Integers and decimals

Integer part:

Decimal counting method; One (a), ten, one hundred, one thousand, ten thousand ... are all called counting units. Where "one" is the basic unit of counting. 10 1 Yes 10, 10 Yes 100 ... The forward speed between every two adjacent counting units is 10. This counting method is called decimal counting method.

How to read integers: read from the previous level, read the name of the level (100 million, 10 thousand), and don't read the zero at the end of each level. One or more zeros in other numbers read only one "zero".

How to write integers: write from the previous level, and write 0 if there is no unit.

Rounding method: Find the divisor and see what the digit with the highest mantissa is. If it is less than 5, it is rounded; If it is 5 or above, it will be rounded and the mantissa will be forward 1. This method of finding approximate value is called rounding.

Comparison of integer sizes: the number with more digits is larger, the number with the same digit is larger, and the number with the same digit is larger than the number with the second digit, and so on.

Decimal part:

Divide the integer 1 into 10, 100, 1000 ... such parts are one tenth, percentage, one thousandth ... These fractions can be expressed in decimals. For example, the record of110 is 0. 1, and the record of 7/ 100 is 0.07.

The first digit to the right of the decimal point is called the tenth digit, and the counting unit is one tenth (0.1); The second number is called the percentile, and the counting unit is one hundredth (0.0 1) ... The maximum counting unit of the decimal part is one tenth, and there is no minimum counting unit. There are several figures in the decimal part, which are called several decimals. For example, 0.36 is two decimal places and 3.066 is three decimal places.

Decimal reading: integer reading, decimal reading, sequential decimal reading.

Decimal writing: the decimal point is written in the lower right corner of the unit.

The essence of decimals: Add 0 at the end of decimals, and the size of 0 remains unchanged. simplify

The movement of decimal point position causes the change of size: it moves to the right and expands to the left, which is1.1230,000 times.

Decimal size comparison: if the integer part is big, it will be big; If the integers are the same, ten digits will be big; And so on.

Fractions and percentages

■ The meaning of fractions and percentages

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 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 "%". Generally, the percentage only indicates the multiple relationship between two quantitative relationships, and cannot be related to the company name.

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.

■ Fraction type

According to the different conditions of numerator, denominator and integer, it can be divided into true fraction, false fraction and band fraction.

■ The relationship between fraction and division and the basic properties of fraction.

1, division is an operation with operation sign; 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.

■ Simplified points and general points

1, where both numerator and denominator are fractions of 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.

■ Countdown

Two numbers whose product is 1 are reciprocal.

To find the reciprocal of a number (except 0), just switch the numerator and denominator of this number.

3. The reciprocal of 1 is1,and 0 has no reciprocal.

■ Score comparison

1, 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.

■ Correlation among percentage, folding and percentage:

For example, a 70% discount is 30%, a 75% discount is 75%, and the percentage is several ten thousandths. For example, a discount of 10% means poor quality. 0%, 65% is 65%.

■ Taxes and interest:

Tax rate: the ratio of taxable amount to various incomes.

Interest rate: the percentage of interest to principal. Calculated by the bank on an annual or monthly basis.

Calculation formula of interest: interest = principal × interest rate × time.

There are three main differences between percentages and fractions:

1. has different meanings. Percent is "a number indicating that one number is the percentage of another number." It can only represent the multiple relationship between two numbers, not a specific quantity. For example, it can be said that 1 meter is 20% of 5 meters, and it cannot be said that "a rope is 20% meters long." Therefore, the percentage cannot be followed by the company name. The score is "divide the unit'1'into several parts on average, indicating the number of such parts or parts". Fraction can not only represent the multiple relationship between two numbers, such as: A is 3, B is 4, A is B? ; It can also express a certain amount, such as: э meters, etc.

2. The scope of application is different. Percentages are often used for investigation, statistics, analysis and comparison in production, work and life. Fractions are usually used for measurement and calculation when integer results are not available.

3. Different writing forms. Percentages are usually not expressed in fractional form, but in percent sign "%". Such as: 45%, writing: 45%; The denominator of percentage is fixed as 100, so no matter how many common divisors there are between the numerator and denominator of percentage, it is not irreducible; Percentages of molecules can be natural numbers or decimals. The numerator of a fraction can only be a natural number, and its expressions include true fraction, false fraction and banded fraction. The calculation result is not that simplest fraction is generally reduced to simplest fraction, but that the false score is converted into a banded score.

Divisibility of numbers

■ The meaning of separability

When the integer A is divided by the integer b(b≠0), the quotient is exactly an integer with no remainder, so we say that A is divisible by B (or that B is 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).

