Importance of (1) fractional multiplication:

1, the fractional multiplication of integers has the same meaning as integer multiplication, and it is a simple operation to find the sum of several identical addends. "Fraction times integer" means that the second factor must be an integer, not a fraction.

2. Multiplying a number by a fraction means finding the fraction of a number.

"A number multiplied by a fraction" means that the second factor must be a fraction, not an integer. The first factor is anything. )

(2) Calculation rules of fractional multiplication:

1, the arithmetic of decimal times integer is: numerator times integer, denominator remains unchanged.

(1) In order to make the calculation simple, it can be reduced first and then calculated. (integer and denominator divisor) (2) divisor is to subtract the greatest common factor from the following integer and denominator. (Integer cannot be multiplied by denominator, and the calculation result must be the simplest fraction).

2. The arithmetic of fractional multiplication is: use the product of molecular multiplication as numerator and the product of denominator multiplication as denominator. (numerator times numerator, denominator times denominator)

(1) If the fractional multiplication formula contains a band fraction, the band fraction should be converted into a false fraction before calculation.

(2) The method of fractional simplification is to divide the numerator and denominator by their greatest common factor at the same time.

(3) In the process of multiplication, the divisor is to cross out two divisible numbers in the numerator and denominator, and then write the divisor above and below respectively. (After reduction, the numerator and denominator can no longer contain common factors, so the calculated result is the simplest score).

(4) The basic nature of the fraction: the numerator and denominator are multiplied or divided by the same number at the same time (except 0), and the size of the fraction remains unchanged.

(3) the relationship between products and elements:

A number (except 0) is multiplied by a number greater than 1, and the product is greater than this number. A×b=c, when b >; In 1, c> answer.

A number (except 0) is multiplied by a number less than 1, and the product is less than this number. A×b=c, when b < 1

A number (except 0) is multiplied by a number equal to 1, and the product is equal to this number. A× b = c, and when b = 1, c = a.

When comparing the sizes of factors and products, we should pay attention to the special situation when the factor is 0.

(d) mixed operation of fractional multiplication

1, the mixed operation order of fractional multiplication is the same as that of integers. Multiply first, then divide, and then add and subtract. If there are parentheses, count them first and then count them outside.

2. The law of integer multiplication is also applicable to fractional multiplication; Algorithms can make some calculations simple.

Multiplicative commutative law: a× b = b× a multiplicative associative law: (a× b )× c = a× (b× c)

Multiplicative distribution law: a× (b c) = a× b a× c

(V) Meaning of reciprocal: Two numbers whose product is 1 are reciprocal.

1 and reciprocal are two numbers, which are interdependent and cannot exist alone. A number cannot be called reciprocal. (It must be clear who is the reciprocal of who)

2. The only criterion to judge whether two numbers are reciprocal is whether the product of the multiplication of two numbers is "1". For example: A× B = 1, then a and b are reciprocal.

3. the method of finding the reciprocal: ① finding the reciprocal of the fraction: exchanging the positions of the numerator and denominator.

② Find the reciprocal of an integer: 1 of an integer. ③ Find the reciprocal of the score: first turn it into a false score, and then find the reciprocal.

(4) Find the reciprocal of the decimal: first find the number of components, and then find the reciprocal.

4. The reciprocal of1is itself because 1× 1 = 1.

0 has no reciprocal, because the product of any number multiplied by 0 is 0, and 0 cannot be used as the denominator.

5. The reciprocal of the true score is a false score, and the reciprocal of the true score is greater than 1 and itself.

The reciprocal of the error score is less than or equal to 1. The reciprocal of the score is less than 1.

(6) fractional multiplication is applied to solve problems.

1, what is the score of a number? (by multiplication)

Given the quantity of unit "1", find the fraction of the quantity of unit "1" and multiply it.

2. Find the quantity of the unit "1" skillfully: in a sentence with a score, the quantity before the score is the corresponding quantity of the unit "1", or the quantity after the word "accounting", "yes" and "ratio" is the unit "1".

