Decimal number 1, divide the integer 1 into 10, 100, 1000 ... These fractions can be expressed in decimals. 2. One decimal place indicates a few tenths, two decimal places indicate a few percent, and three decimal places indicate a few thousandths.

Classification of decimals 1. Divide by integer: pure decimal, decimal 2. Divided by decimal parts: finite decimals and infinite decimals can be divided into infinite circulating decimals and infinite circulating decimals can be divided into pure circulating decimals and mixed circulating decimals.

Integer and decimal number sequence table integer part decimal part decimal part

... billion, million, million.

Numbers ... hundreds of billions, billions, billions, millions, hundreds, thousands, hundreds, dozens? Ten percentage points, thousands, tens of thousands ...

The unit of counting ...100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000. ...

Multi-digit reading method and writing method 1, multi-digit reading method: start from the high position and read one level down; When reading hundreds of millions or tens of thousands of numbers, you should read them according to the reading method of each level, and then add the word "hundred million" or "ten thousand" at the back; The zero at the end of each level is not read, and other numbers have one zero or several zeros in succession, and only one "zero" is read. 2, multi-digit writing: starting from the high position, writing down one level at a time; Write 0 on any number without units.

How to read and write decimal 1 How to read decimals: Generally, the integer part reads integers, the decimal part reads "dots", and the decimal part only reads numbers in sequence. 2. Decimal writing: When writing decimals, the integer part is written as an integer, the decimal point is written in the lower right corner of each digit, and the decimal part is written on each digit in turn.

Rewrite the omitted mantissa 1, and rewrite it into a number with the unit of 10000 or 100000: put the decimal point at the right end of the 10000 or 100000 of the multi-digit number. 2. The mantissa after omitting "ten thousand" or "hundred million": also called rounding to "ten thousand" or "hundred million"; Accurate to "ten thousand" or "one hundred million". The mantissa after omitting the "10,000" digit is to approximate the number on the thousand digits by "rounding".

Topic: Understanding Numbers (2)- Divisibility of Numbers

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The meaning of divisibility The integer A is divisible by the integer B (b≠0), and the quotient of divisibility is exactly an integer without remainder, so we say that A can be divisible by B (that is to say, B can be divisible by A).

When the quotient obtained by dividing A by B is an integer or a finite decimal, and the remainder is 0, we say that A can be divided by B (or B can divide A), where A and B can be natural numbers or decimals (B cannot be 0).

The connection and difference between divisibility and division. All their results have no remainder, which is their similarity. Division includes division, which is a special case of division.

The divisor and multiple are 1. If the number A is divisible by the number B, it is called a multiple of B, and B is called a divisor of A. 2. The divisor of a number is finite, in which the smallest divisor is 1 and the largest divisor is itself. 3. The number of multiples of a number is infinite, the smallest of which is itself, and it has no maximum multiple.

Odd and even numbers 1 and divisible by 2 are called even numbers. For example: 0, 2, 4, 6, 8, 10 ... Note: 0 is also an even number 2, and numbers that are not divisible by 2 are called cardinality. For example: 1, 3, 5, 7, 9 ...

The characteristics of divisible 1, and the characteristics of numbers divisible by 2: 0, 2, 4, 6, 8 in units. 2. The characteristics of numbers that can be divisible by 5: 0 or 5 in a unit. 3. The feature that a number can be divisible by 3: the sum of the numbers on each digit of a number can be divisible by 3, and this number can also be divisible by 3.

The sum of prime numbers is 1, and a number has only 1 and its two divisors. This number is called prime number. 2. A number has other divisors besides 1 and itself. This number is called a composite number. 3. 1 is neither a prime number nor a composite number. 4. Natural numbers can be divided into 1, prime numbers and composite numbers according to the number of divisors. Natural numbers can be divided into odd and even numbers according to whether they are divisible by 2.

The decomposition of prime factor 1, each composite number can be written as the multiplication of several prime numbers, which is called the prime factor of this composite number. For example, 18=3×3×2, and 3 and 2 are called prime factors of 18. 2. Multiplying several prime factors to represent a composite number is called prime factor decomposition. Short division is usually used to decompose prime factors. 3. The greatest common divisor and the least common multiple of several numbers under special circumstances. (1) If among several numbers, the larger number is a multiple of the smaller number and the smaller number is a divisor of the larger number, then the larger number is their least common multiple and the smaller number is their greatest common divisor. (2) If several numbers are pairwise coprime, their greatest common divisor is 1, and their least common multiple is the product of these numbers.

