Unit 1 Fractional Multiplication
I. Fractional multiplication
(A) the significance of fractional multiplication:
1, fractional multiplication by integer has the same meaning as integer multiplication. Is a simple operation to find the sum of several identical addends.
For example: 65×5 means what is the sum of five 65 s? 1/3×5 means what is the sum of five 1/3?
2. Multiplying a number by a fraction means finding the fraction of a number.
For example, 1/3×4/7 is what is 4/7 of 1/3.
4×3/8 means how much is 3/8 of 4.
(2), 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. Note: When multiplying with a fraction, the fraction should be converted into a false fraction before calculation.
3. In order to simplify the calculation, the points that can be reduced are reduced first and then calculated. Try to cut as much as possible, and if you don't cut, you won't cut. The prime factor frequently tested is1/kloc-0 /×11=121; 13× 13= 169; 17× 17=289; 19× 19=36 1)
4. Decimal times fraction. You can convert decimals into fractions first, or you can convert fractions into decimals and then calculate them (decimal calculation is recommended).
(3) The law of comparative size in multiplication
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.
(4) The operation order of fractional mixed operation is the same as that of integer. 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 = a c+b c
Second, solve the problem of fractional multiplication (know the quantity (multiplication) of the unit "1", that is, what is the fraction of the unit "1")
1. Draw a line segment diagram: (1) Relationship between two quantities: Draw two line segment diagrams, draw a unit quantity first, and pay attention to the left alignment of the two line segments. (2) The relationship between the part and the whole: draw a line segment.
2. Find the company "1": the company "1" comes before the rate in the rate sentence;
Or behind "Zhan", "Yes", "Bi" and "Quite".
3. Skills of writing quantitative relations:
(1) "De" is equivalent to "X", "Zhan", "Equivalent", "Yes" and "Bi" are all "="
(2) The word "de" before the score: the quantity in the unit of "1" × score = specific quantity.
For example, if the number A is 20, what is the 1/3 of the number A? The formula is 20× 1/3.
4. Look at the question of whether the scoring rate is more or less before; The relationship of "more or less" comes before the score:
(less than): quantity of "1" ×( 1- decimal) = specific quantity;
For example, the number A is 50, and the number B is less than the number A 1/2. What is the number B?
The formula is: 50×( 1- 1/2)
(Ratio): the quantity of "1" ×( 1+ score) = specific quantity.
For example, Xiaohong has money from 30 yuan, and Xiaoming has 3/5 more than Xiaohong. How much money does Xiaohong have?
The formula is: 50×( 1+3/5).
3. How many times is a number? Use a number × times;
4. Find the fraction of a number: use a number × fraction.
5. What's the score? Use fraction × number.
6. How to find the fraction of the total quantity of a known part and another part;
(1), unit quantity "1"× (1-decimal) = other part quantity (recommended).
(2) The number of unit "1"-the number of known parts which account for several fractions of unit "1" = the number of required parts.
For example, do it on page 15 and practice the seventh question on page 16 of the textbook (sometimes there will be the keyword "among them" in the question).
Position and direction of the second unit (2)
First, the method of determining the position of the object: 1, first find the observation point; 2. Re-determine the direction (see the included angle of the direction); 3, finally determine the distance (see scale)
Second, the key to describing the road map is to select observation points, establish direction markers and determine the direction and distance.
3. Relativity of positional relationship: 1, and 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.
Fourth, the relative position: east-west; North-south direction; East by south-west by north.
Unit 3 Fractional Division
Third, the 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. Find out who is the reciprocal of who.
2. Reciprocal method:
(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. The reciprocal of1is1; Because/kloc-0 /×1=1; 0 has no reciprocal, because 0 multiplied by any number will get 0 (denominator cannot be 0).
4. 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.
5. Use, a×2/3=b× 1/4 to find out what A and B are. A×2/3=b× 1/4 is regarded as equal to 1, that is, the reciprocal sum of 2/3 is calculated, and the reciprocal of 1/4 is calculated.
1, the meaning of fractional division:
Multiplication: factor × factor = product
Division: product/one factor = another factor
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.
For example, the meaning of 1/2÷3/5 means that the product of two factors is known as 1/2, and one factor is 3/5, and the operation of finding the other factor is known.
2, the calculation rules of fractional division:
Dividing by a number that is not zero is equal to multiplying the reciprocal of this number.