■ Dividers and multiples

1. If the number A is divisible by the number B, it is called a multiple of B and B and a divisor ... 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, a number divisible by 2 is called an even number. 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 ...

■ Characteristics of separability

1, the unit is the divisible number of 2: 0, 2, 4, 6, 8.

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.

■ Prime numbers and composite numbers

1, a number only has 1 and its own two divisors, and this number is called a prime number (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 prime numbers and composite numbers according to the number of divisors.

5. Natural numbers can be divided into odd numbers and even numbers according to whether they are divisible by 2.

■ Decomposition prime factor

1, every composite number can be written as the product 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.

The common factor of several numbers is called the common factor of these numbers. The largest one is called the greatest common factor of these numbers. Two numbers whose common factor is only 1 are called prime numbers. The common multiple of several numbers is called the common multiple of these numbers. The largest one is called the greatest common multiple of these numbers.

4. 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.

■ Operation characteristics of odd and even numbers:

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

2. Odd+odd = even, odd+even = odd, even+even = even; Odd-odd = even,

Odd-even = odd, even-odd = odd, even-even = even; Odd× odd = odd, odd× even = even, even× even = even.

Integer, primary school, fractional elementary arithmetic

■ Four algorithms

1, addition a, integer and decimal: the same digits are aligned, starting from the low place, the full decimal becomes a b, and the denominator score is the same: the denominator is unchanged, and the numerator is added; Fractions with different denominators: divide first, then add.

2. Subtraction A, integer and decimal: the same number is aligned. If you subtract from the low place, which number is not enough, subtract one from ten, and then subtract b, with the denominator unchanged and the numerator reduced. Fractions with different denominators: divide first, then subtract.

3. Multiplication A, Integer and Decimal: Multiply the multiplicand with the number on each bit of the multiplier, and the last bit of the number will match the last bit. 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. What can be reduced should be reduced first, and the result should be simplified.

4. Division A, Integer and Decimal: How many digits are there in the divisor? Look at the first few digits of the dividend first (if it is not enough, look at one digit), and write the quotient on which digit other than the dividend. 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.

■ Operating rules

Additive commutative law A+B = B+A

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

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

a-(b-c)=a-b+c

Multiplicative commutative law a×b=b×a

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

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

Division property a \u( b×c)= a \u b \u c

a \(b \c)= a \b×c

(a+b)c = a \c+b \c

(a-b)÷c=a÷c-b÷c

The quotient invariant property m≠0 a÷b=(a×m)÷(b×m) =(a÷m)÷(b÷m)

■ Variation law of product: In multiplication, if one factor remains unchanged and another factor is expanded (or reduced) several times, the product is also expanded (or reduced) by the same multiple.

Summary: One factor magnifies factor A, another factor magnifies factor B, and the product magnifies factor AB.

One factor subtracts one factor, another factor subtracts B, and the product subtracts AB.

■ Law of Quotient Invariance: In division, the dividend and divisor are expanded (or reduced) by the same multiple at the same time, and the quotient remains unchanged.

Generalization: the dividend enlarges (or reduces) a factor, and the quotient enlarges (or reduces) a factor, while the divisor remains the same.

The dividend is constant, the divisor is enlarged (or reduced) by a factor, but the quotient is reduced (or expanded) by a factor.

■ Using the changing law of product and the property of quotient invariance, some calculations can be simplified. But pay attention to the remainder in the division with remainder.

For example: 8500÷200= Divider and divisor can be reduced by 100 times at the same time, that is, 85472 =, the quotient remains unchanged, but the remainder 1 minus 100, then the original remainder should be 100.

Simple equation

■ Use letters to represent numbers.

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

■ Precautions for using letters to represent numbers

1. When a number is multiplied by letters, letters and letters, the multiplication sign can be abbreviated as ""or omitted. 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.

■ Formulas containing letters and their evaluation

Pay attention to the writing format when finding the value of a formula containing letters or evaluating it with a formula.

■ Equality and equality

The expression of equality is called equality.

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.

■ Solutions of equations and solutions of equations

The value of the unknown that makes the left and right sides of the equation equal is called the solution of the equation.

The process of solving an equation is called solving an equation.

■ When solving a text problem with a column equation, if the required unknowns in the problem have been expressed in letters, you don't need to write them again when solving, otherwise, set the required unknowns as X first.

■ Method of solving equations

1, directly using the relationship between parts in the four operations to solve. For example, x-8= 12

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

Minus-Minus = Differential Minus = Minus-Differential Minus = Difference+Minus

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

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

2. First, consider the term containing the unknown x as a number, and then solve it. Such as 3x+20=4 1.

Think of 3x as a number before solving it.

