I. Fractional multiplication

(a) the calculation rules of fractional multiplication:

1, fraction multiplied by integer: the product of numerator multiplied by integer is numerator, and the denominator remains unchanged. (Integer and denominator divisor)

2. Fraction and fractional multiplication: use the product of molecular multiplication as the numerator and the product of denominator multiplication as the denominator.

3. In order to simplify the calculation, the points that can be reduced are reduced first and then calculated.

Note: When multiplying with a fraction, the fraction should be converted into a false fraction before calculation.

(2) Law: (When the multiplication is relatively large)

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

A number (except 0) multiplied by a number (except 0) is less than 1, and the product is less than this number.

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

(3) The operation order of fractional mixed operation is the same as that of integer.

(4) The commutative law, associative law and distributive law of integer multiplication are also applicable to fractional multiplication.

Multiplicative commutative law: a×b=b×a

Law of multiplicative association: (a×b)×c=a×(b×c)

Multiplication and distribution law: (a+b) × c = AC+bcac+BC = (a+b) × c.

Second, the problem of fractional multiplication (see the difficulty decomposition for details)

(Know the quantity (multiplication) of the unit "1" and what is the fraction of the unit "1")

1, find the unit "1": before the rate in the rate sentence; Or "occupy", "be" and "compare"

2. Find several times a number: a number × several times; Find the fraction of a number: a number ×.

3, write quantitative relationship skills:

(1) "de" is equivalent to "x" (multiplication sign).

"Zhan", "Shi", "Bi" and "quite" are equivalent to "=" (equal sign).

(2) Before scoring, it is "Yes":

Number of units "1" × score = number corresponding to the score.

(3) Before the score, it means "more or less":

Number of units "1" × (1 fraction) = corresponding number of fractions.

Second, fractional division.

(1) countdown

The meaning of 1 and reciprocal: two numbers whose product is 1 are reciprocal.

Emphasis: reciprocal, that is, reciprocal is the relationship between two numbers. They are interdependent and reciprocity cannot exist alone. Make it clear who is the reciprocal of who.

2. How to find the reciprocal: (Don't write an equal sign between the original number and the reciprocal)

(1) Find the reciprocal of the fraction: exchange the position of the denominator of the numerator.

(2) Find the reciprocal of an integer: treat an integer as a fraction with a denominator of 1, and then exchange the positions of the denominator of the numerator.

(3) Find the reciprocal of the band score: turn the band score into a false score, and then find the reciprocal.

(4) Find the reciprocal of decimals: Turn decimals into fractions, and then find the reciprocal.

3. Because 1× 1= 1, the reciprocal of 1 is1;

There is no reciprocal because the number of 1 obtained by multiplying 0 by 0 cannot be found.

4. For any number a(a≠0), its reciprocal is1/a; The reciprocal of non-zero integer a is1/a; The reciprocal of the fraction b/a is a/b;

5. The reciprocal of the true score is greater than1; The reciprocal of the false score is less than or equal to1; The reciprocal of the score is less than 1.

(2) Fractional division

1, the meaning of fractional division:

Fractional division has the same meaning as integer division, which refers to the operation of knowing the product of two factors and one of them and finding the other factor.

2. The calculation rule of fractional division: dividing by a number that is not 0 is equal to multiplying the reciprocal of this number.

3. Regularity (when fractional division is relatively large):

(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 is equal to 1, the quotient is equal to the dividend.

4. "[]" is called bracket. In an equation, if there are both parentheses, you should count the parentheses first and then the parentheses.

(3) Solving problems by fractional division (see decomposition of difficult points for details)

(Unknown unit "1") (divided by division): What fraction of the known unit "1"? Find the number of units "1". )

1, the relationship between quantity and fractional multiplication is the same:

(1) is "Yes" before the score:

Number of units "1" × score = number corresponding to the score.

(2) Before the score, it means "more or less":

Number of units "1" ×( 1 fraction) = number corresponding to the fraction.

2. solution: (suggestion: solve by equation)

Equation (1): Let the unknown quantity be x according to the quantitative relation and solve it by equation.

(2) Arithmetic (division): the amount corresponding to the score ÷ the corresponding score = the amount of the unit "1".

3. Find the fraction of one number to another: only use one number to represent another number.

4. Find out how much one number is more (less) than another:

① Find one more fraction: large number ÷ decimal number–1.

② Decimal: 1- decimal ÷ large number

Or (1) find a fraction (large number-decimal number) ÷ decimal number.

② Find less scores: (large number-decimal number) ÷ large number.

(D) Ratio and ratio application

1, the meaning of ratio: the division of two numbers is also called the ratio of two numbers.

2. In the ratio of two numbers, the number before the comparison sign is called the first item of the ratio, and the number after the comparison sign is called the last item of the ratio. The quotient obtained by dividing the former term by the latter term is called the ratio (the ratio is usually expressed in fractions, but it can also be expressed in decimals or integers).

take for example

15 : 10 = 15÷ 10= 1.5

∶ ∶ ∶ ∶

The ratio of the former to the latter.