Topic: Understanding Numbers (3)- Fractions and Percentages

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Meaning of fraction and percentage 1, meaning of fraction: divide the unit "1" into several parts on average, and the number representing such one or several parts is called a fraction. In the fraction, the number indicating how many parts the unit "1" is divided into on average is called the denominator of the fraction; A number indicates how many copies have been made, which is called the numerator of the fraction; One of them is called fractional unit. 2. The meaning of percentage: The number indicating that one number is the percentage of another number is called percentage. Also called percentage or percentage. Percentages are usually not written in the form of fractions, but expressed in concrete "%". 3. Percent indicates the multiple relationship between two quantities, and the unit of measurement cannot be written behind it. 4, into a number: a few percent is a few tenths.

According to the different conditions of numerator, denominator and integer part, the types of fractions can be divided into true fraction, false fraction and with fraction.

The relationship between fractions, decimals and percentages and the conversion between fractions and decimals.

The relationship between fraction and division and the basic property of fraction 1, connection: the numerator of fraction is equivalent to the dividend of division; Denominator is equivalent to divisor; Fractional value is equivalent to quotient difference: division is an operation with operation symbols; The score is a number. Therefore, it should generally be said that dividends are equivalent to a molecule, but it cannot be said that dividends are a molecule. 2. Because there is a close relationship between fraction and division, the basic properties of fraction can be obtained according to the properties of "constant quotient" in division. 3. The numerator and denominator of the score are multiplied or divided by the same number (except 0), and the size of the score remains unchanged. This is called the basic nature of fraction, which is the basis of divisor and total score.

The divisor and general fraction are both 1, and the fraction whose numerator and denominator are prime numbers is called simplest fraction. 2. Turning a fraction into a fraction equal to it, but with smaller numerator and denominator, is called a reduced fraction. 3. Reduction method: divide the denominator by the common divisor of the denominator (except 1); Usually, we have to separate it until we get the simplest score. 4. Changing scores of different denominators into scores of the same denominator is equal to the original score, which is called the total score. 5. General division method: first find the least common multiple of the original denominator, and then turn each fraction into a fraction with this least common multiple as the denominator.

Two numbers whose reciprocal is 1 and whose product is 1 are reciprocal. 2, 2, to find the reciprocal of a tree (except 0), just switch the numerator and denominator of this number. 3. The reciprocal of 1 is1,and 0 has no reciprocal.

The size of the fraction is 1, and the fraction with the same denominator, the larger the numerator, the greater the fraction. 2. The scores with the same numerator are larger with smaller denominator. 3. Fractions with different denominators and numerators are usually divided first, converted into fractions with a common denominator, and then compared. 4. If the scores to be compared have scores, compare their integer parts first, and the score with the larger integer part is larger; If the integer parts are the same, then compare their decimal parts, and the decimal part with the largest decimal part is the largest.

Subject: the operation of numbers (1)-the significance and law of elementary arithmetic.

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Meaning addition of four operations: subtraction of two numbers into one number: knowing the sum of two numbers and one of them, finding the other addend: a, multiplying a number by an integer is a simple operation to find the sum of several identical addends; B, a number multiplied by a decimal or a fraction is the operation of finding a fraction of this number: knowing the product of two factors and one of them, find the other factor.

The four operation principles are 1, addition A, integer and decimal: the same digits are aligned, starting from the low place, one decimal is B, the denominator is the same, and the molecules are added; Fractions with different denominators: divide first, then add. 2. Subtract A, Integer and Decimal: the same number of digits are aligned, which number of digits is not enough to subtract from the low order, and one equals ten and then B is subtracted. Fractions with the same denominator: denominator unchanged, numerator subtraction; Fractions with different denominators: divide first and then subtract 3. A. Integer and decimal multiplication: multiply the multiplicand by the number on each digit of the multiplier, and the last digit of the number will match the last digit. Finally, add the product, the factor is decimal, and the decimal places of the product are the same as those of the two-digit factor. B. Fraction: The product of numerator multiplication is numerator, and the product of denominator multiplication is denominator. If the divisor can be reduced, the result should be simplified. 4. division a, integer and decimal: how many divisors are there? First look at the first few digits of the dividend (if it is not enough, look at one digit), and which one except the dividend is written on the quotient. The divisor is a decimal, which is converted into an integer and divided by it. The decimal point of quotient is aligned with the decimal point of dividend. B, the number A divided by the number B (except 0) is equal to the reciprocal of the number A divided by the number B.

