1. Reading and writing methods of positive numbers and negative numbers:

① When writing positive numbers, you can add+or omit+,but when reading positive numbers, if you add+,you must read orthography; If the+sign is omitted, orthography should also be omitted.

(2) When writing negative numbers, be sure to write numbers, and when reading, be sure to read negative numbers.

(2)0 is neither positive nor negative, it is the dividing point between positive and negative numbers.

2. The straight line that can represent positive numbers, zero numbers and negative numbers is called the number axis.

3. The concept of (1) number axis: The straight line defining the origin, positive direction and unit length is called number axis.

(2) A thermometer can also be regarded as a number axis.

4.( 1) On the number axis, the order from left to right is the order of numbers from small to large.

(2) All negative numbers are on the left of 0, that is, all negative numbers are less than 0; All positive numbers are on the right of 0, that is, positive numbers are greater than 0. So negative numbers are smaller than positive numbers.

(3) Comparing the sizes of two negative numbers, you can first compare the sizes of two corresponding positive numbers, and the corresponding positive numbers are larger and the negative numbers are smaller.

5. Tip: When water freezes, the temperature is 0 degrees Celsius. The meaning of 0 here is not nothing, but a specific number.

6. Tips: When using positive numbers and negative numbers to represent quantities with opposite meanings, you should first indicate which quantity is positive (or negative). If the rise is represented by a positive number, then the decline must be represented by a negative number.

Unit 2: Cylinders and Cones

1. A cylinder consists of two bottom faces and one side face.

2.( 1) The two circular surfaces of a cylinder are called the bottom surface.

(2) Name of the bottom part: The center, radius, diameter and perimeter of the bottom circle of a cylinder are called the center, radius, diameter and perimeter of the bottom respectively.

(3) Bottom features: The bottom of the cylinder is two identical circles.

3.( 1) The curved surface around the cylinder is called the side surface.

(2) Features: The side of the cylinder is curved.

4.( 1) The distance between the two bottom surfaces of a cylinder is called the height of the cylinder.

(2) A cylinder has countless heights.

5. Cut the cylinder parallel to the bottom surface, and the section is two circles with the same size as the bottom surface; Cut the cylinder perpendicular to the bottom surface along the diameter of the bottom surface, and the cross section is two identical rectangles.

6. The side of the cylinder is unfolded into a rectangle, the length of which is equal to the circumference of the bottom of the cylinder, and the width is equal to the height of the cylinder.

7. Take any point on the circumference of the upper and lower bottom surfaces of the cylinder as A and B respectively, connect AB (so that AB is not the height of the cylinder), and cut the side surface of the cylinder along AB. The cylinder is a parallelogram when unfolded.

8. Tip: The bottom of a cylinder is round, but the surface is not oval.

9. Tips: When cutting along the height, the side development of the cylinder is rectangular.

10. From the top and bottom of the cylinder, you will get a circle; Seen from the front or side of the cylinder, you will get a rectangle (or square).

1 1. If the side of the cylinder is unfolded into a rectangle, the circumference of the bottom surface of the cylinder is about 3 times the diameter of the bottom surface. If the side of a cylinder is square, the height of the cylinder is about 3 times the diameter of its bottom surface.

12. Side area of cylinder = perimeter of bottom surface? Tall man. If the letter S represents the lateral area of a cylinder, C represents the circumference of the bottom and H represents the height, then the formula for calculating the lateral area of a cylinder is S=Ch.

13.( 1) Given the diameter and height of the cylinder bottom, we can use the formula: S=? Dh directly calculates the side area of the cylinder.

(2) Given the radius and height of the cylinder bottom, we can use the formula: S=2? Rh directly calculates the side area of the cylinder.

14. The surface area of a cylinder refers to the sum of the side area of the cylinder and its two bottom areas.

15. Surface area of cylinder = side area of cylinder+bottom area? 2, expressed in letters as S table =S side +2S bottom.

16.( 1) Given the radius and height of the cylinder bottom, we can use the formula: s table =2? rh+2？ R2 directly calculates the surface area of the cylinder.

(2) When the diameter and height of the bottom surface of the cylinder are known, the surface area of the cylinder can be calculated according to the formula: s table =? dh+？ (d？ 2)2 directly calculate the surface area of the cylinder.

(3) Given the circumference and height of the bottom surface of the cylinder, the surface area of the cylinder can be calculated according to the formula: s table =Ch+? (C/2？ )2=Ch+C2/4？ Find the surface area of a cylinder.

