Unit 1: location 1. Determine the position of a point with a number pair. The first number represents a column and the second number represents a row. As shown in (3, 5) (third column, fifth row)
2. The left and right translation of the graph: the column changes, but the line remains the same. Graphics move up and down: lines change, but columns remain the same.
Unit 2 Fractional Multiplication
1. Significance of fractional multiplication: 2. Multiplying the score by the score is to find the score of a number. For example, × indicates what a quarter of the total is.
1, fractional multiplication of integers has the same meaning as integer multiplication, and they are both simple operations to find the sum of several identical addends. For example, what is the sum of five when x 5 means?
Second, the calculation rule of fractional multiplication: 1, fractional multiplication with integer: the product of numerator and integer multiplication 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. In order to simplify the calculation, what can be reduced should be reduced first and then calculated.
Note: When multiplying with a fraction, the fraction should be converted into a false fraction before calculation.
The basic property of a fraction: when the numerator and denominator are multiplied or divided by the same number (except 0), the value of the fraction remains unchanged.
Third, the law of relative 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.
Fourth, the operation order of decimal mixed operation is the same as that of integer.
5. 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) Law of multiplicative distribution: (a+b )× c = a× c+b× c.
Sixth, solve the problem of fractional multiplication.
(Know the quantity of unit "1" and multiply it by the fraction of unit "1" (specific quantity).) How much is a number = a number × a fraction.
1, find the unit "1": before the fraction in the fraction sentence; Or behind "accounting", "yes" and "ratio"; 2. See if there are more or less problems;
3. Skills of writing quantitative relations: (1) "de" is equivalent to "×", "Zhan", "Shi" and "Bi" are equivalent to "="
(2) The "score" before the score: the quantity in the unit of "1" × score = specific quantity.
(3) Before the score, it means "more or less": unit quantity "1"× (1-score) = specific quantity; Quantity in "1" ×( 1+ fraction) = specific quantity.
(When the specific quantity is known, find the unit "1" and divide it by. )
Third, reciprocal 1, the meaning of reciprocal: the product is 1, and the two numbers are reciprocal. The reciprocal of 1 is1; 0 has no emphasis on reciprocal: reciprocal, that is, reciprocal is the relationship between two numbers, which are interdependent, and reciprocal cannot exist alone. Make it clear 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 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.
Unit 3: Fractional Division
I. Fractional division
1, the meaning of fractional division: fractional division is the inverse operation of fractional multiplication, that is, the operation of finding another factor by knowing the product of two numbers and one factor. Dividing by a number is the reciprocal of this number, and dividing by a few is the fraction of this number.
Multiplication: factor × factor = product division: product ÷ one factor = another 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.
When the fractional division is relatively large, the law is: when the divisor is greater than 1, the quotient is less than the dividend; When the divisor is less than 1 (not equal to 0), the quotient is greater than the dividend; When the divisor equals 1, the quotient equals 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
Third, the ratio and the application of ratio
1 and the division of two numbers are 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. The last item of the ratio cannot be 0.
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).
2. 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.
3. 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.
4. Relationship and difference between ratio, division and fraction: (Difference) Division is an operation, fraction is a number, and ratio indicates the relationship between two numbers.
The front part of the ratio is equivalent to the dividend in division and the numerator in fraction; The latter term of the ratio is equivalent to the divisor in division and the denominator in fraction; The comparison number is equivalent to the division number in division and the fractional line in fraction; The ratio is equivalent to the quotient of division and the fractional value.
Note: The scores of the two teams in sports competitions are 2: 0, etc. This is only a form of scoring, and does not represent the division of two numbers.
(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 first term and the last term of the ratio are both integers and prime numbers, so this ratio is the simplest integer ratio. According to the basic properties of ratio, the ratio is transformed into the simplest integer ratio.
3. 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 = 3/2 = 3: 2.
5. Proportional allocation: allocate a quantity according to a certain proportion. This method is usually called proportional distribution.
Unit 5: Percentage
First, 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 connection and difference between percentage and score: connection: both of them can express the double ratio relationship of 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 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.
Second, the percentage and fraction, decimal exchange
(1) 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.
(b) Percentage and score of reciprocity
1, percentage component number:
First, rewrite the percentage into a fraction with the mother letter 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.