Subject: Number Operation (2)- Algorithms and Simple Algorithms

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Canada-France exchange method a+b = b+a

Law of association (a+b)+c = a+(b+c)

A-B-C = A-(B+C)

Multiplicative exchange law a×b=b×a

The law of association (a×b)×c=a×(b×c)

Distribution law (a+b) × c = a× c+b× c

Invariant properties of divisor m≠0 a÷b=(a×m)÷(b×m) =(a÷m)÷(b÷m)

Subject: Digital Operation (3)- Elementary Arithmetic

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Four mixed operations have only one operation level-from left to right.

There are two levels of operation-the second level of operation is calculated first.

Parentheses are the only enclosed symbols, first inside and then outside.

There are two kinds of brackets: the first kind of parenthesis (left parenthesis)

Re-neutralization (bracket solution)

Outside the back (outside the bracket)

The application method of four operations In the elementary arithmetic of integers, decimals and fractions, we should choose the most reasonable and simple method for operation.

Topic: Number Operation (4)- Text Question

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According to the relationship between numbers, text titles can be calculated correctly by grasping the key words in the narrative and listing the formulas.

Topic: Basic knowledge of algebra (1)- using letters to represent numbers

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It is a basic feature of algebra to express the meaning of numbers with letters. It is not only simple and clear, but also can express the general law of quantitative relationship.

The letter indicates that the number is 1, and the letter indicates any number. Example: Xiaohong is one year old, and her mother is 24 years older than her. Her mother's age can be expressed as (a+24) years old.

2. Use letters to express common quantitative relations: for example, distance, time and speed are expressed as s=vt, v=s÷t and T = S ÷ V..

3. Use letters to represent algorithms and examples of properties; Additive commutative law A+B = B+A additive commutative law (A+B)+C = A+(B+C)

4. Use letters to represent calculation formulas and rules. For example: circumference of a circle: c=2∏r or c=∏d Area of a circle: s=∏r2.

Notes on using letters to represent the number 1. When numbers are multiplied by letters, letters and letters, the multiplication sign can be abbreviated as "?" Or omit it. Numbers are multiplied by numbers, and the multiplication sign cannot be omitted.

2. When 1 is multiplied by any letter, omit "1".

When a number is multiplied by a letter, write the number before the letter.

In literacy and evaluation with letters, attention should be paid to the writing format when seeking formula values with letters or evaluating with formulas.

Topic: Fundamentals of Algebra (2)- Simple Equation

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An equation and an equation represent an equal relationship. Equations with unknowns are called equations. There are two conditions to judge whether a formula is an equation: first, it contains unknowns; The second is the equation. So the equation must be an equation, but the equation is not necessarily an equation.

The solution of the equation and the value of the unknown quantity that makes the left and right sides of the equation equal are called the solution of the equation. The process of solving an equation is called solving an equation.

Simple equation appendix+appendix = and one appendix = and-the solution of another appendix

Minus-Minus = Difference Minus = Difference Minus

Subtraction = difference+subtraction

Multiplier × Multiplier = Product One factor = Product ÷ Another factor

Divider = quotient divisor = dividend = quotient

Dividend = divisor × quotient

Topic: Basic knowledge of algebra (III)-the nature and significance of ratio and proportion

I. Meaning and nature of ratio and proportion

Bibi example

It means that dividing two numbers means that two ratios are equal.

The basic property is that both the former term and the latter term are multiplied or divided by the same number (except 0). The product of two external terms is equal to the product of two internal terms with the same ratio.

Second, the relationship between ratio, fraction and division

The ratio of the former to the latter.

Fractional values of numerator and denominator of fractional line "-"

Divider quotient of division and divider

Third, the difference and connection between seeking ratio and simplifying ratio.

Result in a meaningful and just way

Find a number (integer, decimal, fraction), which is obtained by dividing the previous item by the later item.

Simplified ratio converts the ratio of two numbers into the simplest integer ratio. The former and the latter use a ratio (the former and the latter) to multiply or divide by the same number (except 0).

Fourth, 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 numbers corresponding to two quantities must be Y/x=k (certain).

The product of two numbers corresponding to two quantities in inverse proportional relation must be Xy=k (certain).

