Summary of knowledge point concept
1. The meaning of multiplying decimal by integer: a simple operation to find the sum of several identical addends; The significance of multiplying a number by a pure decimal is to find a few tenths, a few percent and a few thousandths of this number.
2. Decimal multiplication rules
First, calculate the product according to the calculation rules of integer multiplication, and then look at the factor * * *, how many decimal places there are, just count a few from the right side of the product and point to the decimal point; If the number of digits is not enough, make up with "0".
Step 3 Decimals
The significance of fractional division is the same as integer division, that is, knowing the product of two factors and one of them, and finding the other factor.
4. Calculation rules of fractional division where divisor is an integer.
First of all, according to the law of integer division, the decimal point of quotient should be aligned with the decimal point of dividend; If there is a remainder at the end of the dividend, add "0" after the remainder to continue the division.
5. Divider is a calculation rule of fractional division.
First, move the decimal point of the divisor to make it an integer, then move the decimal point of the divisor to the right by several digits (the number of digits is not enough, make up "0"), and then calculate according to the division rule that the divisor is an integer.
6. Approximate quantity of products:
Rounding is an accurate counting reservation method, which is essentially the same as other methods. But the special point is that the difference between the reserved part and the actual value cannot exceed half of the last order of magnitude: if there is a probability of 0 ~ 9, the error sum of this method is the smallest for a large number of reserved data.
7. Mutualization of Numbers
(1) Number of decimal components
There are several decimals, so writing a few zeros after 1 as the denominator and removing the decimal point of the original decimal as the numerator can reduce the number of offer points.
(2) Fractions become decimals
Divide the numerator by the denominator Those that are divisible are converted into finite decimals, and some that are not divisible are converted into finite decimals. Generally three decimal places are reserved.
(3) Decimal system
A simplest fraction, if the denominator does not contain other prime factors except 2 and 5, this fraction can be reduced to a finite decimal; If the denominator contains prime factors other than 2 and 5, this fraction cannot be reduced to a finite decimal.
(4) Decimal percentage
Just move the decimal point two places to the right and add hundreds of semicolons at the end.
(5) Decimal percentage
To convert percentages to decimals, simply remove the percent sign and move the decimal point two places to the left.
(6) Percentage of scores
Fractions are generally converted into decimals first (three decimal places are generally reserved when they are not used up), and then decimals are converted into percentages.
(7) Decimal percentage
First, rewrite the percentage as the number of components, and put forward a quotation that can be simplified to the simplest score.
8. Classification of decimals
(1) Finite Decimal: The number of digits in the decimal part is a finite decimal, which is called a finite decimal. For example, 4 1.7, 25.3 and 0.23 are all finite decimals.
(2) Infinite decimal: The digits in the decimal part are infinite decimal, which is called infinite decimal. For example: 4.33...3. 145438+05926 ...
(3) Infinite acyclic decimal: the decimal part of a number with irregular arrangement and infinite digits. Such a decimal is called an infinite acyclic decimal.
(4) Cyclic decimal: the decimal part of a number, in which one or several numbers appear repeatedly in turn, is called cyclic decimal. For example: 3.555 … 0.0333 …12.1095438+009 …; The decimal part of cyclic decimal is called the cyclic part of cyclic decimal. For example, the period of 3.99 ... is "9", and the period of 0.5454 ... is "54".
9. Cyclic segment: If an infinite decimal place is followed by a number that starts from a certain place to the right and ends at a certain place, it is called cyclic decimal, and this number is called cyclic segment. A cyclic decimal can be converted into a fraction by writing it as terms and an infinite geometric series.
10. simple equation: the equation AX B = C (A, b, c are constants) is called a simple equation.
Equation: An equation with an unknown number is called an equation. (Note that the equation is an equation and contains unknowns, both of which are indispensable. )
Equations are different from arithmetic. An arithmetic formula is a formula, which consists of an operation symbol and a known number, and it represents an unknown number. An equation is an equation, and the unknown in the equation can participate in the operation. Only when the unknown is a specific numerical value can the equation be established.
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The value of the unknown that makes the left and right sides of the equation equal is called the solution of the equation.