3. The law of 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.
"[]" is called a bracket. In an equation, if there are both parentheses, you should count the parentheses first and then the parentheses.
Second, fractional division to solve the problem
1, solution: (1) Equation: Set the unknown quantity as x according to the quantitative relation and solve it with the equation.
Solution: Let the unknown quantity be X (which must be solved), and then let the equation be X× fraction = specific quantity.
For example, 20 cocks are 1/3 of the number of hens. How many hens are there? (Unit 1 is the number of hens, unit 1 unknown. ) solution: suppose there are x hens. The column equation is: X× 1/3=20.
(2) Arithmetic (division): If the quantity of the unit "1" is unknown, divide by division:
That is, a fraction of the known unit "1", find the quantity of the unit "1".
Corresponding fractional amount ÷ corresponding fractional amount = unit "1"
For example, 20 cocks are 1/3 of the number of hens. How many hens are there? (Unit 1 is the number of hens, unit 1 unknown. ) divided by the formula: 20÷ 1/3.
2. See if there are any questions that are more or less than the previous scores;
The relationship of "more or less" comes before the score:
(less than): specific quantity ÷ (1 minute rate) = quantity in the unit "1";
For example, there are 50 peach trees, less than apple trees 1/6. How many apple trees are there?
The formula is: 50( 1- 1/6)
(Biduo): specific quantity ÷ (1+ score) = unit "1".
For example, a commodity is now in 80 yuan, which is higher than the original price 1/7. What is the original price?
The formula is 80÷( 1+ 1/7).
3. Find a number that is a fraction of another number: divide one number by another, and the result is written as a fraction.
For example, boys are 20 and girls are 15, and the number of girls accounts for a fraction of the number of boys.
The formula is: 15÷20= 15/20=3/4.
4. How to calculate how much one number is more than another:
Using the difference of two numbers, the amount of the unit "1" = fraction.
That is, ① Find a fraction more than another number: use (large number-decimal number) ÷ another number (divided by that number), and the result is written as a fraction.
For example, how much is 5 more than 3? (5-3)÷3=2/3
(2) Find how many fractions one number is less than another number: use (large number-decimal) ÷ another number (divided by that number) and write the result as a fraction.
For example, how much is 3 less than 5? (5-3)÷5=2/5
Note: A few more points does not mean a few less points, because the units are different.
5. Engineering problem: Taking the total workload as "1", how long it takes to complete a project is 1 ÷ efficiency, that is, 1÷ (work efficiency = 1/ time).
For example, a project takes 5 days to complete, 10 days to complete, and 3 days to complete. How many days can three people complete? Formula:1÷ (1/5+110+1/3)
Fourth unit ratio
(A), the meaning of the ratio
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.
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).
15 ∶ 10 = 3/2
The ratio of the former to the latter.
3. The ratio can represent the relationship between two identical quantities, that is, the multiple relationship. Exodus: The length is several times the width.
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 in the form of 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:
The ratio of the former item to the latter item is ":"
Divider divisor divisor quotient
Fractional value of fractional dividing line "-"
7. Difference between ratio, division and fraction: Division is an operation, and 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.
9. In the sports competition, the scores of the two teams are 2: 0, etc. This is only a form of scoring, and does not represent the division of two numbers.
10, find the ratio: the last item is divided by the previous item, and the result is written as a score (if the score is not reduced, the score will not be reduced).
For example:15:10 =15÷10 =15/10 = 3/2.
(B) 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:
(2) Using the method of calculating the ratio. Note: The final result should be written in the form of ratio.
For example:15:10 =15 ÷10 =15/10 = 3/2 = 3: 2.
Or15:10 =15 ÷10 = 3/2. The simplest integer ratio is 3: 2.
5. If there is a unit in the ratio, then the unit should be the same when simplifying the calculation of the ratio, and the result is no unit.
6. Proportional allocation: allocate a quantity according to a certain proportion. This method is usually called proportional distribution. There are generally two ways to solve the problem.
1, solved by fractional rate: proportional distribution is usually based on the total amount, that is, the conversion component rate. We must first find out the total number of copies, then find out how many copies account for a fraction of the total number of copies, and finally multiply the total number by a fraction.
For example, there is 25 grams of sugar water, and the sugar water ratio is 1:4. How many grams of sugar and water are there respectively?