3. The ratio can represent the relationship between two identical quantities, that is, the multiple relationship. You can also use the ratio of two different quantities to represent a new quantity.

For example: distance-speed = time.

4. Discrimination rate and ratio

Ratio: indicates the relationship between two numbers, which can be written as ratio or fraction.

Ratio: equivalent to quotient, it is a number, which can be an integer, a fraction or a decimal.

According to the relationship between fraction and division, the ratio of two numbers can also be written as a fraction.

6, the relationship between ratio and division, fraction:

7. The difference between ratio, division and fraction: except is an operation, fraction is a number, and ratio represents the relationship between two numbers.

8. According to the relationship between ratio and division and fraction, it can be understood that the latter term of ratio cannot be 0.

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

(5) the basic nature of the ratio

1, according to the relation of ratio, division and fraction:

The property that the quotient is invariant: the dividend and divisor are multiplied or divided by the same number at the same time (except 0), and the quotient is invariant.

The basic property of a fraction: when the numerator and denominator of the fraction are multiplied or divided by the same number at the same time (except 0), the value of the fraction remains unchanged.

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

2. The simplest integer ratio: the first and last terms of the ratio are integers and prime numbers, so this ratio is the simplest integer ratio.

3. According to the basic properties of the ratio, the ratio can be reduced to the simplest integer ratio.

4. Simplified ratio:

(1) Simplify with the Basic Properties of Ratio

① Divide the first and last terms of the ratio by their common factors.

② the ratio of two fractions: multiply the last item in the previous paragraph by the least common multiple of the denominator at the same time, and then simplify it by simplifying the integer ratio.

③ Proportion of two decimal places: move the decimal point position to the right, first change it into integer proportion and then simplify it.

(2) Using the method of calculating the ratio. Note: The final result should be written in the form of ratio.

5. Proportional allocation: allocate a quantity according to a certain proportion. This method is usually called proportional distribution.

If the ratio of two quantities is known, let these two quantities be.

6. The distance is fixed, and the speed ratio is inversely proportional to the time ratio. (For example, for the same distance, the speed ratio is 4: 5 and the time ratio is 5: 4).

The total amount of work is certain, and the work efficiency is inversely proportional to the working hours.

(For example, the total amount of work is the same, the working time ratio is 3:2, and the working efficiency ratio is 2:3)

Three. per cent

(A) the meaning and writing of percentage

1, meaning of percentage: indicates that one number is a percentage of another number.

Percentage refers to the ratio of two numbers, so it is also called percentage or percentage.

2, the main contact and difference between percentage and score:

(1) connection: both of them can express the ratio relation of two quantities.

(2) the difference:

① Different meanings: the percentage only represents the multiple ratio of two numbers, not the specific quantity, so it can't take units;

Fraction can not only represent a specific number, but also the relationship between two numbers, indicating that it can take units when there are numbers.

② The percentage of molecules can be integers or decimals;

The numerator of a fraction cannot be a decimal, only a natural number other than 0.

3, the percentage of writing: usually not written in the form of fractions, but after the original molecule to add "%"to indicate.

(b) Exchange of percentages and decimals:

1, decimal percentage: the decimal point is moved to the right by two places, followed by hundreds of semicolons.

2. Decimal percentage: move the decimal point two places to the left and remove the percent sign at the same time.

(c) Percentage and score of reciprocity

1, percentage component number:

Divide the percentage into components first, and then rewrite the percentage into component number 100, which can be simplified to the simplest fraction.

2. Percentage of scores:

(1) Using the basic properties of the fraction, the denominator of the fraction is enlarged or reduced, and the fraction with the letter 100 is written as a percentage.

(2) Convert fractions into decimals (except infinity, three decimals are usually reserved), and then convert decimals into percentages.

(d) Conversion between ordinary fractions, decimals and percentages.

Part II Graphics and Geometry

circle

First, know the circle.

1. Definition of a circle: A circle is a plane figure surrounded by a curve.

2. Center of the circle: Fold a circular piece of paper twice, and the point where the crease intersects the center of the circle is called the center of the circle.

Usually represented by the letter o. Its distance to any point on the circle is equal.

3. Radius: The line segment connecting the center of the circle and any point on the circle is called radius. Generally represented by the letter R.

Separate the two feet of the compass, and the distance between the two feet is the radius of the circle.

4. Diameter: The line segment whose two ends pass through the center of the circle is called diameter. Usually represented by the letter d.

The diameter is the longest line segment in a circle.

5. The center of the circle determines the position of the circle, and the radius determines the size of the circle.

6. In the same circle or equal circle, there are countless radii and countless diameters. All radii are equal and all diameters are equal.

7. In the same or equal circle, the length of the diameter is twice that of the radius, and the length of the radius is the diameter.

Expressed in letters: d=2r or r=d/2.

8, axisymmetric graphics:

If a graph is folded in half along a straight line, the graphs on both sides can completely overlap, and this graph is axisymmetric.

The straight line where the crease lies is called the symmetry axis. (Any straight line passing through the center of the circle or a straight line with diameter)

9. Rectangles, squares and circles are symmetrical figures, and they all have axes of symmetry. These figures are all axisymmetric figures.

