The sixth grade mathematics volume 1 knowledge points of unit 1 and unit 2
First unit
negative number
1. negative number: on the number axis, all negative numbers are to the left of 0, and all negative numbers are less than natural numbers.
Positive number: Numbers greater than 0 are called positive numbers (excluding 0).
(0) is neither positive nor negative, it is the dividing line between positive and negative numbers. Second unit
Cylinders and cones
1. Features of cylinder: (1) Features of bottom surface: The bottom surface of cylinder is two completely equal circles.
(2) Characteristics of the side surface: The side surface of the cylinder is a curved surface.
(3) Characteristics of height: There are countless heights of a cylinder.
2. Height of cylinder: The distance between two bottom surfaces is called height.
3. The development diagram of the cylinder side: when developed along the height, the development diagram is (rectangular); The length of this rectangle is equal to (the circumference of the bottom of the cylinder) and the width of the rectangle is equal to (the height of the cylinder). The area of this rectangle is equal to (the side area of the cylinder), because
That is, the rectangular area = length × width, so the lateral area of the cylinder = bottom circumference × height. When the bottom circumference and the height are equal, the development diagram along the height is (square); The unfolded graph is a parallelogram when it is not unfolded along the height.
4. lateral area of cylinder: lateral area of cylinder = perimeter of bottom × height, expressed in letters: S side =Ch.
H=S side ÷C
C= S side ÷h
S side =∏dh=2∏rh
5. Surface area of cylinder:
Surface area of cylinder = lateral area+bottom area ×2.
That is, S table = S side +S bottom × 2 = CH+∏ (C ∏ ∏ 2) × 2 = ∏ DH+∏ (D ∏ 2) × 2 = 2 ∏ RH+∏ R ×
In order to avoid calculation errors, it is best to use the formula step by step. )
6. Application of cylindrical surface area in practice: surface area of uncovered bucket = side area+bottom area.
Surface area of oil drum = side area+two bottom areas.
Surface area of chimney ventilation pipe = transverse area
Right side area: lampshade, drain pipe, paint column, ventilation pipe, roller, toilet paper shaft, potato chip box packaging.
Side area+one bottom area: glass, bucket, pen container, hat, swimming pool side area+two bottom areas: oil bucket, rice bucket and pot bucket.
7. Cylinder volume: V = Sh h = V÷S = V÷h V =∏RH (commonly known as R).
V=∏(d÷2) h (known d)
V = ∏ (c ∏ ∏ 2) h (called c)
8. Cut a cylinder into several parts and make it into an approximate cuboid. In this process, the shape has changed.
The volume hasn't changed. The surface area is increased by 2rh.
9. Features of the cone: (1) Features of the bottom surface: The bottom surface of the cone is a circle.
(2) Characteristics of the side surface: The side surface of the cone is a curved surface.
(3) Characteristics of height: The cone has height.
10, the height of the cone: the distance from the apex of the cone to the center of the bottom surface is the height of the cone.
1 1. Volume of cone: The volume of a cylinder is equal to three times the volume of a cone with the same height as its bottom surface, and vice versa.
The volume is equal to one third of the volume of a cylinder with the same height as its bottom. V cone = 1/3 V column = 1/3 Sh
V cone = 1/3 ∏rh V cone = 1/3 ∏(d÷2)h V cone =1/3 ∏ (c ∏悏 II.
12, the relationship between cylinder and cone:
(1) The volume of a cone with the same height as the equal bottom of a cylinder is one third of the volume of the cylinder.
(2) Between a cone with equal volume and height and a cylinder (equal bottom and equal height), the bottom area of the cone is three times that of the cylinder.
(3) Between a cone with equal volume and bottom area and a cylinder with equal height, the height of the cone is three times that of the cylinder.
13. Cones in life: sand piles, funnels and hats.
Typical question:
1, and the side of the cylinder is square, and its height is ∏ times the diameter of the bottom surface.
That is, h=C=∏d, and its lateral area is S-side = h.
2. The radius of the bottom surface of the cylinder is enlarged by 2 times, the height is unchanged, the surface area is enlarged by 2 times, and the volume is enlarged by 4 times.
3. The radius of the bottom surface of the cylinder is enlarged by 2 times, the height is also enlarged by 2 times, the surface area is enlarged by 4 times and the volume is enlarged by 8 times.
