The sixth grade mathematics knowledge Volume II Beijing Normal University Edition 1
1, the relationship between "point, line, surface and body" is:
The motion of the point forms a line; The movement of the line forms the surface; The rotation of the surface forms a body.
2, the characteristics of cylinder:
(1) The two bottom surfaces of a cylinder are two circles with the same radius, and the side surfaces are curved surfaces.
(2) The distance between the two bottoms is called the height of the cylinder.
(3) A cylinder has countless heights, all of which are of equal length.
(4) A cylinder is a cube obtained by rotating a rectangle around its length or width by 360 degrees, so the section after cutting along the height line is rectangular.
3, the characteristics of the cone:
(1) The base of a cone is a circle, and there is a vertex at the position opposite to the base.
(2) The side of the cone is a curved surface.
(3) The cone has only one height.
(4) The cone is a cube obtained by rotating a right-angled triangle around a right-angled side by 360 degrees, so the section cut along the height line is an isosceles triangle.
4. Cut along the height of the cylinder, and the side development diagram of the cylinder is rectangular (or square) (if it is not cut along the height, it may be parallelogram).
Lateral area of cylinder = perimeter of bottom × height, expressed in letters: S side =Ch.
Application of the formula of cylinder side area;
(1) Given the perimeter and height of the bottom surface, the side surface area can be calculated by the formula: S side = ch.
(2) When the diameter and height of the bottom surface are known, the lateral area can be obtained by using the formula: s side = π DH;
(3) When the radius and height of the bottom surface are known, we can use the formula: S-side =2πrh to calculate the side surface area.
Calculation method of cylinder surface area: If the side surface area of the cylinder is represented by S side, the bottom of S represents the bottom area, D represents the bottom diameter, R represents the bottom radius and H represents the height, then the surface area of the cylinder is: S table =S side +2S bottom or S table =πdh+πd2/2 or S table =2πrh+2πr2.
Special application of calculation method of cylindrical surface area;
(1) The surface area of a cylinder only includes the side area and a cylindrical object with a bottom area, such as a bucket without a cover.
(2) The surface area of a cylinder only includes the side area, such as a chimney, an oil pipe and other cylindrical objects.
5. Volume of cylinder: the size of space occupied by cylinder.
6. Derivation of cylindrical volume formula;
Deduction of the area formula of the circle reviewed in the sixth grade: The more copies of the circle are equally divided, the closer the figure is to a parallelogram or rectangle. The base of the parallelogram is equivalent to half the circumference of the circle, and the height is equivalent to the radius of the circle; The length of a rectangle is equal to half the circumference and the width is equal to the radius of the circle. So the area of a circle = π× radius× radius = π× radius 2.
Just like the derivation of the area formula of a circle, you can also cut the cylinder along the fan shape at the bottom of the cylinder and the height of the cylinder, and divide it into several equal parts. The finer the better. Then, piece it together into a three-dimensional figure similar to a cuboid. The shape has changed, but the volume has not changed, so you can find that the bottom area of the assembled cuboid is equal to the bottom area of the cylinder, the height of the cuboid is also equal to the height of the cylinder, and the volume of the cuboid = bottom area × height. Therefore,
Volume of cylinder = bottom area × height If V stands for volume of cylinder, S stands for bottom area and H stands for height, then V=Sh.
Example: Fill in the blank: The derivation process of cylindrical volume formula is based on the mathematical idea of (transformation), in which (shape) has changed and (volume) has not changed. The number is higher than the cylinder (height) and its bottom area is equal, so the volume formula of the cylinder is (bottom area × height).
Application of cylindrical volume formula;
(1) When calculating the volume of a cylinder, if the bottom area and height are given in the question, you can use the formula: V=Sh.
(2) Given the radius and height of the bottom surface of the cylinder, the volume can be calculated by the formula: v = π r2h;
(3) Given the diameter and height of the cylinder bottom, the volume can be calculated by the following formula: v = π (d/2) 2h;
(4) Given the circumference and height of the cylinder bottom, the volume can be calculated by the following formula: v = π (c/2π) 2h;
The volume of cylindrical container = bottom area × height, which is V=Sh in letters.
