the statistical chart, and mark the legend.

(3) statistical chart of broken lines

(1) A broken-line statistical chart represents a number with a unit length. According to the number of points, the points are connected by line segments in turn. It can not only express quantity, but also clearly express the change of quantity.

(2) Drawing method of statistical graph of broken line:

A. collating data;

B. Draw a vertical axis and a horizontal axis, and express a certain quantity in units of length;

C, tracking each point according to the quantity, and then connecting each point with line segments in turn;

D write the name and drawing date of the statistical chart, and mark the legend.