Quantitative measurement

A unit of length is used to measure the length of an object. Commonly used length units are: kilometers, meters, decimeters, centimeters and millimeters.

? Second, the length unit:

1 km = 1 000m1m = 10 decimeter.

1 decimeter = 1 0cm1cm =10mm

1 m = 100 cm 1 m = 1000 mm

Third, the area unit is used to measure the size of the surface or plane figure of an object. Public area units: square kilometers, hectares, square meters, square decimeters and square centimeters.

Four, the calculation of land area, usually in hectares. The square with a side length of 100 meters covers an area of 1 hectare.

Five, measuring a large area of land, usually in square kilometers. Square land with side length 1000 m, area 1 km2.

Area unit of intransitive verbs: (100)

1 km2 = 1 00ha1hectare =10000m2

1 m2 = 100 square decimeter 1 square decimeter = 100 square centimeter

Unit of volume is used to measure the space occupied by objects. Commonly used unit of volume are: cubic meter, cubic decimeter (liter) and cubic centimeter (milliliter).

Eight. Unit of volume: (1000)

1 m3 = 1000 cubic decimeter

1L = 1000ml

Understanding, perimeter and area of plane graphics

First, connect two points with a ruler to get a line segment; Extending one end of the line segment indefinitely can get a ray; Extend both ends of a line indefinitely and you can get a straight line. Line segments and rays are both parts of a straight line. A line segment has two endpoints and its length is limited. A ray has only one endpoint, a straight line has no endpoint, and both rays and straight lines are infinitely long.

Second, two rays from a point form an angle. The size of the angle is related to the size of both sides, and has nothing to do with the length of the sides. The unit of measurement of angle size is ().

Third, the classification of angles: angles less than 90 degrees are acute angles; An angle equal to 90 degrees is a right angle; An angle greater than 90 degrees and less than 180 degrees is an obtuse angle; The angle equal to 180 degrees is a flat angle; An angle equal to 360 degrees is a fillet.

Four, two straight lines intersecting at right angles are perpendicular to each other; Two straight lines that do not intersect in the same plane are parallel to each other.

5. A triangle is a figure surrounded by three line segments. Each line segment constituting a triangle is called an edge of the triangle, and the intersection of every two line segments is called the vertex of the triangle.

6. Triangle can be divided into acute triangle, right triangle and obtuse triangle according to angle.

According to different sides, it can be divided into equilateral triangle, isosceles triangle and arbitrary triangle.

7. The sum of the internal angles of a triangle is equal to 180 degrees.

8. In a triangle, the sum of any two sides is greater than the third side.

Nine, in a triangle, there is at most one right angle or at most one obtuse angle.

X. A quadrilateral is a figure surrounded by four sides. Common special quadrangles are parallelogram, rectangle, square and trapezoid.

Xi。 A circle is a curved figure. The distance from any point on the circle to the center of the circle is equal, and this distance is the length of the radius of the circle. The line segment passing through the center of the circle with both ends in the circle is called the diameter of the circle.

Twelve, there are some graphics, folded in half along a straight line, and the graphics on both sides of the straight line can completely overlap. This graph is an axisymmetric graph. This straight line is called the axis of symmetry.

Thirteen, the sum of all the edges of a figure is the perimeter of the figure.

Fourteen, the size of the surface of the object or the closed plane figure is called their area.

Fifteen. Derivation of calculation formula for plane figure area;

The derivation process of 1 parallelogram area formula

① A parallelogram can be transformed into a rectangle by cutting and translating.

② The length of rectangle is equal to the base of parallelogram, the width of rectangle is equal to the height of parallelogram, and the area of rectangle is equal to the area of parallelogram.

③ Because: rectangular area = length× width, so: parallelogram area = bottom× height. Namely: S=ah.

2 Derivation process of triangle area formula

① Two identical triangles can form a parallelogram.

② The base of a parallelogram is equal to the base of a triangle, the height of a parallelogram is higher than that of a triangle, and the area of a triangle is equal to half the area of a parallelogram with equal base and equal height.

(3) Because the parallelogram area = base × height, and the triangle area = base × height ÷2. Namely: S=ah÷2.

3 Derivation process of trapezoidal area formula

① Two identical trapezoids can be used to form a parallelogram.

② The base of parallelogram is equal to the sum of the upper and lower base of trapezoid, the height of parallelogram is higher than that of trapezoid, and the area of trapezoid is equal to half of that of parallelogram.

③ Because: parallelogram area = bottom × height, so: trapezoid area = (upper bottom+lower bottom) × height ÷2. Namely: S=(a+b)h÷2.

Draw a picture to illustrate the derivation process of the formula of circular area

① Divide the circle into several equal parts, cut it open and spell it into an approximate rectangle.

② The length of a rectangle is equivalent to half of the circumference, and the width is equivalent to the radius of the circle.

