Mathematics knowledge points in the second volume of the sixth grade 1
First unit negative number
1, the origin of negative numbers:
In order to express two quantities with opposite meanings (such as profit and loss, income and expenditure). ), learning 0 1 3.42/5 is not enough. So there is a negative number, the profit is positive and the loss is negative; Take income as positive and expenditure as negative.
2. Negative number: the number less than 0 is called negative number (excluding 0), and the number to the left of 0 on the number axis is called negative number.
If a number is less than 0, it is called a negative number.
There are countless negative numbers, including (negative integer, negative fraction, negative decimal)
Negative numbers are written as:
The number is preceded by a minus sign "-"and cannot be omitted.
For example: -2, -5.33, -45, -2/5.
Positive number:
Numbers greater than 0 are called positive numbers (excluding 0), and numbers to the right of 0 on the axis are called positive numbers.
If a number is greater than 0, it is said to be positive. There are countless kinds of positive numbers, including (positive integer, positive fraction and positive decimal)
How to write a positive number: you can add a plus sign before the number or omit it.
For example: +2, 5.33, +45, 2/5
4,0 is neither positive nor negative, it is the dividing line between positive and negative numbers.
6. Compare the size of two numbers:
(1) Use the number axis:
negative number
② Use the meaning of positive and negative numbers: the larger the positive number, the larger the number, and the smaller the number. Negative numbers are relatively large, the large number is small, and the small number is large.
Mathematics knowledge points in the second volume of the sixth grade
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.
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.
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-interest tax payable = interest-interest × interest tax rate = interest ×( 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
Mathematics knowledge point 3 in the second volume of the sixth grade
Unit 3 Cylinders and Cones
I. Cylinder
1. Formation of a cylinder: rotate one side of a rectangle as an axis to form a cylinder.
Cylinders can also be obtained by curling rectangles.
Two ways:
1. The perimeter with the length of the rectangle as the base, with the width as the height;
2. Take the width of the rectangle as the perimeter of the bottom and the length as the height.
Among them, the cylinder volume obtained by the first method is larger.
2. The height of a cylinder is the distance between two bottoms. A cylinder has countless heights, and their values are equal.
3, the characteristics of cylinder:
(1) Features of the bottom surface: The bottom surface of a 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.
4, cylinder cutting:
① Crosscutting: the cross section is circular, and the surface area is increased by 2 times of the bottom area, that is, S increase =2πr?
② Vertical cutting (over-diameter): the cross section is rectangular (if h=2R, the cross section is square), the length of the rectangle is the height of the cylinder, the width is the diameter of the bottom surface of the cylinder, and the surface area is increased by two rectangles, that is, the increase of S =4rh.
5, the side of the cylinder:
① Spread along the height, and the spread diagram is rectangular. If h=2πr, the spread diagram is square.
(2) Do not expand along the height, and the expanded figure is a parallelogram or irregular figure.
③ You can't get a trapezoid no matter how you unfold it.
6, cylinder related calculation formula:
Bottom area: s bottom =πr?
Bottom circumference: C bottom =πd=2πr
Transverse area: S side =2πrh.
Surface area: s table =2S bottom +S side =2πr? +2πrh
Volume: V column =πr? h
Common test questions:
① Knowing the bottom area and height of a cylinder, find the lateral area, surface area, volume and bottom perimeter of the cylinder.
② Knowing the circumference and height of the bottom surface of the cylinder, find the lateral area, surface area, volume and bottom area of the cylinder.
③ Knowing the circumference and volume of the bottom surface of the cylinder, find the lateral area, surface area, height and bottom area of the cylinder.
④ Knowing the area and height of the bottom surface of the cylinder, find the lateral area, surface area and volume of the cylinder.
⑤ Given the lateral area and height of the cylinder, find the radius, surface area, volume and bottom area of the cylinder.
The solution to the above common problems is usually to find the radius and height of the bottom of the cylinder, and then calculate it according to the relevant calculation formula of the cylinder.
Surface area of uncovered oil drum = side area+surface area of oil drum with one bottom area = 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+bottom area: glass, bucket, pen container, hat, swimming pool.
Side area+two bottom areas: oil barrel, rice barrel and tank.
Second, the cone
1, cone formation: rotate the right-angled side of the right-angled triangle as the axis to get the cone. A cone can also be obtained by sector curling.
