The first unit of the sixth grade mathematics volume is heavy and difficult.
1, the origin of negative numbers:
It is not enough to learn 0 1 3.4 2/5 in order to express two quantities with opposite meanings (such as profit loss and income and expenditure). 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:
Put a negative sign in front of the number? -? Number, can't 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 number 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: can you add a plus sign before the number? +? Number, can also be omitted.
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.
Negative numbers are less than 0, positive numbers are greater than 0, negative numbers are less than positive numbers, and positive numbers are greater than negative numbers.
5. Number axis:
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.
1/3 > 1/6- 1/3 & lt； - 1/6
The second unit of the sixth grade mathematics volume is difficult and difficult.
(1) Discount and percentage
1, discount: used for goods, the current price is a few percent of the original price, called discount. Common name? Discount? .
A few fold is a few tenths, that is, dozens of percent. For example, 20% = 8/ 10 = 20%,
65% discount = 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:
Interest after tax = interest-taxable interest amount = interest-interest? Interest tax rate = interest? (1- interest tax rate)
Interest after tax = 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
The third unit of the sixth grade mathematics volume is difficult and difficult
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 the bottom area, that is, the S increase =2? r？
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) expands along the height, and the expansion diagram is rectangular. If h=2? R, the expanded figure is a 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: bottom =? r？
Bottom circumference: C bottom =? d=2？ r
Lateral area: S side =2? right hand
Surface area: s table =2S bottom +S side =2? r？ +2? right hand
Volume: V column =? r？ 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: bottom =? r？
Bottom circumference: C bottom =? d=2？ r
Volume: V cone = 1/3? r？ 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.
? What are the difficulties in the second volume of sixth grade mathematics?