3. What is speed?

Speed is the distance traveled per unit time. Speed = distance/time/time = distance/speed/distance = speed × time

Unit time refers to 1 hour, 1 minute, 1 second and other time units with the size of 1, such as minutes, hours and seconds.

4. How much is A more (less) than B?

More: (A-B) less: (B-A) B

Location and direction of the second unit (II) 1. What is a number pair?

Number pair: consists of two numbers, separated by commas and enclosed in parentheses. The numbers in brackets are the number of columns and rows from left to right, that is, "columns first, then rows".

The function of number pair: to determine the position of a point. Longitude and latitude are the principles.

2, the method of determining the position of the object:

(1), find the observation point first; (2) Redirection (depending on the included angle of the direction); (3) Finalize the distance (see scale).

The key to drawing a road map is to select observation points, establish direction signs and determine the direction and distance.

Relativity of positional relationship: the positions of the two places are relative. When describing the positional relationship between the two places, the observation points are different, the narrative direction is just the opposite, and the degree and distance are just the same.

Relative position: east-west; North-south direction; East by south-west by north.

Unit 3 Division of Fractions

First, the significance of fractional division: fractional division is the inverse operation of fractional multiplication. Knowing the product of two numbers and one of the factors, we can find the other factor.

Second, the calculation rules of fractional division: dividing by a number (except 0) is equal to multiplying the reciprocal of this number.

1, dividend/divisor = dividend × the reciprocal of divisor.

2. When division becomes multiplication, the dividend must not be changed, "÷" becomes "X" and the divisor becomes its reciprocal.

3. When there are decimals and fractions in the fractional division formula, the number of components and false fractions should be changed before calculation.

4. The change law of dividend and quotient:

① Divided by a number greater than 1, the quotient is less than the dividend: a÷b=c When b> is in 1, c

② Divided by a number less than 1, the quotient is greater than the dividend: a÷b=c when b.

③ Divided by a number equal to 1, the quotient equals the dividend: A ÷ B = C When B = 1, c = a..

Third, the mixed operation of fractional division

1, the mixed operation is calculated by trapezoidal equation, and the equal sign is written in the lower left corner of the first number.

2. Operation sequence:

(1) Division: calculation at the same level, from left to right; Or convert all divisions into multiplication before calculation; Or follow the simple method of "dividing by several numbers is equal to multiplying the product of these numbers". Addition and subtraction are primary operations, and multiplication and division are secondary operations.

② Mixed operations: multiplication, division, addition and subtraction without brackets; Parentheses are counted in parentheses first, and then outside parentheses.

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

Fourth unit ratio

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

1. In the comparison formula, the number before the comparison symbol (:) is called the former item, the item after the comparison symbol is called the latter item, the comparison symbol is equivalent to the division symbol, and the quotient of the former item divided by the latter item is called the ratio.

For example, 3: 4: 5 is pronounced 3 to 4 to 5.

2. The ratio represents the relationship between two numbers, which can be expressed by a fraction, written in the form of a fraction, and read as several to several.

Example:12: 20 =12 ÷ 20 = = 0.612: 20.

Distinguish ratio and ratio: ratio is a number, usually expressed as a fraction, and can also be an integer or a decimal.

A ratio is a formula that represents the relationship between two numbers. It can be written as a ratio or a fraction.

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

4. Simplified proportion: the simplified result is still a proportion, not a number.

(1), the two terms before and after the ratio are divided by the greatest common divisor at the same time.

(2) The simplified method of the ratio of two fractions is to multiply the last term in the previous paragraph by the least common multiple of the denominator, and then simplify the integer ratio. You can also find the ratio and write it out in the form of ratio.

(3), the ratio of two decimal places, move the position of the decimal point to the right, and also convert it into an integer ratio first.

5. Find the ratio: write the ratio symbol as a division symbol and then calculate it. The result is a number (or fraction), which is equivalent to quotient, not ratio.

6. The difference between ratio, division and fraction:

Division: Divider divisor symbol () Divider (not 0) Quotient invariant division is an operation.

Fraction: numerator fraction line (-) Denominator (cannot be 0) The basic property of a fraction is a number.

Ratio: the basic attribute ratio (not 0) of the item after the preceding comparison symbol (:) indicates the relationship between two numbers.