17. Tip: Finding the surface area of cylindrical objects such as ventilation pipes, chimneys and oil pipes is actually finding their lateral area.

18. Tip: After the cylinder is cut into n sections, its surface area increases by 2(n- 1) base areas.

19. The space occupied by a cylinder is called the volume of the cylinder.

20. The volume of the cylinder = the bottom area? Height, letter formula: V=Sh or V=? r^2h

2 1. Tips: The calculation method of volume is the same as that of volume, except that the data for calculating volume should be measured from the inside.

22. In the calculation process, if the radius, diameter or circumference of the bottom surface of a cylinder is known, then the bottom area should be calculated first, and then the volume should be calculated. The calculation formula is: V=? r^2h,V=？ (d？ 2)^2h,V=？ 【C？ (2? )]^2h

23. Tips: The height of the cylinder is constant, the radius, diameter or perimeter of the bottom surface is enlarged by n times, the volume is enlarged by n times, the radius, diameter or perimeter of the bottom surface is reduced by 1/n, and the volume is reduced by1/(n 2).

24. Tips: In the perspective view of a cylinder, the distance between the centers of two bottom surfaces is the height of the cylinder, but in the plane development view of the cylinder, the width of a rectangle (or the side length of a square) is the height of the cylinder.

25. The radius ratio of two cylinders is1:a (a >; 0), the height ratio is a: 1, so their volume ratio is1:a.

26. A cone consists of a bottom surface and a side surface.

(1) Bottom surface: The circular surface of a cone is its bottom surface, and it has a bottom surface. The center, radius, diameter and perimeter of the cone bottom are called the center, radius, diameter and perimeter of the cone bottom, which are represented by the letters O, R, D and C respectively.

(2) Side: The curved surface around the cone is its side.

(3) Height: The distance from the apex of the cone to the center of the bottom surface is the height of the cone. The height is indicated by the letter H.

(4) The cone has only one height.

(5) A right triangle can be rotated to form a cone.

27. Tips:

(1) The line segment from the apex of the cone to any point on the circumference of the bottom surface is the generatrix of the cone, and the length of the generatrix of the cone is greater than the height of the cone.

(2) Draw a generatrix at will, and expand the side of the cone to get a fan shape, then the side development diagram of the cone is a fan shape.

(3) cutting the cone parallel to the bottom surface, and the section is two identical circles, which are smaller than the bottom surface circle of the cone; Cut the cone perpendicular to the bottom along the height, and the cross section is two identical isosceles triangles.

28. Tip: A semicircle can form a cone, but a whole circle cannot form a cone.

29. The volume of the cone = the bottom area? Tall? 3, expressed in letters: V cone =V column? 3=Sh？ three

30. The relationship between cylinder and cone:

(1) Cylinders and cones with equal bottoms and equal heights: the volume of a cylinder is twice that of a cone; The volume of a cone is two thirds smaller than that of a cylinder.

(2) Cylinders and cones with equal bottoms and equal heights: the height of a cone is three times that of a cylinder, or twice that of a cylinder; The height of the cylinder is 65438+ 0/3 of the height of the cone, or the height of the cylinder is less than 2/3 of the height of the cone.

(3) Cylinders and cones with equal volumes: the bottom area of a cone is 3 times that of a cylinder, or 2 times that of a cylinder; The bottom area of a cylinder is 1/3 of a cone, or the bottom area of a cylinder is 2/3 smaller than that of a cone.

3 1. Tips:

(1) Given the radius and height of the cone bottom, we can directly use the formula: V=? r^2h？ 3 Find the volume of the cone.

(2) Given the diameter and height of the cone bottom, we can directly use the formula: V=? (d？ 2)^2h？ 3 Find the volume of the cone.

(3) Given the circumference and height of the cone bottom, we can directly use the formula: V=? (C？ 2)^2h？ 3 Find the volume of the cone.

32. use V=Sh? When calculating the volume of a cone, don't forget to divide by 3 or multiply by 1/3.

33. Tips: The volume of a cylinder is three times that of a cone, or the volume of a cone is 1/3 of that of a cylinder. It must be based on the premise that the height of cylinder and cone is equal.

34. Of the two cones that rotate around the right-angled side of a right-angled triangle, the cone that rotates around the shorter right-angled side has a larger volume.