4. The radius of the bottom surface of the cylinder is expanded by 3 times, the height is reduced by 3 times, the surface area is unchanged, and the volume is expanded by 3 times.
5. The sum of the volumes of cylinders and cones with equal bottoms and equal heights is 48 cubic centimeters. The volume of this cylinder is
() cubic centimeters, the volume of the cone is () cubic centimeters.
The formula is 48(3+ 1) or 48( 1+ 1/3).
6. The volume difference between a cylinder with equal bottom and equal height and a cone is 24 cubic decimeters. The volume of this cylinder is () cubic decimeter and the volume of the cone is () cubic decimeter.
The formula for finding the volume of a cone is: 24 (3-1) or 24 (1-1/3).
7. A cylinder and a cone have the same volume and the same bottom area. The height of the cylinder is 2 cm, and the height of the cone is () cm.
V column =V cone SH =1/3sh2 =1/3hh = 2 ÷1/3hh = 6.
Summarize and sort out the knowledge points of unit 1 and unit 2 of sixth grade mathematics volume II.
Meaning of integer sum and divisibility of 1. 1
1. When counting objects, the numbers 1, 2,3,4,5 used to represent the number of objects are called integers.
2. Add the "-"sign before the positive integer 1, 2, 3, 4 and 5, and the numbers-1, -2, -3, -4 and -5 are called negative integers.
3. Zero and positive integers are collectively called natural numbers.
4. Positive integers, negative integers and zero are collectively called integers.
5. Integer A is divisible by Integer B. If the quotient of divisibility happens to be an integer without remainder, we say that A is divisible by B, or B is divisible by A. ..
1.2 factor and multiple
1. If the integer A is divisible by the integer B, then A is called a multiple of B and B is called a factor of A..
The number of factors of a number is limited, in which the smallest factor is 1 and the largest factor is itself.
The number of multiples of a number is infinite, and the smallest multiple is itself.
1.3 is a number divisible by 2,5.
1. One-digit numbers are 0, 2, 4, 6, 8, and all numbers are divisible by 2.
2. Among positive integers (except 1), two numbers adjacent to odd numbers are even.
3. In a positive integer, two numbers adjacent to an even number are odd numbers.
4. The unit number is 0, and the number of 5 can be divisible by 5.
5.0 is an even number
1.4 prime number, composite number and factorization factor
1. An integer containing only the factor 1 and itself is called a prime number or prime number.
2. There are other factors besides 1 and itself, and such numbers are called composite numbers.
3. 1 is neither a prime number nor a composite number.
4. Odd numbers and even numbers are collectively called positive integers, and prime numbers, composite numbers and 1 are collectively called positive integers.
5. Every composite number can be written as the product of several prime numbers, which are called prime factors of this composite number.
6. Multiplying a prime factor by a composite number to represent it is called prime factor decomposition.
7. What methods are usually used to decompose prime factors: branch decomposition and short division.
1.5 common factor and maximum common factor
1. The common factor of several numbers is called the common factor of these numbers, and the largest is called the greatest common factor of these numbers.
2. If the smaller number is a factor of the larger number, then the greatest common factor of the two numbers is smaller.
3. If two numbers are prime numbers, then the greatest common factor of these two numbers is
Summary and arrangement of the knowledge points of unit 1 and unit 2 in the sixth grade mathematics volume 3
First, negative numbers:
1, get a preliminary understanding of negative numbers in familiar life situations, read and write positive numbers and negative numbers correctly, and know that 0 is neither positive nor negative.
2. Learn to express some practical problems in daily life with negative numbers, and experience the close relationship between mathematics and life.
I can learn to compare the sizes of positive numbers, 0 and negative numbers with the help of the number axis.
Second, the cylinder and cone
1. Know cylinders and cones and master their basic characteristics. Know the bottom, sides and height of a cylinder. Know the bottom and height of the cone.
2. Explore and master the calculation method of lateral area and surface area of cylinder and the calculation formula of volume of cylinder and cone, and use the formula to calculate the volume to solve simple practical problems.
3. By observing, designing and making cylinder and cone models, we can understand the relationship between plane graphics and three-dimensional graphics and develop students' spatial concept.
Third, the proportion.
1, to understand the meaning and basic properties of proportion, will be solution ratio.
2. Understanding the meaning of positive proportion and inverse proportion can help us find examples of positive proportion and inverse proportion in our life, and we can use the knowledge of proportion to solve simple practical problems.