1+4=5 sugar accounts for 1/5 25× 1/5 sugar, and water accounts for 4/5 25×4/5 water.
2, the same number solution: first of all, we must find out the total number of copies, then find out how many copies each, and finally find out how many copies.
For example, there is 25 grams of sugar water, and the sugar water ratio is 1:4. How many grams of sugar and water are there respectively?
The number of sugar and water is 1+4=5, and one is 25÷5=5. The number of sugar is 1, which is 5× 1, and the number of water is 4, which is 5×4.
Unit 5 Understanding of 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, it is represented by the letter R. If the two feet of a compass are separated, 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. Generally 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 an 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 1/2 of the diameter. Expressed in letters: d=2r or r=d/2.
8. 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.
9. Rectangles, squares and circles are symmetrical figures, and they all have axes of symmetry. These figures are all axisymmetric figures.
The figures with the symmetry axis of 10 and only 1 are: angle, isosceles triangle, isosceles trapezoid, sector and semicircle. Figures with only two axes of symmetry are: rectangles; Figures with only three axes of symmetry are: equilateral triangles; Figures with only four axes of symmetry are: squares; Figures with countless axes of symmetry are: circles and rings.
1 1. Draw the symmetry axis with a pencil and draw a dotted line (triangle) with a ruler. The two ends of this dotted line should slightly exceed the figure.
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: (rolling method) Make a mark on the round paper, aim at the scale of ruler 0, and roll it on the ruler once to get the circumference of the circle. Or use a wire to measure the length of the wire wound on the round paper, which is the circumference of the circle (rope measurement method).
It is found that the ratio of circumference to diameter of a circle (circumference divided by diameter) is a fixed number, that is, a little more than three times. We call it pi, which is expressed by the letter 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). The first person in the world to calculate pi was China mathematician Zu Chongzhi.
(1), the circumference of a circle is always greater than 3 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.
4. The circumference formula of a circle: the circumference of a circle is equal to pi times the diameter, and C= πd is expressed by letters.
(1), it is known that the circumference of a circle is divided by pi, which is expressed in letters.
D = C÷π the circumference of a circle is equal to 2 times π times the radius, and C=2πr is expressed in letters.
(2) Divide pi by twice pi to find the radius of the circumference of a known circle.
Use letters to represent r = C ÷ 2π(r = C/2π).
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: 2π r ÷ 2, that is, c half = π r.
(2) The circumference of a semicircle: equal to half the circumference plus the diameter. Calculation method: the circumference of a half circle =5. 14 r (deduction process chalf = 2π r ÷ 2+d = π r+d = π r+2r = 5.14r).
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. Derivation of circle area formula: (1) The more sectors a circle is divided into (even number), the closer the mosaic image is to a rectangle. The length of a rectangle is equivalent to half the circumference, and the width of a rectangle is equivalent to the radius of the circumference.
(2) The relationship between the spelled figure and the circumference and radius of the circle.
Radius of circle = width of rectangle
Half of the circumference = the length of the rectangle.
3. Calculation method of circular area: Because: rectangular area = length × width.
So: the area of a circle = half of the circumference × the radius of the circle.
That is, s circle = c ÷ 2× r = π r× r = π r
The area formula of the circle: s circle =πr → r = S circle ÷π.
4. The area of the ring: a ring, the radius of the outer circle is represented by the letter R, and the radius of the inner circle is represented by the letter R (R=r+ the width of the ring. )
S ring = πR -πr or the area formula of the ring: S ring = π(R -r) (this formula is recommended).
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, the radius, diameter and circumference of the same circle are expanded by three times, and the square of the area is expanded by three times to get 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, if the radius ratio of two circles is 2: 3, then the diameter ratio and perimeter ratio of these two circles are both 2: 3, while 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. Results of common π values: π = 3.14; 2π = 6.28 ; 5π= 15.7
10, formula of outer and inner circle (inscribed circle) S=0.86r derivation process: S=S positive -S circle = d-π r = 2r× 2r-π r = 4r-π r = r× (4-π) = 0.86r.
1 1, formula S= 1. 14r inside the circumscribed circle: S=S circle -S positive = π r-dr/2× 2 = 2r× r/2× r = π r-2r.
12, the figure surrounded by an arc and two radii passing through both ends of this arc is called a sector. The angle of the vertex at the center of the circle is called the central angle. The area of the sector is related to the central angle and radius.