Quotient invariance: Divider and divisor are multiplied or divided by the same number (except 0) at the same time, and the quotient remains unchanged.

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

Application of Fractional Division and Fractional Ratio

1, the known unit "1".

2. The quantity of unknown unit "1" is divided.

3. The basic quantitative relationship of fractional application problems (by fraction)

(1)A is a fraction of B?

A = b× fraction b = fraction a = fraction a = fraction a = fraction a = fraction a.

(2) How much is A more (less) than B?

4. Proportional distribution: the method of distributing a quantity according to a certain proportion is called proportional distribution.

5. Draw a line graph:

(1) Find out the number of the unit "1", first draw the unit "1", and mark the known and unknown.

(2) Analyze the quantitative relationship. (3) Find the equivalence relation. (4) Column equation.

Draw two line graphs for the relationship between two quantities, and draw a line graph for the relationship between part and whole.

Unit 5 circle

First, the characteristics of the circle

1. Circle is a plane figure surrounded by a closed curve in a plane.

2. Features of the circle: beautiful appearance and easy rolling.

3. center o: the point of the center is called the center. The center is generally represented by the letter o.

After the circle is folded in half for many times, the intersection of creases is at the center of the circle, that is, the center of the circle. The center of the circle determines the position of the circle.

Radius r: The line segment connecting the center of the circle and any point on the circle is called radius. In the same circle, there are countless radii, all of which are equal. The radius determines the size of the circle.

Diameter d: The line segment whose two ends pass through the center of the circle is called the diameter. The same circle has countless diameters, and all the diameters are equal. The diameter is the longest line segment in a circle.

The inner diameter of the same circle or equal circle is twice the radius: d = 2r or r = d ÷ 2.

4. Equal circles: circles with equal radii are called concentric circles, and equal circles can be completely overlapped by translation.

Concentric circles: Two circles with coincident centers and unequal radii are called concentric circles.

5. The circle is an axisymmetric figure: if a figure is folded in half along a straight line, the figures on both sides can completely overlap, and this figure is an axisymmetric figure. The straight line where the crease lies is called the symmetry axis.

Figures with symmetry axis: semicircle, sector, isosceles trapezoid, isosceles triangle and angle.

A figure with two axes of symmetry: a rectangle.

A figure with three axes of symmetry: an equilateral triangle

A figure with four axes of symmetry: a square

Figures with or without symmetry axis: circles and rings.

Step 6 draw a circle

(1) The distance between two feet of a compass is the radius of a circle. (2) Draw a circle: fix the radius, center of the circle and make a circle.

Second, the circumference of the circle: the length of the curve around the circle is called the circumference of the circle, and the circumference is represented by the letter C.

1, the circumference of a circle is always more than three times the diameter.

2. Pi: The ratio of the circumference to the diameter of a circle is a fixed value, which is called Pi and is expressed by the letter π.

Namely: pi = circumference ÷ diameter ≈ 3. 14.

Therefore, the circumference of a circle (c) = diameter (d)×π- circumference formula: c = π d, c = 2π r.

Pi π is an infinite acyclic decimal, and 3. 14 is an approximation.

3. Circumference change law: how many times does the radius expand, how many times does the diameter expand, and the circumference expansion multiple is the same as the radius and diameter expansion multiple.

4, semicircle circumference = half circumference+diameter = π r+d.

Third, the area of the circle.

1, derivation of the formula of circular area

As shown in the figure, divide a circle into several parts along the diameter and cut it into a rectangle. The more copies, the closer the image is to a rectangle.

Radius of circle = width of rectangle

Half of the circumference = the length of the rectangle.

Rectangular area = length × width

So: the area of the circle = half of the circumference (πr)× the radius of the circle (r)

S circle = π r× r = π R2

2. For several figures, the circumference of a circle is the shortest and that of a rectangle is the longest when the areas are equal; On the contrary, in the case of equal perimeters, the area of a circle is the largest, while the area of a rectangle is the smallest.

At the same time, the circular area is the largest. Taking advantage of this feature, baskets and plates are made into circles.