3. Calculate according to the sequence of four operations, deform the equation, and then solve it. For example, 2.5×4-x=4.2,

The product of 2.5×4 should be found first, so that the equation can be transformed into 10-x=4.2, and then the solution can be obtained.

4. Transform the equation by algorithm or property, and then solve it. Such as: 2.2x+7.8x = 20

Firstly, the equation is transformed into (2.2+7.8) x = 20 by using the algorithm or properties, then the equation is transformed into 10x = 20 by calculating the brackets, and finally the solution is made.

Ratio and proportion

■ Ratio and proportional application problems

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".

■ Problem solving strategy

When doing exercises related to solution ratio distribution, we should be good at finding out the ratio of total quantity and distribution, and then convert the ratio of distribution into component quantity or number of copies to answer.

■ Strategies to solve the problem of positive-negative ratio application

1, examine the question and find out two related quantities in the question.

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

The feeling of numbers and symbols

Cultivating students' sense of numbers in mathematics teaching mainly means that students have the ability to express specific data and quantitative relations with numbers; Ability to judge and calculate different arithmetic operations, and experience in choosing appropriate methods (mental calculation, written calculation and using calculator) to implement calculation; You can infer from the data, and check the accuracy and reliability of the data and inference, and so on.

■ The purpose of cultivating students' sense of numbers is to let students learn mathematical thinking and understand and explain practical problems by mathematical methods.

■ The cultivation of number sense is conducive to the improvement of students' ability to ask and solve problems. When students encounter problems, they consciously and actively establish contact with certain mathematical knowledge and skills, so it is possible to construct mathematical models related to specific things. A certain sense of number is an important condition for completing this kind of task. For example, how to number all the athletes participating in the school sports meeting? This is a practical problem, there is no fixed solution, and different formulas can be used. Different arrangement schemes may be different in practicality and convenience. For example, you can distinguish between grades and classes numerically, distinguish between boys and girls, or quickly know what kind of events a player is participating in.

■ The concept of number itself is abstract, and the establishment of the concept of number is not completed at one time. It takes a process for students to understand and master the concept of numbers. In the process of understanding numbers, let students get in touch with situations and examples related to experience and feel and experience in the realistic background, which will enable students to grasp the concept of numbers more concretely and deeply and establish a sense of numbers. In the process of understanding numbers, let students talk about the numbers around them, the numbers used in life, and how to express things around them with numbers. It will make students feel that numbers are around them, and many phenomena can be expressed simply and clearly with numbers. Estimate the number of words in a page, how many pages there are in a book, how many grains there are in a handful of soybeans, and so on. These perceptions and experiences of specific quantities are the basis for students to establish a sense of numbers, which will be of great help to students to understand the meaning of numbers.

■ Students should be encouraged to express the quantitative relations and changing laws in specific situations in their own unique ways, which is the decisive factor in developing students' sense of symbols.

The introduction of letter representation is an important step to learn mathematical symbols and learn to use symbols to express the implied quantitative relations and changing laws in specific situations. Try to introduce from practical problems, so that students can feel the meaning of letters.

First of all, letters are used to represent arithmetic rules, arithmetic laws and calculation formulas. The generalization of the algorithm deepens and develops the understanding of logarithm.

Second, letters are used to express various quantitative relations in the real world and various disciplines. For example, the relationship between speed v, time t and distance s in uniform motion is s=vt.

Thirdly, using letters to represent numbers is convenient for abstracting quantitative relations and changing rules from specific situations and expressing them accurately, which is conducive to further solving problems with mathematical knowledge. For example, we use letters to represent the unknowns in practical problems, and use the equation relations in the problems to list the equations.

■ Letters and expressions have different meanings on different occasions. For example:

5=2x+ 1 indicates a condition that x satisfies. In fact, x only occupies a special number here, and its value can be found by solving the equation.

Y=2x represents the relationship between variables, X is the independent variable, and you can take any number in the definition field, Y is the dependent variable, and Y changes with the transformation of X;

(a+b) (a-b) = a-b stands for a generalized algorithm and an identity;

If A and B represent the length and width of a rectangle and S represents the area of the rectangle, then S=ab represents the formula for calculating the area of the rectangle, which also means that the area of the rectangle changes with the change of length and width.

■ How to cultivate students' sense of symbols

We should try our best to help students understand the meanings of symbols, expressions and relationships in practical problem situations, and develop their sense of symbols in solving practical problems.

