Notes on the first volume of sixth grade mathematics

First unit position

Method of using data to represent position:

First count horizontally, then look at the lines, this number is the first number in the data; Then count vertically to see which column, this number is the second number in the data. (What rows and columns)

Unit 2 Fractional Multiplication

Fraction multiplied by integer:

The product of an integer and a numerator is a numerator, and the denominator remains the same. (If you can subtract points, you can subtract points first and then calculate. )

Score times score:

Molecule times product of numerator, denominator times product of denominator. (If you can subtract points, you can subtract points first and then calculate. )

Fractional multiplication and addition, multiplication and subtraction mixed operation order:

1. In the formula without brackets, if there is only addition, subtraction, multiplication and division, it should be calculated from left to right.

ⅱ. There are multiplication, division and addition and subtraction in the formula without brackets, so multiplication and division should be calculated first and then addition and subtraction.

Ⅲ. For the formula with brackets, count the brackets first, and then count the brackets outside.

(4) law of fractional multiplication

1. Swap the positions of two factors, and the product remains the same. This is the so-called multiplication commutative law.

a×b=b×a

2. Multiply the first two numbers, and then the third number; Or multiply the last two numbers, and then the first number. This is called the law of multiplicative association.

(a×b)×c=a×( b×c)

3. When the sum of two numbers is multiplied by a number, you can multiply it separately and then add it. This is the so-called law of multiplication and division. (a+b)×c=a×c+b×c

When the difference between two numbers is multiplied by a number, you can multiply it first and then subtract it. This is the so-called law of multiplication and division. (a-b)×c=a×c-b×c

. 25×4= 100 125×8= 1000 25×8=200 125×4=500

Law (for size comparison):

1, a number (except 0) multiplied by a number greater than 1, and the product is greater than this number;

2, a number (except 0) multiplied by a number less than 1 (except 0), the product is less than this number;

A number (except 0) is multiplied by 1, and the product is equal to this number. The first number

Who is who's score, the first number is divided by the second number, and the second number is expressed as a score.

Find several times of a number, a number × several times;

Find the fraction of a number, a number × a fraction.

reciprocal

Concept: Two numbers whose product is 1 are reciprocal.

Important: ① The product must be 1.

② It can only be two numbers.

③ Reciprocal refers to the relationship between two numbers, which is not a number.

Unit 3 Fractional Division

Multiplication: factor × factor = product

Division: product/one factor = another factor

The significance of fractional division:

Fractional division, like integer division, refers to the operation of finding another factor by knowing the product of two factors and one of them.

(3) Fractional division:

The number A divided by the number B (except 0) is equal to the reciprocal of the number A multiplied by the number B.

Law (for size comparison):

1, when the divisor is greater than 1, the quotient is less than the dividend;

2. When the divisor is less than 1 (not equal to 0), the quotient is greater than the dividend;

3. When the divisor equals 1, the quotient equals the dividend.

""is called a bracket. In an equation, if there are both parentheses, count the parentheses first.

Solve the problem of "what is the score of a given number, and find this number";

1 "column equation method.

Solve the format of application problems with equations;

1, solution Write the word "thank you" and add a colon. )

Settings. Set an unknown number, set an unknown number according to the topic, and set whatever you ask. )

Look for it. (equivalence relation found)

Column. (According to the equation of equivalence relation, solve the equation)

Answer.

2 "column partition formula

① Analyze the quantitative relationship.

A number × several/several = specific quantity.

Unit quantity "1"× pieces/piece = specific quantity.

Quantity unit "1"= specific quantity/piece.

② Formula calculation.

The concept of ratio: the division of two numbers is also called the ratio of two numbers.

In the ratio of two numbers, the number before the comparison sign is called the first term of the ratio, and the number after the comparison sign is called the last term of the ratio. The quotient obtained by dividing the former term by the latter term is called the ratio.

For example,15:10 =15 ÷10 = 3/2 (the ratio is usually expressed as a fraction and can also be expressed as a decimal or an integer).

∶ ∶ ∶

The ratio of the former to the latter.

Note: 1, according to the relationship between the ratio and division and fraction, it can be understood that the latter term of the ratio cannot be 0;

2. In sports competitions, the scores of the two teams are 2: 0, 1:0, etc. This is just a form of scoring, and does not represent the division of two numbers.

The basic nature of the ratio: the first and last items of the ratio are multiplied or divided by the same number (except 0), and the ratio remains unchanged.

According to the nature of the ratio, the ratio can be changed into the simplest integer ratio. When the front and back terms of the ratio are not integers, expanding the front and back terms of the ratio into integers is the simplest integer ratio.

Application of ratio: the first item+the last item = the total number of copies.

The specific number of the total number * * * the number of copies in the preceding paragraph/the total number * * * = the number of objects in the preceding paragraph.

Total specific amount * * * Number of copies of the last item/total * * * = Number of objects of the last item.