3. Knowing the image with positive proportional relationship, you can draw an image on grid paper with coordinate system according to the given data with positive proportional relationship, and you can calculate or estimate the value of the other quantity in the image according to one of them.
If you know the scale, you will find the scale of the plan and the distance or actual distance on the map according to the scale.
5, understand the phenomenon of zoom in and out, can use the form of grid paper to zoom in or out simple graphics according to a certain proportion, and realize the similarity of graphics.
6. Infiltrate the thought of function, and let students be inspired by dialectical materialism.
Fourth, statistics.
1, can comprehensively use the learned statistical knowledge, accurately extract statistical information from statistical charts, and correctly interpret statistical results.
2, according to the information provided by the statistical chart, make a correct judgment or simple prediction.
Fifth, mathematical wide angle.
1. After going through the inquiry process of "pigeon hole principle" and having a preliminary understanding of "pigeon hole principle", we will use "pigeon hole principle" to solve simple practical problems. 2. Feel the charm of mathematics through the flexible application of pigeon hole principle.
Sixth, sorting out and reviewing
1, the system grasps the basic knowledge about integers, decimals, fractions and percentages, negative numbers, ratios and proportions, and equations. Can skillfully perform four operations of integer, decimal and fraction, can estimate the addition, subtraction, multiplication and division of integer and decimal, and can use the simple algorithm learned to calculate reasonably and flexibly; Be able to solve the learned equations; Get into the habit of checking.
2, consolidate the appearance of commonly used units of measurement, grasp the progress between the units, and can simply rewrite.
3. Master the characteristics of geometry; Skillfully calculate the perimeter, area and volume of some geometric shapes and apply them; Consolidate simple drawing and measuring skills; Consolidating the understanding of axisymmetric graphics will draw a symmetrical axis of graphics and consolidate the understanding of translation and rotation of graphics; The position of an object can be determined by several pairs or according to the direction and distance, and the knowledge of proportion can be mastered and applied.
4, master the preliminary knowledge of statistics, can read and draw simple statistical charts, can make simple judgments and predictions according to the data, can find the possibility of some simple events, and can solve some practical problems in calculating the average.
5. Further feel the interrelation between mathematical knowledge and understand the role of mathematics; Master the common quantitative relations and problem-solving thinking methods, and be able to flexibly use the knowledge learned to solve some simple practical problems in life.
(A) the number of reading and writing
1, integer reading: from high to low, read step by step. When reading the 110 million level, first read according to the reading method of the 100 million level, and then add a word "100 million" or "10 thousand" at the end. The zeros at the end of each stage are not read, and only a few zeros of other digits are read.
2. Writing of integers: from high to low, writing step by step. If there is no unit on any number, write 0 on that number.
3. Decimal reading method: When reading decimals, the integer part is read by integer reading method, the decimal point is read as "dot", and the decimal part reads the numbers on each digit from left to right in sequence.
4. How to write decimals: When writing decimals, the integer part is written as an integer, the decimal point is written in the lower right corner of each digit, and the decimal part is written on each digit in sequence.
5. How to read fractions: When reading fractions, read the denominator first, then the "fraction", and then the numerator. Both numerator and denominator read integers.
6. How to write the score: write the fractional line first, then the denominator, and finally the numerator and the integer.
7. Reading method of percentage: When reading percentage, read the percentage first, and then read the number before the percentage symbol. When reading, read by integers.
8. How to write the percentage: The percentage is usually not written as a fraction, but expressed by adding a percent sign "%"after the original molecule.
(2) The number of rewrites
In order to facilitate reading and writing, a large multi-digit number is often rewritten as a number in units of "10,000" or "100 million". Sometimes, if necessary, you can omit the number after a certain number and write it as an approximation.
1, exact number: in real life, for the convenience of counting, a larger number can be rewritten as a number in units of ten thousand or one hundred million. The rewritten number is the exact number of the original number. For example, 1254300000 is rewritten into ten thousand, and the number is125430000; Rewrite it to the number 12 in hundreds of millions. 54.3 billion.
2. Approximation: According to the actual needs, we can also use a similar number to represent a larger number and omit the mantissa after a certain number. For example: 13024900 15 The mantissa after omitting 100 million is1300 million.