2. The height of a cone is the distance between two vertices and the bottom. Unlike a cylinder, a cone has only one height.
3, the characteristics 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.
4, cone cutting:
① Crosscutting: the section is circular.
② Vertical cutting (passing through the apex and diameter): the cutting surface is an isosceles triangle, the height of which is the height of the cone, and the bottom is the diameter of the bottom of the cone, and the area is increased by two isosceles triangles.
That is, s increase =2rh.
5, cone related calculation formula:
Bottom area: s bottom =πr?
Bottom circumference: C bottom =πd=2πr
Volume: V cone = 1/3πr? h
Common test questions:
① Given the area and height of the bottom of the cone, find the volume and circumference of the bottom surface.
② Given the circumference and height of the bottom of the cone, find the volume and bottom area of the cone.
③ Given the circumference and volume of the bottom of the cone, find the height and bottom area of the cone.
The solution to the above common problems is usually to find the radius and height of the cone bottom, and then calculate it according to the relevant calculation formula of the cylinder.
Third, the relationship between cylinder and cone
1, the height of cylinder and cone is equal, and the volume of cylinder is three times that of cone.
2. The volume of cylinder and cone is the same, and the height of cone is three times that of cylinder.
3. The volume of cylinder and cone is very large, and the bottom area of cone (note: it is the bottom area rather than the bottom radius) is three times that of cylinder.
4. The cylinder and the cone have equal bottoms and heights, and the volume difference is 2/3Sh.
Problem summary
① Direct use formula: the surface area, lateral area, bottom area and volume can be clearly obtained through analysis.
Obviously, the change of radius leads to the change of bottom perimeter, lateral area, bottom area and volume.
Analyze the radius, bottom area, bottom perimeter, lateral area, surface area and volume ratio of two cylinders (or two cones).
② Transformation of the relationship between cylinder and cone: including the problem of cutting into maximum volume (between cube, cuboid and cylinder cone).
③ the problem of cross section
(4) Submerged volume: (The volume of the rising part of the water surface is the volume of the articles immersed in the water, which is equal to the bottom area of the water holding volume multiplied by the rising height) The volume is a cylinder or cuboid, a cube.
⑤ Equal volume conversion problem: melting a cylinder and making it into a cone, or pouring the solution in the cylinder into a cone, this is a problem with constant volume. Be careful not to multiply by 1/3.
The sixth grade, the second volume, mathematics knowledge points 4
Unit 4 Proportion
1 and the meaning of the ratio (1) The division of two numbers is also called the ratio of two numbers.
(2) "Bi:" is a comparative symbol, pronounced "Bi". The number before the comparison symbol is called the first item of comparison, and the number after the comparison symbol is called the last item of comparison. The quotient obtained by dividing the former term by the latter term is called the ratio.
(3) Compared with division, the former term of ratio is equivalent to dividend, the latter term is equivalent to divisor, and the ratio is equivalent to quotient.
(4) The ratio is usually expressed in fractions, decimals or even integers.
(5) The latter term of the ratio cannot be zero.
(6) According to the relationship between fraction and division, we can know that the former term of ratio is equivalent to numerator, the latter term is equivalent to denominator, and the ratio is equivalent to fractional value.
2. Basic properties of ratio: The first and second items of ratio are multiplied or divided by the same number (except 0) at the same time, and the ratio remains unchanged, which is called the basic properties of ratio.
3. Find the ratio and simplify the ratio:
The method of finding the ratio: divide the former term of the ratio by the latter term, and the result is that a numerical value can be an integer, a decimal or a fraction.
According to the basic properties of the ratio, the ratio can be reduced to the simplest integer ratio. Its result must be the simplest ratio, that is, the first term and the last term are prime numbers.
4. Proportional distribution:
In agricultural production and daily life, it is often necessary to allocate a quantity according to a certain proportion. This distribution method is usually called proportional distribution.
Methods: First, find out the scores of each part in the total, and then find out what the scores of the total are.
5. Meaning of proportion: Two expressions with equal proportion are called proportion.
The four numbers that make up a proportion are called proportional terms.
The two items at both ends are called external items, and the two items in the middle are called internal items.
6. Basic properties of proportion: In proportion, the product of two external terms is equal to the product of two internal terms. This is the basic nature of the so-called proportion.
7, the difference between ratio and proportion
The ratio of (1) indicates the division of two quantities, which has two terms (i.e. the former and the latter); Proportion refers to two formulas with equal proportions, which have four items (namely, two internal items and two external items).