Number of objects mentioned in the preceding paragraph ÷ Number of copies/total number mentioned in the preceding paragraph * * * = specific amount of total number * * *

Number of objects in the last item ÷ Number of objects/total number in the last item * * * = specific number of total number * * *

Unit 4 circle

When a circle is folded in half several times, some creases will appear. The point where these creases intersect at the center of the circle is called the center of the circle (fixed point). Generally, it is represented by the letter O. The line segment connecting the center of the circle with any point on the circle is called radius, which is generally represented by the letter R. The line segment passing through the center of the circle with both ends on the circle is called diameter, which is generally represented by the letter D.

In the same circle, all radii are equal and all diameters are equal.

The diameter of the same circle is twice the radius, and the radius is half the diameter. d=2r r= 1/2d

A circle is an axisymmetric figure. A straight line with a diameter is the symmetry axis of a circle, and there are countless symmetry axes of a circle.

The ratio of the circumference to the diameter of any circle is a fixed number, which we call pi by letters.

(pai) means. It is an infinite acyclic decimal, = 3. 1415926535-but in practical application, we usually only take its approximate value, that is, = 3.14.

If c is used to represent the circumference of a circle, there is C= d or c = 2 r.

The area formula of the circle: the area of the circle = r× r.

= r2

Key points: ①r2 stands for r× r.

② Unity of length unit and area unit.

(3) When calculating, you can't write the area formula.

Annular region: large circle region-small circle region (or outer circle region-inner circle region)

Central angle: the angle of the vertex at the center of the circle is called the central angle. The circumferential angle is 360.

Unit 5 Percentage

Concept: Numbers like the above, such as 18%, 50% and 64.2%, are called percentages.

Percent indicates the percentage of one number to another. Percentage is also called percentage after percentage.

Percentages are usually not expressed by fractions, but by adding a percent sign "%"after the original molecules. For example:

90% writing: 90%

(2) Read Percent: When reading Percent, read Percent first, then read the number before the percent sign and read it as an integer.

(3) Writing of percentage: percentage is usually not written in the form of fraction, but expressed by adding a percent sign "%"after the original molecule.

(4) The difference between percentage and fraction: percentage can only express the relationship between two numbers, while fraction can not only express the relationship between numbers, but also express a specific quantity with the name of the unit.

(5) Conversion of percentage, decimal and fraction.

Decimal to Percent: Move the decimal point two places to the right and add a percent sign after the number.

Decimal Percentage: Delete the percent sign and move the decimal point to the left by two places.

Percentage component number: the component mother is 100, and the offer score can be reduced. If the numerator of the percentage is decimal, we should first rewrite the percentage into a fraction with integer components according to the basic properties of the fraction, and then simplify the fraction.

There are two ways to convert fractions into percentages: one is to convert the denominator of fractions into fractions 100 according to their basic properties, and the other is to convert fractions into decimals and then use decimal percentages. (When using the second method, it is inexhaustible, and usually three decimal places are reserved. )

Solve the problem by percentage:

Percentage of what = quantity of what/total quantity * * * *100%

When solving percentage application problems, we should pay attention to figuring out who is better than who. The standard of comparison is different, and the unit "1" is also different. When solving problems, we should pay attention to finding out who sees the unit "1".

Because of the different standards of comparison, A is a few percentage points more than B, but the same percentage is indispensable than A. ..

(9) In real life, people often use "increase by a few percent", "decrease by a few percent" and "save by a few percent" to indicate the range of increase or decrease. (In whom "1")

A few percent increase means that the increase accounts for a few percent of the original.

A few percent reduction means a few percent reduction of the plan.

What percentage is saved means what percentage is saved.

Taxes are mainly divided into consumption tax, value-added tax, business tax and personal income tax. The tax paid is called tax payable, and the ratio of tax payable to various incomes (sales, turnover-) is called tax rate.

There are many ways to deposit money in the bank, such as current account, lump-sum deposit and withdrawal, and lump-sum deposit and withdrawal. The money deposited in the bank is called the principal; When withdrawing money, the excess money paid by the bank is called interest, and the ratio of interest to principal is called interest rate.

According to national regulations, the interest earned from deposits should be taxed at the rate of 20%, which is the so-called "interest tax". The interest we get from withdrawing money from the bank is after-tax interest. There is no tax in debt interest.

Interest = principal × interest rate× time

The interest rate is determined by the bank and stipulated by the People's Bank of China. Interest rate reflects a period of economic development and consumption. According to the changes of national economic development, the interest rate of bank deposits is sometimes adjusted.

Unit 6 Statistics

Features of bar graph: The bar graph can clearly see the quantity.

Features of broken-line statistical chart: broken-line statistical chart can not only see the quantity, but also see the change of quantity.

(2) Use the area of the whole circle to represent the total number, and use the size of each sector in the circle to represent the percentage of each part in the total number. This kind of statistical chart is called departmental statistical chart. Features: We can clearly show the relationship between the number of each part and the total through the fan-shaped statistical chart.

Unit 7 Mathematics Wide Angle

The problem solved here can be solved by equation method. (Unknown as few settings as possible)

Solve the format of application problems with equations;

1, solution Write the word "thank you" and add a colon. )

Settings. Set an unknown number, set an unknown number according to the topic, and set whatever you ask. )

Look for it. (equivalence relation found)

Column. (According to the equation of equivalence relation, solve the equation)

Answer.

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