3. Rounding method: If the highest digit of the mantissa to be omitted is 4 or less, the mantissa is removed; If the digit with the highest mantissa is 5 or more, the mantissa is truncated and 1 is added to its previous digit. For example, the mantissa after omitting 3.459 billion is about 350,000. After omitting 472509742 billion, the mantissa is about 4.7 billion.
4. Size comparison
(1) Comparing the sizes of integers: Comparing the sizes of integers, the number with more digits will be larger; If the digits are the same, look at the highest digit; The greater the number in the highest place, the greater the number; The number in the highest bit is the same. Just look at the next bit, and the bigger the number, the bigger it is.
(2) Compare the sizes of decimals: look at their integer parts first, and the bigger the integer part, the bigger the number; If the integer parts are the same, the tenth largest number is larger; One tenth of the numbers are the same, and the number with the largest number in the percentile is the largest. ...
(3) Comparing scores: the scores with the same denominator and the scores with large numerator are larger; For numbers with the same numerator, the score with smaller denominator is larger. If the denominator and numerator of a fraction are different, divide the fraction first, and then compare the sizes of the two numbers.
(3) the number of mutual
1, decimal component number: there are several decimals, so as long as you write a few zeros after 1 as the denominator and remove the decimal point after the original decimal point as the numerator, you can reduce the number of quotation points.
2. Fractions become decimals: numerator divided by 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. 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 to the right by two places, followed by hundreds of semicolons.
5. Decimal percentage: Decimal percentage, just remove the percent sign and move the decimal point two places to the left.
6. Convert fractions into percentages: usually, first convert fractions into decimals (three decimal places are usually reserved when they are not used up), and then convert decimals into percentages.
7. Decimal percentage: First rewrite the percentage into component numbers, and make a quotation that can be converted into the simplest fraction.
(4) Divisibility of numbers
1, decompose a composite number into prime factors, usually by short division. Divide this complex number by a prime number until the quotient is a prime number, and then write the divisor and quotient in the form of multiplication.
2. The way to find the greatest common divisor of several numbers is to divide the common divisors of these numbers continuously until the quotient obtained is only the common divisor of 1, and then multiply all the common divisors to get the product, which is the greatest common divisor of these numbers.
3. The method of finding the least common multiple of several numbers is: divide by the common divisor of these numbers (or part of them) until it is coprime (or pairwise coprime), and then multiply all the divisors and quotients to get the product, which is the least common multiple of these numbers.
4. Two numbers that become coprime relations: 1 and any natural number coprime; Two adjacent natural numbers are coprime; When the composite number is not a multiple of the prime number, the composite number and the prime number are coprime; When the common divisor of two composite numbers is only 1, these two composite numbers are coprime.
(5) Approximate points and general points
Reduction method: divide the denominator by the common divisor of the denominator (except 1); Usually, we have to separate it until we get the simplest score.
General division method: first find the least common multiple of the denominator of the original fraction, and then turn each fraction into a fraction with this least common multiple as the denominator.
decimal
1, the meaning of decimal
Divide the integer 1 into 10, 100, 1000 ... a tenth, a percentage, a thousandth ... can be expressed in decimals.
One decimal place indicates a few tenths, two decimal places indicate a few percent, and three decimal places indicate a few thousandths. ...
Decimal system consists of integer part, decimal part and decimal part. The point in the number is called the decimal point, the number to the left of the decimal point is called the integer part, and the number to the right of the decimal point is called the decimal part.
In decimals, the series between every two adjacent counting units is 10. The propulsion rate between the highest decimal unit "one tenth" of the decimal part and the lowest unit "one" of the integer part is also 10.
2. Classification of decimals
Pure decimals: Decimals with zero integer parts are called pure decimals. For example, 0.25 and 0.368 are pure decimals. With decimals: decimals whose integer part is not zero are called with decimals. For example, 3.25 and 5.26 are all decimals.
Finite decimals: The digits in the decimal part are finite decimals, which are called finite decimals. For example, 4 1.7, 25.3 and 0.23 are all finite decimals.
Infinite decimal: The digits in the decimal part are infinite decimal, which is called infinite decimal. For example: 4.33...3. 145438+05926 ...
Infinite acyclic decimal: the decimal part of a number with irregular arrangement and unlimited digits. Such decimals are called infinite cyclic decimals. For example: ∈
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 …12.15438+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". 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 ...