(3) The mutual transformation between common fractions and decimals and percentages.
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%. (Generally, the powder yield is 70% and 80%, and the oil yield is 30% and 40%. )
(2) Discount: What percentage of the original price of a commodity is sold, which is called discount. Commonly known as "discount". A few percent discount means a few tenths, that is, dozens of percent. For example, 20% discount = 0.8 = 80%, and 65% discount = 0.65 = 65%.
2. The fraction: 10% is one tenth, that is, 10%. 35% is 3.5%, which means 35%.
(3) Pay taxes 1. Paying taxes: Paying taxes means handing over a part of the collective or individual income to the state according to the relevant provisions of the national tax law.
2. Significance of tax payment: tax payment is one of the main sources of national fiscal revenue. The state uses the collected taxes to develop economy, science and technology, education, culture and national defense security. The tax paid is called the tax payable. The ratio of taxable amount to income is called tax rate. Taxable amount = total income × tax rate
(4) Interest 1. Deposits can be divided into demand, lump-sum deposit and lump-sum withdrawal.
2. Significance of saving: People often deposit temporarily unused money in banks or credit cooperatives, which can not only support national construction, but also make personal use of money safer and more planned, and increase some income.
The money in the bank is called the principal. The extra money paid by the bank when withdrawing money is called interest. The ratio of interest to principal is called interest rate. Interest = principal × interest rate× time
Note: If interest tax is required, after-tax interest = interest ×( 1- interest tax rate). Interest on national debt and educational deposits is not taxed.
Unit 6: Statistics
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.
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. )
Unit 7: Mathematical Wide Angle
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.
Second, the solution to the problem of "chickens and rabbits in the same cage": equation method.
Formula: commonly used quantitative relationship.
1, number of copies × number of copies = total number of copies/number of copies = total number of copies/number of copies = number of copies.
2. 1 multiple× multiple = multiple1multiple = multiple/multiple = 1 multiple
3. Speed × time = distance/speed = time/distance/time = speed.
4. Unit price × quantity = total price ÷ unit price = total quantity ÷ quantity = unit price
5. Work efficiency × working hours = total workload ÷ work efficiency = working hours
Total workload ÷ working time = working efficiency
6. Appendix+Appendix = sum, and-one addend = another addend.
7. Minus-Minus = Minus-Minus = Minus+Minus = Minus
8. Factor × factor = product ÷ one factor = another factor.
9. Dividend = quotient dividend = divisor quotient × divisor = dividend
15. Encountered distance = speed and× meeting time = meeting distance ÷ sum of speed and speed = meeting distance ÷ meeting time.
16, concentration problem Solute weight+solvent weight = solution weight100% = concentration solution weight × concentration = solute weight/concentration = solution weight.
17, profit and discount profit = selling price-cost profit rate = profit-cost×100% = (selling price-cost-1) × 100% price = principal× price percentage interest = principal× time.
Formula for representing geometric objects by letters
The length of a rectangle is represented by a, the width by b, the circumference by c and the area by s.
c=2(a+b) s=ab
The side length a of a square is represented by, the perimeter is represented by c, and the area is represented by S.
c=4a s=a？
The base a of the parallelogram is denoted by, the height is denoted by h, and the area is denoted by s.
S = ah
The base of a triangle is represented by a, the height by h and the area by s.
s=ah/2
The upper base of the trapezoid is represented by a, the lower base is represented by h, the center line is represented by m, and the area is represented by s.
s=(a+b)h/2 s=mh
The radius of a circle is represented by R, the diameter by D, the circumference by C and the area by S.
c= πd=2πr s=π r？
The radius of the sector is represented by r, n is the degree of the central angle, and the area is represented by s.
s=π nr？ /360
The length of a cuboid is represented by a, the width by b, the height by h, the surface area by s and the volume by v.
v=sh s=2(ab+ah+bh) v=abh
The side length of the cube is represented by a, the perimeter of the bottom surface is represented by s, and the volume is represented by v.
s=6a？ v=a？
The height of the cylinder is represented by H, the circumference of the bottom is represented by C, the area of the bottom is represented by S, and the volume is represented by V. 。
S side =ch s table =s side +2s bottom v=sh.
The height of the cone is H, the bottom is S and the volume is V. 。
v=sh/3