13, s norm =S cycle × n/360; S sector ring =S ring ×n/360
14, the sector is also an axisymmetric figure with an axis of symmetry.
15, the result of perimeter and area of common radius and diameter.
Radius radius square diameter perimeter area
1 1 2 6.28 3. 14
2 4 4 12.56 12.56
3 9 6 18.84 28.26
4 16 8 25. 12 50.24
5 25 10 3 1.4 78.5
6 36 12 37.68 1 13.04
7 49 14 43.96 153.86
8 64 16 50.24 200.96
9 8 1 18 56.52 254.34
10 100 20 62.8 3 14
1.5 2.25 3 9.42 7.065
2.5 6.25 5 15.7 19.625
3.5 12.25 7 2 1.98 38.465
4.5 20.35 9 28.26 63.585
5.5 30.25 1 1 34.54 94.985
7.5 56.25 15 47. 1 176.625
Unit 6 percentage
First, the meaning and writing of percentage
Meaning of (1) 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 connections and differences between percentage and score:
Connection: Both can express the proportional relationship between two quantities.
Differences: ① Different meanings: Percent only represents the multiple ratio of two numbers, and cannot represent the specific quantity, so it cannot take units;
Fraction can represent both a specific number and the relationship between two numbers, and the specific number can be expressed in units.
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 do not write in fractional form, but add "%"after the original molecule to indicate, read as%.
Second, the percentage and fraction, decimal exchange
(1) Exchange of percentages and decimals:
1, decimal percentage: move the decimal point to the right by two places (the number of digits is not enough to make up 0), and add hundreds of semicolons at the end.
2. Decimal percentage: move the decimal point to the left by two places (the number of digits is not enough to make up 0), and remove the percentage sign at the same time.
(b) Percentage and score of reciprocity
1, percentage component number: first rewrite the percentage into a fraction, capital 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. (This method is recommended)
(3) Conversion between ordinary fraction and fractional percentage;
Third, solve the problem by percentage.
(a) general application problems
1, commonly used percentage calculation method:
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%.
2. Find out what percentage of one number is another, divide one number by another, and write the result as a percentage.
For example, boys are 20 and girls are 15. What percentage of boys are girls?
The formula is: 15 ÷ 20 = 15/20 = 75%.
3. Given the quantity (multiplication) of the unit "1", what percentage is the unit "1"? The relationship between quantity and fractional multiplication is the same:
Before (1) percentage is "de": quantity of unit "1" × percentage = percentage corresponding quantity.
(2% is a "more or less" quantitative relationship:
Unit quantity "1"× (1percentage) = percentage corresponding quantity.
4. The unit "1" is unknown (by division). What percentage of the unit "1" is known? Find the unit "1". The method is the same as the fractional method.
Solve: (1) equation: set the unknown quantity as x according to the quantitative relationship and solve it by equation.
(2) Arithmetic (division): percentage corresponding quantity ÷ corresponding percentage = unit "1".
5. The method of finding the percentage more (less) than another number is the same as the method of fraction. Only the result should be written as a percentage. See if there are any problems above or below the percentage;
Before the percentage is "more or less":
(less than): specific quantity ÷ (1- percentage) = quantity "1";
For example, there are 50 kilograms of rice, 50% less than flour trees, and how many kilograms of flour there are.
The formula is: 50÷( 1-50÷)
(Biduo): specific quantity ÷ (1+ percentage) = quantity in the unit "1".
For example, workers manufactured 1 10 parts, which was 10 more than originally planned. How many parts were originally planned?
The formula is:110 ÷ (1+10 ÷)
6. How to find the percentage of one number more than another: The method is the same as the fractional method.
What percentage is the amount in the unit "1" based on the difference between two numbers?
That is, ① Find the percentage of one number more than another: use (large number-decimal number) ÷ another number (divided by that number), and the result is written as a percentage.
How much is A more than B? Method a, (A-B)-B (recommended)
Method b, A-B-100
For example, the teacher planned to change 40 assignments, but actually changed 50 assignments. What percentage did he actually change more than planned?
The formula is: (50-40) ÷ 40 = 0.25 = 25%
(2) Find how many fractions one number is less than another number: use (large number-decimal) ÷ another number (divided by that number) and write the result as a percentage.