(2) The ratio has basic properties, which is the basis of simplifying the ratio; Proportion also has a basic nature, which is the foundation of solution ratio.
8. Proportional quantity: two related quantities, one change and the other change. If the ratio (that is, quotient) of the corresponding two numbers in these two quantities is certain, these two quantities are called proportional quantities, and their relationship is called proportional relationship.
X/y=k (certain) is represented by letters.
9. Inverse proportional quantity: two related quantities, one change and the other change. If the product of the corresponding two numbers in these two quantities is certain, these two quantities are called inverse proportional quantities, and their relationship is called inverse proportional relationship.
X×y=k in the letter (sure).
10, a method for judging whether two quantities are directly proportional or inversely proportional:
The key is to see that the quotient of two relative numbers in these two related quantities must still be a product, and if the quotient is certain, it is proportional; If the product is constant, it is inversely proportional.
1 1. scale: the ratio of the distance on a picture to the actual distance is called the scale of this picture.
12, classification of scale
(1) Digital scale and line scale (2) Reduced scale and enlarged scale
13, distance on the map:
Map distance/actual distance = scale
Actual distance × scale = map distance
Distance on the map/scale = actual distance
14, application steps of scale drawing:
(1) Write the name of the graph,
(2) determine the scale;
(3) Calculate the distance on the map according to the scale;
(4) Drawings (unit length of drawings)
(5) Mark the actual distance and write down the place names.
(6) mark the scale
15. Magnification and reduction of graphics: same shape, different sizes.
16, solve the problem by proportion:
According to the invariants in the problem, find out two related quantities, correctly judge the proportional relationship between the two related quantities, and list the corresponding equations according to the positive and negative proportional relationship and solve them.
17. Common quantitative relations: (directly proportional or inversely proportional)
Unit price × quantity = total price
Single output × quantity = total output
Speed × time = distance
Efficiency × working hours = total amount of work
18、
Given the distance on the map and the actual distance, we can find the scale.
Given the scale and distance on the map, you can find the actual distance.
Given the scale and the actual distance, you can find the distance on the map.
When calculating, the units of drawing distance and real distance must be unified.
19, the total number of hectares sown is fixed. Is the number of hectares sown per day inversely proportional to the number of days to be used?
Answer: the number of hectares sown every day × days = the total number of hectares sown.
It is known that the total number of hectares sown is fixed, that is, the product of the number of hectares sown every day and the number of days to be used is fixed, so the number of hectares sown every day is inversely proportional to the number of days to be used.
Mathematics knowledge points in the second volume of the sixth grade 5
Unit 5 Mathematical Wide Angle-Pigeon Nest Problem
1 and pigeon's nest principle is an important and basic combination principle, which plays a very important role in solving mathematical problems.
② Using formulas to solve problems:
Number of objects = number of pigeons = quotient ... remainder
At least number = quotient+1
2. Calculation method of touching two identical colored balls.
(1) Make sure to touch two balls of the same color, and the number of balls touched is at least more than the number of colors 1.
Number of objects = number of colors × (at least-1)+ 1.
② Extreme thinking: First touch two balls of different colors with the most unfavorable touch method, and then no matter what color you touch, you can guarantee that there must be two balls of the same color.
③ Formula:
Two colors: 2+ 1=3 (pieces)
Three colors: 3+ 1=4 (pieces)
Four colors: 4+ 1=5 (pieces)
Sixth grade, the second volume of mathematics knowledge points induction related articles;
★ Summary of the knowledge points reviewed at the end of the sixth grade mathematics.
★ Essentials of Sixth Grade Mathematics Final Knowledge by People's Education Press (Volume II)
★ Summary of knowledge points that must be memorized in the second volume of sixth grade mathematics
★ Sort out and summarize the knowledge points of the first volume of mathematics in the sixth grade.
★ Summary of preliminary knowledge points of mathematical geometry in grade six
★ Summary of Mathematics Knowledge Points in the Sixth Grade of Primary School
★ Mathematics knowledge points for the first to sixth grades necessary for junior high school exams.
★ The arrangement of mathematics knowledge points in the first to sixth grades of junior high school
★ A Complete Collection of Mathematics Learning Methods and Skills in the Sixth Grade of Primary School
★ Statistical knowledge points of mathematics in sixth grade of primary school.