3. Variation law of circular area: how many times the radius expands, how many times the diameter and circumference also expand, and the multiple of circular area expansion is the square of the radius and diameter expansion multiple.

4. Annular area = great circle-small circle = π R2-π R2.

Sector area = π R2× n ÷ 360 (n stands for the degree of the central angle of the sector)

Runways: The perimeter of each runway is equal to the perimeter of the circle formed by two semi-circular runways plus the sum of two straight runways. Because the lengths of two straight runways are the same, the starting lines of two adjacent runways are different, and the distance between them is 2×π× runway width.

When the radius of a circle increases by one centimeter, the circumference increases by 2π one centimeter.

The diameter of the circle increases by b cm, and the circumference increases by πb cm.

6. The inscribed circle of any square, that is, the diameter of the largest circle, is the side length of the square, and their area ratio is 4: π.

7. Public data

π=3. 142π=6.283π=9.424π= 12.565π= 15.7

Percentage of Unit 6 (1)

The meaning of 1. percentage: The number indicating that one number is the percentage of another number is called percentage. Percentages are also called percentages or percentages, and percentages cannot have units.

Note: Percent is specially used to express the special ratio relation, which indicates the ratio of two numbers.

1, the difference and connection between percentage and score;

(1) connection: both can be used to express the proportional relationship between two quantities.

(2) Difference: the meaning is different: the percentage only indicates the proportional relationship, not the specific quantity, so it can't take the unit. Fractions not only indicate the proportional relationship, but also express the specific quantity in units. The numerator of percentage can be decimal, and the numerator of fraction can only be integer.

Note: Percentages are widely used in life, and the problems involved are basically the same as fractions. Fractions with denominator of 100 are not percentages, and the denominator must be written as "%",so it is wrong to say that fractions with denominator of 100 are percentages. The two zeros of "%"should be lowercase, not to be confused with the number before the percentage. Generally speaking, attendance, survival rate, qualified rate and correct rate can reach 100%, rice yield and oil yield can not reach 100%, and the completion rate and percentage increase can exceed 100%. Generally, the powder yield is 70% and 80%, and the oil yield is 30% and 40%.

2. The relationship between decimals, fractions and percentages

(1) Percentalized Decimal: Move the decimal point to the left by two places and delete "%".

(2) Decimal percentage: move the decimal point two places to the right and add "%".

(3) Percentilization score: First, write the percentage as a score with the denominator of 100, and then simplify it to the simplest score.

(4) Fractional percentage: divide the numerator by the denominator to get a decimal, and then convert it into a percentage.

(5) Decimal fraction: Simplify fractions with decimal parts of 10, 100, 1000, etc.

(6) Fractional decimal: numerator divided by denominator.

Second, the percentage of application problems.

1, find common percentages, such as: compliance rate, pass rate, survival rate, germination rate, attendance rate, etc. Is to find the percentage of one number to another.

2. Find out how much one number is more (or less) than another. In real life, people often use how much to increase, how much to decrease and how much to save to express increase or decrease.

What percentage of A is greater than B: (a-b) B.

How much is B less than A: (a-b) A.

3. Find the percentage of a number. A number (unit "1") × percentage.

3. What percentage of a number is known? Find this number.

Partial quantity/percentage = a number (unit: "1")

5. Classification of percentage application problems

What percentage of (1)B is A-(A-B) × 100% = what percentage?

(2) What is the percentage that A is greater than B-(A-B) ÷ B × 100%?

(3) How much is A less than B-(B-A) ÷ B × 100%?

Unit 7 Significance of departmental statistical chart

1, the meaning of sector statistical chart: the total number is represented by the area of the whole circle, and the relationship between the number of parts and the total number is represented by the area of each sector in the circle, that is, the percentage of the number of parts to the total number, so it is also called percentage chart.

2. Advantages of commonly used statistical charts:

(1) Bar charts directly display the numbers of each quantity.

(2) The statistical chart of broken lines not only shows the increase and decrease of quantity intuitively, but also clearly shows the number of each quantity.

(3) The fan-shaped statistical chart intuitively shows the relationship between the part and the whole.