How much is B less than A? Method a, (A-B) A (recommended)
Method b, 100-B-A
For example, Zhang San's household electricity consumption 100 degrees, Li Si's household electricity consumption is 90 degrees, and what percentage is Li Si's household electricity consumption less than Zhang San's?
( 100-90)÷ 100=0. 1= 10﹪
Note: A few percent more does not mean a few percent less, because the units are different.
7. If A is more or less than B, how much is B less or more than A? Use a? ( 1 A？ ).
8. Find the price after falling a and rising a: 1× (1-a )× (1+a) (assuming the original price is "1"). Find the change range (what is the percentage of the price after the price reduction to the price after the price increase) and use 1- the percentage of the price increase after the price reduction.
Unit 7: departmental statistical chart
1. The significance of sector statistics chart: the total number is represented by the area of the whole circle, and the relationship between the number of each part and the total number is represented 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. (Write the percentage on the statistical chart)
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. )
4. Application: 1. Will observe the statistical chart.
2. What mathematical information did you get?
Answer (1), what percentage of the total is * * *?
② * * accounts for the largest proportion, while * * accounts for the smallest proportion;
3. What other math questions do you ask? What are the percentages of * * and * * * respectively?
Mathematical Wide Angle: Numbers and Shapes
1, the total number of points in each picture can be regarded as the product of the multiplication of two identical numbers, and these formulas can also be expressed in the form of square numbers. 1+3 = 221+3+5 = 321+3+5+7 = 42: the sum of consecutive odd numbers starting from1is equal to the square of odd numbers.
2. The sum of consecutive even numbers starting from 2 is equal to even number plus even number squared (i.e., (n2+n), or equal to even number multiplied by a number greater than even number 1, i.e., n×(n+ 1).
Supplementary content (location)
1, we use number pairs (number pairs: composed of two numbers, separated by commas and enclosed by brackets. The number in brackets is the number of columns and rows from left to right, that is, "column first and then row") to determine the position of the point. As shown in (3, 5): (third column, fifth row)
Vertical lines are called columns (viewed from left to right) and horizontal lines are called rows (viewed from front to back). First the column, then the number of rows.
2. Translation is represented by "up", "down", "front", "back", "left" and "right", and the status quo of the graph remains unchanged during translation.
3. The graphic is translated left and right: the lines are unchanged; The graph is translated up and down: the columns are unchanged.
Supplementary content (the problem of "chickens and rabbits in the same cage")
First, the characteristics of the problem of "chickens and rabbits in the same cage":
There are two or more unknowns in the topic, and it is required to find a single quantity of each unknown according to the total amount.
The second is to solve the problem of "chickens and rabbits in the same cage"
1, hypothesis method (1) if all rabbits (2) if all chickens;
(Generally, it is assumed that they are all large numbers (multiple feet), and then the difference between two feet is calculated. Divide the big difference by the small difference to get the decimal number (several feet), and finally subtract the total head to get the big number. (We call it setting big and small, setting small and setting big)
For example, 34 students go boating, with 4 people in the big boat and 2 people in the small boat. The rented 12 boat was just full, so I asked how many boats the big boat and the small boat rented.
Hypothetical method:
(1) Assuming that all ships are very large, take 12×4=48 (people).
② Then the difference between the actual number of people and the number of people made by big ships is 48-34= 14 (person).
(3) In fact, the big boat takes 4-2=2 (people) more than the small boat.
(4) The big difference is small (that is, the number of ships), 14÷2=7 (pieces).
⑤ When the total number of ships decreases, the number of ships is 12-7=5 (pieces). (Note the unit)
2. Equation method: For example, 34 students go boating, 4 in the big boat and 2 in the small boat. The rented 12 boat was just full, so I asked how many big boats and small boats I rented.
Solution: If there are x ships, there are 12-X ships.
4x+2x (12-x) = 34 4x is the seating capacity of the big ship, 4 is the seating capacity of the big ship, and 2×( 12-X) is the seating capacity of the small boat. There are (12-X) boats on each boat, with a total of 30 people. 2×( 12-X) is calculated by multiplication and division to get 24-2X.
So 4x+2x 12-x) = 34.
4X+2× 12-2×X=34
4X+24-2 X=34
2 X+24=34
2 X=34-24
2 X= 10
X=5
12-5=7 (article)
A: Rent five big boats and seven small boats.