Verb (abbreviation for verb) scale

The ratio of the distance on the picture to the actual distance is called the scale of this picture. Namely: distance on the map: actual distance = scale. Usually the scale is written as the ratio of 1.

Topic: Basic knowledge of algebra (IV)-Application of ratio and proportion

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In industrial production and daily life, it is often necessary to allocate a quantity according to a certain proportion. This distribution method is usually called "proportional distribution".

When solving the problems related to proportional distribution, we should be good at finding out the ratio of total quantity and distribution, and then convert the ratio of distribution into component quantities or shares to answer.

The positive and negative comparison examples should use the problem-solving strategy of 1, examine the questions and find out two related quantities in the questions.

2. Analyze and judge whether the two related quantities in the problem are directly proportional or inversely proportional.

3, unknown, column ratio formula

4. Solution ratio style

5. Test and write the answers

Subjects: application problems (1)- simple application problems and compound application problems.

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A simple application problem consists of two known conditions and a problem, which is called a simple application problem. It is the basis of compound application problems. Solving problems should be based on the definition of four operations, and their sum, difference, product and quotient should be found.

Compound application problem 1, compound application problem is composed of two or more simple application problems, so its quantitative relationship is also complicated, and it can only be solved through two or more operations.

2. When solving compound application problems, the common thinking methods are "analysis method" and "synthesis method".

3. The analysis method is based on the problem required by the application problem, using the knowledge that a problem must have two conditions, and gradually pushing it to the known conditions, that is, the idea of "exploring the reasons".

4. The comprehensive law is to proceed from the known conditions and gradually push to the solution of the problem, that is, the idea of "seeking the result from the cause"

However, when solving problems, the two methods are often combined, that is, comprehensive analysis is adopted, and sometimes the quantitative relationship is analyzed with the help of line segment diagram, so as to find a solution.

The general steps to solve application problems are 1. Find out the meaning of the problem-find out the known conditions and problems by examining the questions.

2, quantitative relationship analysis-analysis of known conditions, the relationship between conditions and problems, to determine the methods and steps to solve the problem.

3. Column Calculation-List formulas and calculate numbers.

4. Check and write the answer-Check, check and write the answer.

Topic: Application Problems (2)-Typical Application Problems

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Typical application problems generally refer to application problems with unique structural characteristics and specific solutions. The main problems in teaching materials are the application of averaging, normalization and satisfaction. When solving typical application problems, we should also pay attention to the analysis of quantitative relations, and at the same time pay attention to summarizing the structural characteristics and problem-solving rules of each type of typical application problems, so that when analyzing the meaning of the problems, we can make our thinking more agile and broaden our thinking.

Topic: Application Problems (III)-Solving Application Problems with Equations.

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The characteristic of solving application problems with tabular equations is to represent unknown quantities with letters, list equations according to the equal relationship between quantities in the problem, and then solve them. Solving application problems with column equations is the practical application of simple equations and an important mathematical method. It can broaden the thinking, turn the difficult into the easy and improve the flexibility of solving problems.

Step 1: Find out the meaning of the problem, find out the unknown, and express it by x.2. Find the equivalence relation according to the meaning of the question, list equation 3, solve equation 4, test and write the answer.

The common method of finding equivalence relation according to the meaning of the topic 1. According to the common quantitative relationship, the equivalent relationship is established.

2. According to the learned calculation formula,

3. According to the key narrative sentences in the question, the basic equivalence relation is determined as a whole.

4. Analyze the quantitative relationship by using line graph and list method, and establish the equivalence relationship.

The thinking method is to use positive thinking to solve application problems, that is, according to the narrative order of the problems, the position quantity represented by X is temporarily regarded as known, and it participates in column operation like the known quantity.

Topic: Application Questions (4)-Application Questions of Fractions and Percentages

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The key to solving the application problems of fractions and percentages is: (1) determining the standard quantity (unit "1"); (2) Find the relationship between quantity and price, and then solve it in the form of columns.

Classification: 1. Find the fraction (or percentage) of one number to another. 2. Find the fraction (or percentage) of a number. 3. Find the fraction (or percentage) of a number. 4. Engineering problems.

The problem of fractional multiplication is known. Find the sum (or percentage) of a number and use multiplication. That is, "a number x the sum of fractions (or a few percent). Quantity in "1" × fraction = component

Fractional division application problem 1. What is the fraction and (or percentage) of a number? Find this number and divide it by division, which is "how much/how much". Component score = quantity in "1"

Find the fraction (or percentage) of one number to another, and then divide it. Namely: "one number ÷ another number". Component ÷ Unit Quantity "1"= score.