Mixed cycle decimal: the cycle section does not start from the first digit of the decimal part. This is called mixed cyclic decimal. 3. 1222 …… 0.03333 ……
When writing a cyclic decimal, for simplicity, the cyclic part of the decimal only needs one cyclic segment, and a dot is added to the first and last digits of this cyclic segment. If there is only one number in the circle, just click a point on it. For example: 3.777 ... 0.5302 ...
mark
1, the meaning of the score
Divide the unit "1" into several parts on average, and the number representing such a part or parts is called a fraction.
In the score, the middle horizontal line is called the dividing line; The number below the fractional line is called the denominator, indicating how many copies the unit "1" is divided into on average; The number below the fractional line is called the numerator, indicating how many copies there are.
Divide the unit "1" into several parts on average, and the number representing one part is called fractional unit.
2. Classification of scores
True fraction: The fraction with numerator less than denominator is called true fraction. The true score is less than 1.
False fraction: Fractions with numerator greater than denominator or numerator equal to denominator are called false fractions. False score is greater than or equal to 1. With fraction: False fraction can be written as a number consisting of integer and true fraction, which is usually called with fraction. 3 Reduction and comprehensive score
Changing a fraction into a fraction equal to it, but with smaller numerator and denominator, is called divisor. The denominator of a molecule is a fraction of a prime number, which is called simplest fraction.
Dividing the scores of different denominators by the scores of the same denominator equals the original score, which is called the total score.
4) Percentage
1, a number indicating that one number is a percentage of another number is called percentage, also called percentage or percentage. Percentages are usually expressed as "%". The percent sign is a symbol indicating percentage.
Proportion refers to two equal expressions called proportion. In proportion, the product of two external terms is equal to two internal terms. This is called the basic nature of proportion.
According to the basic nature of proportion, if any three items in the proportion are known, another unknown item in this proportion can be found. Finding the unknown term in the proportion is called the solution ratio.
For example: x: 320 =1:10x = 320×10x = 320 ÷10x = 32.
The knowledge of the second volume of mathematics in grade six
Second Unit Percentage 2
(1) Discount and percentage
1, discount: used for goods, the current price is a few percent of the original price, called discount.
Commonly known as "discount".
A few fold is a few tenths, that is, dozens of percent. For example, 20% = 8/ 10 = 20%,
50% off =6. 5/ 10=65/ 100=65﹪
The key to solving the discount problem is to first convert the discount number into percentage or fraction, and then solve it according to the problem-solving method of finding a percentage (fraction) of a number.
The goods are now 20% off: the current price is 20% off the original price.
The goods are now 50% off: the current price is 65% of the original price.
2, into the number:
A few percent is a few tenths, that is, dozens of percent. For example, ten percent =110 =10.
Eighty-five percent =8. 5/ 10=85/ 100=80﹪
To solve the problem of a number, the key is to first convert the number into percentage or fraction, and then solve it according to the method of finding more (less) numbers than the number.
The purchase price of clothes increased this time 10%: the purchase price of clothes increased this time 10%.
The wheat harvest this year is 85% of last year's.
(2), tax rate and interest rate
1, tax rate
(1) tax payment: tax payment is to pay a part of 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.
(3) Taxable amount: The tax paid is called taxable amount.
(4) Tax rate: The proportion of tax payable to various incomes is called tax rate.
(5) Calculation method of tax payable:
Taxable amount = total income × tax rate
Income = tax payable ÷ tax rate
2. Interest rate
(1) deposits can be divided into demand deposits and lump-sum deposits.
(2) The 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.
(3) Principal: The money deposited in the bank is called principal.
(4) Interest: The excess money paid by the bank when withdrawing money is called interest.
(5) Interest rate: The ratio of interest to principal is called interest rate.
(6) Interest calculation formula:
Interest = principal × interest rate× time
Interest rate = interest/time/principal × 100%
(7) Note: If you want to pay interest tax (interest on national debt and education savings is not taxed), then:
After-tax interest = interest-taxable interest amount = interest-interest x interest tax rate = interest x( 1- interest tax rate)
After-tax interest = principal × interest rate × time ×( 1- interest tax rate)
Shopping strategy:
Cost estimation: according to the actual problems, choose a reasonable estimation strategy and make an estimation.
Shopping strategy: according to the actual needs, analyze and compare several common preferential strategies, and finally choose the most favorable scheme.
Reflection after learning: the benefits of using strategies in doing things