10, only 1 has an axis of symmetry: angle, isosceles triangle, isosceles trapezoid, sector and semicircle.

A figure with only two axes of symmetry is a rectangle.

A figure with only three axes of symmetry is an equilateral triangle.

A figure with only four axes of symmetry is a square.

Figures with countless axes of symmetry are: circles and rings.

Second, the circumference of the circle.

1, circumference of a circle: The length of the curve around a circle is called the circumference of a circle. It is represented by the letter c.

2, pi experiment:

Make a mark on the circular paper, aim at the scale of ruler 0, and roll it on the ruler once to find out the circumference of the circle. It is found that the general rule is that the ratio of the circumference to the diameter of a circle is a fixed number (π).

3. Pi: The ratio of the circumference to the diameter of any circle is a fixed number, which we call Pi. Represented by the letter π(pai).

(1) The circumference of a circle is always greater than three times its diameter, and this ratio is a fixed number. Pi π is an infinite acyclic decimal. In the calculation, π ≈ 3. 14 is generally taken.

(2) When judging, the ratio of circumference to diameter of a circle is π times, not 3. 14 times.

(3) The first person in the world to calculate pi was Chinese mathematician Zu Chongzhi.

4, the circumference of the circle formula

Draw a circle in the square, and the diameter of the circle is equal to the side length of the square.

Draw a circle in a rectangle. The diameter of a circle is equal to the width of a rectangle.

6. Distinguish the circumference of a semicircle from that of a semicircle:

(1) Half circumference: equal to the circumference of a circle ÷2.

Calculation method: 2πr÷2 is π r.

(2) The circumference of a semicircle: equal to half the circumference plus the diameter.

Calculation method: πr+2r

Third, the area of the circle.

1, area of the circle: the size of the plane occupied by the circle is called the area of the circle. It is represented by the letter s.

2. The figure surrounded by an arc and two radii passing through both ends of the arc is called a sector. The angle of the vertex at the center of the circle is called the central angle.

3. Derivation of circle area formula:

(1), with the gradual approach of the transformation idea: reflect the circle into a square and turn the curve into a straight line; Turn the new into the old, the unknown into the known, the complex into the simple, and the abstract into the concrete.

(2) The more sectors (even numbers) a circle is divided into, the closer the mosaic image is to a rectangle.

(3) The relationship between the spelled figure and the circumference and radius of the circle.

4. Area of the ring:

A ring, the radius of the outer circle is r, and the radius of the inner circle is r, (R=r+ the width of the ring. )

S ring = πR? -πr？ or

The area formula of the ring: s ring =π(R? -r？ )。

5, a circle, how many times the radius is expanded or reduced, the diameter and circumference are also expanded or reduced by the same multiple.

And the area is expanded or reduced by a multiple of the multiple.

For example:

In the same circle, the radius is tripled, the diameter and circumference are tripled, and the area is expanded nine times.

6. Two circles: radius ratio = diameter ratio = circumference ratio; And the area ratio is equal to the square of this ratio.

For example:

The radius ratio of the two circles is 2: 3, so the diameter ratio and perimeter ratio of the two circles are 2: 3 and the area ratio is 4: 9.

7. The ratio of the area of any square to its inscribed circle is a fixed value, that is, 4∶π.

8. When the perimeters of rectangle, square and circle are equal, the area of circle and square is in the middle, and the area of rectangle is the smallest. On the contrary, when the area is the same, the circumference of a rectangle is the longest, the square is in the middle and the circumference of a circle is the shortest.

9. Determine the starting line:

(1), the length of each runway = the circumference of the circle formed by two semi-circular runways+the length of two straights.

(2) The straight line length of each runway is equal, and the circumference of each circle determines the total length of each runway. (So the starting line is different)

(3) The distance between every two adjacent runways is 2×π× runway width.

(4) Every time the radius of a circle increases by one centimeter, its circumference increases by 2πa cm; When the diameter of a circle increases by one centimeter, the circumference increases by one centimeter.

1 1, commonly used π value results:

2π = 6.28 3π = 9.42

4π = 12.56 5π = 15.7

6π = 18.84 7π = 2 1.98

8π = 25. 12 9π = 28.26

10π = 3 1.4 16π = 50.24

25π = 78.5 36π = 1 13.04

64π = 200.96 96π = 30 1.44

Fan statistics chart

First, the meaning of the pie chart:

The total number is expressed by the area of the whole circle, and the relationship between the number of parts and the total number is expressed by the area of each sector in the circle.

That is, the percentage of each part in the total (so it is also called percentage chart).

Second, the advantages of commonly used statistical charts:

1, bar chart: you can clearly see the quantity of various quantities.

2. Broken line statistical chart: We can not only see the number of various quantities, but also clearly see the increase and decrease of the quantity.

3. Department chart: It can clearly reflect the relationship between the quantity of each part and the total.

Third, the size of the sector: in the same circle, the size of the sector is related to the size of the central angle of this sector. The bigger the central angle, the bigger the sector. (So the percentage of the sector area to the circle area is the percentage of the central angle of the sector to the peripheral angle. )