Engineering problem application problem 1, the total workload is expressed as "1", and the work efficiency is expressed as "a fraction" of the total workload completed in unit time. According to the total amount of work and work efficiency, find out the time for cooperation to complete the work.

2. Relationship among three quantities: work efficiency × working time = total workload (unit "1"), total workload (unit "1"), working time = work efficiency ÷ total workload (unit "1") and work efficiency = working time.

Subject: Measurement of quantity

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The meaning of quantity, measurement and measurement unit The things that can be measured, such as quantity, length, size, weight and speed, are the characteristics of objective things and are called quantity. 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.

Commonly used units of measurement and their propulsion rate 1, currency, length, area, area, volume, volume, weight unit and their propulsion rate. (omitted) 2. Common time units and their relationships. (omitted)

Chemical polymerization 1, chemical polymerization 2, chemical polymerization 3, the relationship between chemical polymerization and polymerization between similar units of measurement

Distance measurement method 1, tool measurement 2, estimation

Subject: Basic knowledge of geometry (1)- Lines and angles

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A straight line has no end points and extends infinitely in two directions, so it cannot be measured.

A line segment has two endpoints. The line segment between two points on a straight line is called a line segment and can be measured.

The ray has only one endpoint, and one end of the line segment extends infinitely to get the ray, which cannot be measured.

When two straight lines intersect at right angles, they are called perpendicular to each other, and one of them is called perpendicular to the other.

Two straight lines that never intersect in the same plane.

The figure formed by two rays from a point is called an angle. The size of the angle is related to the size of both sides, but not to the length of both sides of the angle.

Classification of angles (omitted)

Topic: Basic knowledge of geometry (II)-Plane graphics

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Triangle 1 and triangle are figures surrounded by three line segments. Draw a vertical line from the vertex of a triangle to its opposite side. The line segment between the vertex and the vertical foot is called the height of the triangle. A triangle has three heights. 2. The sum of the internal angles of a triangle is 180 degrees. 3. Triangle can be divided into acute triangle, right triangle and obtuse triangle. 4. Triangle can be divided into isosceles triangle, equilateral triangle and equilateral triangle.

Quadrilateral 1, quadrilateral is an ideal figure surrounded by four line segments. 2. The sum of the internal angles of any quadrilateral is 360 degrees. 3. The characteristics of quadrilateral (omitted) 4. Rectangular and square are special parallelograms; A square is a special rectangle.

A circle is a curved figure on a plane. The same circle or the same circle has the same diameter, and the diameter is equal to twice the radius. A circle has countless axes of symmetry. The center of the circle determines the position of the circle, and the radius determines the size of the circle.

A figure enclosed by two radii of a central angle and the arc it subtends. The sector is an axisymmetric figure.

Axisymmetric graph 1. If a graph is folded in half along a straight line, the graphs on both sides can completely overlap. This graph is called an axisymmetric graph; This suffocation is called symmetry axis. 2. Line segments, angles, isosceles triangles, rectangles, squares, etc. They are all axisymmetric figures, and the number of their symmetry axes is different.

The perimeter and area are 1, and the length of a plane figure is called perimeter. 2. The size of a plane figure or the surface of an object is called the area. 3. The formula for calculating the perimeter and area of common figures is as follows: (omitted)

A complex figure with an area of 1 and consisting of two or more simple figures is called a composite figure. 2. Problem solving methods: combined summation method and empty difference method.

Topic: Basic knowledge of geometry (III)-3D graphics

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Classification 1. Three-dimensional graphics are divided into cylindrical and conical. Cylinders are divided into cuboids, cubes and cones.

Differences and connections between cuboids and cubes.

Characteristics of cylindrical cone

Surface area and volume of three-dimensional graph 1, lateral area 2, surface area 3, volume 4, volume 5, conversion between volume and unit of volume.

Quadrature formula 1, surface area formula 2, volume formula.

Topic: Basic knowledge of statistics.

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Statistical table 1, what is statistical table 2, statistical table classification 3, steps and methods of making statistical table.

Statistical chart 1, statistical chart definition 2, statistical chart classification 3, how to make bar statistical chart 4, how to make broken line statistical chart 5, how to draw fan statistical chart.