It is necessary to train the symbol operation and carry out a certain number of symbol operations appropriately and in stages. But it does not advocate too much formal operation training.

The development of students' sense of symbols cannot be achieved overnight, but should run through the whole process of mathematics learning and develop gradually with the improvement of students' mathematical thinking.

Calculation of quantity

■ Quantity, length, size, weight, speed, etc. Among things, the characteristics of these measurable objective things 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.

■ number+company name = name number

Only one unit name is called a monomer.

There are two or more unit names called composite numbers.

High-level units, such as changing meters into centimeters, low-level units, such as changing centimeters into meters.

■ Numbers with only one unit name are called odd numbers. Such as: 5 hours, 3 kilograms (only one unit)

There are two or more unit names called composite numbers. Such as: 5 hours and 6 minutes, 3 kilograms and 500 grams (using two units)

56 square decimeter =(0.56) square meter is to convert a single number into a single number.

560 square decimeter =(5) square meter (60 square decimeter) is an example of converting a single number into a compound number.

■ High-level units are relative to low-level units. For example, "meter" is a high-level unit relative to decimeter and a low-level unit relative to kilometer.

■ Table of common calculation formulas

(1) rectangular area = length × width, and the calculation formula is S = A B.

(2) Square area = side length × side length, and the calculation formula is s = a× a.

(3) The circumference of a rectangle is (length+width) × 2, and the calculation formula is s=(a+b)× 2.

(4) Square perimeter = side length × 4, and the calculation formula is s= 4a i.

(5) The area of a parallelogram = bottom× height, and the calculation formula is s = a h. 。

(6) Triangle area = base × height ÷2, and the calculation formula is s=a×h÷2.

(7) Trapezoidal area = (upper bottom+lower bottom) × height ÷2, and the calculation formula is s=(a+b)×h÷2.

(8) cuboid volume = length× width× height, and the calculation formula is v=a bh.

(9) The area of a circle = π× radius square, and the calculation formula is s=лr2.

(10) cube volume = side length × side length× side length, and the calculation formula is v=a3.

(1 1) The volume of cuboids and cubes can be written as the bottom area × height, and the calculation formula is v=sh.

(12) cylinder volume = bottom area × height, and the calculation formula is v = s h.

■ 1 year 1 February (3 1 day includes1,3, 5, 7, 8, 10,1February, and 30 days includes 4, 6, 9 and/kloc.

■ The leap year is a multiple of 4, and the whole hundred years must be a multiple of 400.

■ There are 365 days in a normal year and 366 days in a leap year.

■ 1- 100 is the first century, and 190 1-2000 is the twentieth century.

Understanding and calculation of plane graphics

■ triangle

1, a triangle is a figure surrounded by three line segments. It has stability. 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 according to angle.

4. Triangle can be divided into isosceles triangle, equilateral triangle and equilateral triangle according to its sides.

■ quadrilateral

1, quadrilateral is a figure surrounded by four line segments.

2. The sum of the internal angles of any quadrilateral is 360 degrees.

3. Only one set of quadrilaterals with parallel opposite sides is called trapezoid.

4. Two groups of parallelograms with parallel opposite sides are called parallelograms, which are easy to deform. Rectangular and square are special parallelograms; A square is a special rectangle.

■ circle

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

■ Axisymmetric figure

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.

■ perimeter and area

1, 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 commonly used graphics is as follows:

(1) rectangular area = length × width, and the calculation formula is S = A B.

(2) Square area = side length × side length, and the calculation formula is s = a× a.

(3) The circumference of a rectangle is (length+width) × 2, and the calculation formula is s=(a+b)× 2.

(4) Square perimeter = side length × 4, and the calculation formula is s= 4a i.

(5) The area of a parallelogram = bottom× height, and the calculation formula is s = a h. 。

(6) Triangle area = base × height ÷2, and the calculation formula is s=a×h÷2.

(7) Trapezoidal area = (upper bottom+lower bottom) × height ÷2, and the calculation formula is s=(a+b)×h÷2.

(8) Diameter: d = 2r Radius: r = d÷2

Circumference: circle c = πd d = C÷π.

C circle = 2πr r = C÷π÷2

Area of circle: S circle = πr2 area of circular ring: S ring =π×(R2-R2)

The circumference of a semicircle: c semicircle =πr+2r.

Area of semicircle: s semicircle =πr2÷2

■ Combined graphic area

1, a more complex graph composed of two or more simple graphs, is called a composite graph.

2. Problem solving methods: combined summation method and empty difference method.

Remember to study hard.