③ Because: rectangular area = length× width, so: circular area =πr×r=πr2. Namely: S=πr2

Sixteen, the perimeter and area calculation formula of plane graphics:

Rectangular perimeter = (length+width) × 2

Rectangular area = length × width

Square perimeter = side length × 4

Square area = side length × side length

Parallelogram area = base × height

Triangle area = base × height ÷ 2

Cognition, perimeter and area of three-dimensional graphics

Cuboids and cubes have 6 faces, 12 sides and 8 vertices. Cubes are special cuboids.

Second, the characteristics of the cylinder: one side, two bottom surfaces, and countless heights.

Third, the characteristics of the cone: a side, a bottom, a vertex and a height.

4. Surface area: The sum of all the areas of a three-dimensional figure is called the surface area of this three-dimensional figure.

Verb (abbreviation of verb) Volume: The size of the space occupied by an object is called the volume of the object. The volume that a container can hold other objects is called the volume of the container.

Six, three relationships between cylinder and cone:

? ① Equal bottom and equal height: volume 1︰3?

? ② Equal bottom and equal volume: height 1︰3?

? ③ Equal contour volume: the bottom area is 1︰3.

Seven, equal bottom and equal height cylinders and cones:

① The volume of a cone is 1/3 of that of a cylinder,

② The volume of a cylinder is three times that of a cone,

③ The volume of the cone is 2/3 smaller than that of the cylinder,

(4) The volume of a cylinder is twice as large as that of a cone.

8. Cylinders and cones with equal bottoms and equal heights: cone 1, difference 2, columns 3 and 4.

Nine, three-dimensional graphics formula derivation:

1 What figure do you get when the side of the cylinder is unfolded? What is the relationship between each part of this figure and the cylinder? (Derivation process of cylindrical lateral area formula)

① A rectangle is generally obtained when the side of a cylinder is unfolded. ?

② The length of the rectangle is equivalent to the circumference of the bottom of the cylinder, and the width of the rectangle is equivalent to the height of the cylinder.

③ Because: rectangular area = length × width, so: cylindrical lateral area = bottom circumference × height.

④ A square may be obtained when the side of the cylinder is unfolded.

The side length of a square = the circumference of the bottom of the cylinder = the height of the cylinder.

When we study the formula for calculating the volume of a cylinder, we can deduce it by converting the cylinder into a three-dimensional figure (approximation) that we have learned before. Please tell me the name of this three-dimensional figure and its relationship with the relevant parts of the cylinder.

(1) Divide the cylinder into several equal parts, cut it and put it together into an approximate cuboid.

② The bottom area of the cuboid is equal to that of the cylinder, and the height of the cuboid is higher than that of the cylinder.

③ Because cuboid volume = bottom area × height, cylinder volume = bottom area × height. Namely: V=Sh.

Please draw a picture to illustrate the derivation process of cone volume formula.

① Find an empty cone and an empty cylinder with equal bottom and height.

(2) Fill the cone with sand, pour it into the cylinder, and find that it is filled just three times. Pour the sand in the jar into the cone and find that it has just been poured out three times.

③ It is found through experiments that the volume of a cone is equal to one third of the volume of a cylinder with equal bottom and equal height; The volume of a cylinder is equal to three times the volume of a cone with equal bottom and equal height. Namely: V= 1/3Sh.

X. formula for calculating the sum of side length, surface area and volume of three-dimensional graphics:

Name calculation formula

Sum of sides of a cuboid = (length+width+height) × 4

Cuboid surface area cuboid surface area = (length× width+length× height+width× height) ×2

Cuboid volume cuboid volume = length × width × height

Sum of cube sides = side length × 12

Cubic surface area Cubic surface area = side length × side length ×6

Cubic volume Cubic volume = side length × side length× side length

Side area cylinder side area = bottom circumference x height.

Cylinder surface area Cylinder surface area = side surface area+bottom area ×2

Cylinder volume cylinder volume = bottom area × height

Conical volume Conical volume = 1/3SH

(b) graphics and conversion

First, the methods to change the position of graphics are translation, rotation and so on. When changing the position, the vertices, line segments and curves corresponding to each figure should be synchronously translated and rotated by the same angle.

2. Without changing the shape of a graph, only changing its size usually makes the elements of each graph, such as the length and width of a rectangle and the bottom and height of a triangle, enlarge or shrink at the same time.

3. Symmetric figures mean that the figures on both sides of the symmetry axis can completely overlap after being folded in half, but not exactly the same.

(3) graphics and location

First, in real life and situations, when facing the short distance of teaching, we usually use up, down, front and back to describe the specific location.

Second, when we face maps and orientation maps, we usually use east, west, south, north, south, east and north to describe the direction. Combined with the displayed scale, the specific distance is calculated, and the position is determined by combining the direction and distance.