If the solutions of two equations are the same, then these two equations are called homosolution equations.
13. The same solution principle of the equation:
Add or subtract the same number or the same equation from both sides of the (1) equation, and the obtained equation is the same solution equation as the original equation.
(2) The equation obtained by multiplying or dividing the same number whose two sides are not zero is the same as the original equation.
14. Solving the equation: Solving the equation, the process of solving the equation is called solving the equation.
15. the significance of solving application problems by using column equations;
The method of solving application problems with equations and finding the unknown quantity of application problems.
16. Steps to solve application problems with column equations
(1) Find out the meaning of the problem and determine the unknown number, which is represented by X;
(2) Find out the equal relationship between the quantities in the questions;
(3) column equation, solving equation;
(4) check or check, and write the answer.
17. the method of solving application problems with column equations
(1) synthesis method
Firstly, the known number (quantity) and unknown number (quantity) in the application problem are listed as related algebraic expressions, and then the equivalent relationship between them is found out, and then the equation is listed. This is a thinking process from part to whole, and its thinking direction is from known to unknown.
(2) Analysis method
First find out the equivalence relation, and then according to the need of establishing equivalence relation, list the known number (quantity) and unknown number (quantity) in the application problem into related algebraic expressions and then list the equations. This is a thinking process from the whole to the part, and its thinking direction is from unknown to known.
18. Application of solving equations: Application of common equation solutions in primary schools;
(1) general application questions;
(2) Sum and difference times;
(3) Calculation of the perimeter, area and volume of the geometric shape;
(4) Application questions of scores and percentages;
(5) ratio and proportion application problems.
19. area formula of parallelogram:
Bottom× height (the derivation method is shown in the figure); If "h" is used for height, "a" for base and "s" for parallelogram area, then s is parallel to four sides =ah.
20. Triangle area formula:
S△= 1/2*ah(a is the base of the triangle, and H is the height corresponding to the base).
2 1. trapezoid area formula
The area formula of (1) trapezoid is (upper bottom+lower bottom) × height ÷2.
Expressed in letters: (a+b)×h÷2
(2) Another formula: centerline × height.
Expressed in letters: l h.
(3) Trapezoids with mutually perpendicular diagonals: Diagonal× Diagonal÷2
Extended data
1. decimal classification
(1) Pure Decimal: A decimal with zero integer is called a pure decimal. For example, 0.25 and 0.368 are pure decimals.
(2) With decimals: decimals whose integer part is not zero are called with decimals. For example, 3.25 and 5.26 are all decimals.
(3) Pure cyclic decimal: the cyclic segment starts from the first digit of the decimal part, which is called pure cyclic decimal. For example: 3.111.5656 ...
(4) Mixed cycle decimal: the cycle section does not start from the first digit of the decimal part, which is called mixed cycle decimal. 3. 1222 ...0.03333 ... When writing a cyclic decimal, you only need to write a cyclic segment for the cyclic part of the decimal, and point a point at the first and last bit of this cyclic segment. If there is only one number in the circle, just click a point on it.
2. Representation method of cyclic profile
Decimal scores are divided into two categories.
One kind: pure cyclic decimal, and the cyclic node is a molecule; Write several 9s as denominator, and how many people write several 9s in the loop.
The other: mixed cyclic decimal (that is, this problem), with acyclic number MINUS decimal part as the molecule; Write a few 9s, and then write a few 0s as the denominator. If the number of cycles is several, write a few 9s, and if the number of cycles (fractional part) is several, write a few 0s.
3. Area of parallelogram
The area of parallelogram is equal to the product of two adjacent sides multiplied by the sine value of the included angle;
4. Area of triangle
(1)S△= 1/2*ah(a is the base of the triangle, and H is the height corresponding to the base).
(2) s delta =1/2acsinb =1/2bcsina =1/2absinc (three angles ∠A∠B∠C, opposite sides are A, B and C respectively, see trigonometric function).
(3)S△=abc/(4R) (R is the radius of the circumscribed circle)
(4)S△=[(a+b+c)r]/2 (r is the radius of the inscribed circle)
(5)S△=c2sinAsinB/2sin(A+B)