6. The application of cylindrical container formula is the same as that of cylindrical volume formula.
7. The volume of the cone: the size of the space occupied by the cone.
The volume of the cone = 1/3× bottom area× height If V is used to represent the volume of the cone, S is the bottom area and H is the height,
The letter formula is: 1/3Sh.
Application of cone volume formula;
(1) When calculating the volume of a cone, if the bottom area and height are given in the question, you can directly use the formula "v= 1/3Sh".
(2) When calculating the volume of the cone, if the radius and height of the bottom surface are given in the question, 1/3πr? h
(3) When calculating the volume of a cone, if the diameter and height of the bottom surface are given in the question, you can use 1/3π(d/2)? h
(4) When calculating the volume of a cone, if the perimeter and height of the bottom are given in the question, you can use 1/3π(c/2r)? h
Beijing normal university printing plate second grade mathematics knowledge volume 2
1 means that two ratios are equal, which is called ratio.
Such as: 3: 4 = 9: 12.
2. There are four items in the proportion, namely two internal items and two external items.
In 3: 4 = 9: 12, 3 and 12 are called external proportional terms, and 4 and 9 are called internal proportional terms. None of the four digits of the ratio can be 0.
3. Basic properties of proportion: In a proportion, the product of two external terms is equal to the product of two internal terms.
4. Scale: The ratio of the distance on the map to the actual distance is called the scale of this map.
Distance on the map ÷ actual distance = out of scale
Map distance = actual distance × scale
Actual distance = distance on map/scale.
5. Classification of scale:
Scale is divided into reducing scale (scale; 1)。
According to different forms of expression, scale can also be divided into line scale and numerical scale.
6. Scaling of graphics: When a picture is enlarged or reduced, it can only be similar if it is drawn in the same scale.
Beijing normal university printing plate sixth grade mathematics knowledge volume 3
Unit 3 the movement of graphics
The knowledge of graphic transformation in this volume is further deepened on the original basis, and it is required to draw translation, rotation and axisymmetric graphics on grid paper, specifically:
First turn: indicate which point to turn around, clockwise or counterclockwise, and how many degrees to turn (90 degrees, 180 degrees, 270 degrees).
For example, rotate graph B clockwise/counterclockwise by 90 degrees around point O to get graph C;
Direction of rotation around the center point:
Clockwise: that is, in the direction of the clock, from top to bottom, then down and finally up.
Counter-clockwise: opposite to clockwise, from top to left, then down, and finally up.
The second translation: several translations about the direction of the Ming Dynasty (up, down, left and right).
For example, the graph A is translated up/down/left/right by 4 squares to get the graph B;
The third is symmetrical figure: it is necessary to explain which straight line is the symmetrical figure of which figure.
For example, taking the straight line MN as the axis of symmetry, an axisymmetric figure D of Figure C is drawn.
Unit 4 Positive and Inverse Proportion
1, there are a lot of interdependent variables in life, one quantity changes, and the other quantity changes accordingly.
2. Proportion:
Two related quantities, one of which changes and the other changes with it. If the ratio 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.
If the letters X and Y are used to represent two related quantities and the letter K is used to represent their ratio (certain), the positive proportional relationship can be expressed as: y/x=k (certain).
Judging whether two quantities are proportional: although some related quantities change with the change of another quantity, the ratio of their corresponding numbers is not necessarily proportional, such as the sum and difference of the minuend, the area and side length of the square, etc.
The proportional image is a straight line.
3, the meaning of inverse proportion:
Two related quantities, one variable and the other variable. 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.
If the letters X and Y are used to represent two related quantities and K is used to represent their products, the inverse relationship can be expressed as: X Y = K (certain).
To judge whether two quantities are inversely proportional: first, consider whether these two quantities are related; See if the product of these two quantities is determined; Finally draw a conclusion.
The inverse scale image is a smooth curve.
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