I. Fractional multiplication and fractional division
1.
The significance of fractional multiplication: a simple operation to find the sum of several identical fractions
2.
Significance of fractional division: Know the product of two multipliers and one of them, and find the operation of the other multiplier.
3.
Algorithm of fractional multiplication:
( 1)
Fraction and integer multiplication: numerator and integer multiplication, denominator unchanged. For example:
(2)
Fraction multiplied by fraction: the product of numerator multiplied by numerator is numerator, and the product of denominator multiplied by denominator is denominator. What can be reduced is reduced first. Such as;
4.
Algorithm of fractional division:
( 1)
A number divided by an integer (except 0) is equal to this number multiplied by the reciprocal of this integer.
For example:
(3)
A number divided by a fraction is equal to the number multiplied by the reciprocal of the fraction.
For example:
(4)
Dividing by a number (except 0) is equal to multiplying the reciprocal of this number.
5.
If the product of two numbers is 1, then we call one of them the reciprocal of the other. For example, the reciprocal of 1/2 is 2, and the reciprocal of 2 is 1/2. These two numbers are reciprocal. The reciprocal of 1 is 1, and 0 has no reciprocal.
6.
Practical problems of fractional multiplication and division
( 1)
Find the fraction of a number and multiply it.
For example, how much is three quarters of five? 5×
=
(2) Know the fraction of a number, find this number, divide it, or solve the equation.
For example:
It is known that 3/7 of a number is 15. What's this number?
15÷
Second, the mixed operation of fractions
1.
The order of fractional mixing operation is the same as that of integer mixing operation: multiply first, then divide, and then add and subtract. If there are brackets, the ones inside the brackets are counted first, and then the ones outside the brackets are counted.
2.
Operating rules:
(1) Multiplicative Distribution Law:
(2) Multiplicative associative law:
(3) Multiplicative commutative law:
Using the algorithm, you can simply perform the mixed operation of fractions.
Third, the understanding, surface area, volume and volume of cuboids.
1.
A cuboid has six faces, generally rectangular (in special cases, two opposite faces are square), and the areas of the opposite faces are equal; There are 8 vertices, 12 edges, and 12 edges can be divided into 3 groups: 4 long, 4 wide and 4 high.
2.
A cube has six faces, all of which are squares with equal areas; There are 8 vertices, 12 sides, and each side has the same length.
3.
Cubes are special cuboids. (Length, width and height are equal)
4.
Sum of sides of a cuboid = (length+width+height) ×4
5.
Sum of cube sides = side length × 12
6.
The total area of six faces of a cuboid is called its surface area. The areas of two opposite sides of a cuboid are equal, and the areas of the front and back sides are equal to length × height; Left and right area = width × height;
Up and down area = length × width
7.
Surface area of cuboid = length × width ×2
+Length× Height× 2
+Width× Height× 2
s=a×b×2
+a×h×2+b
× height× 2
8.
The total area of six faces of a cube is called its surface area, and the areas of all six faces are equal.
9.
Surface area of cube = side length × side length ×6
10.
The size of the space occupied by an object is called its volume. Commonly used unit of volume are: cubic centimeter, cubic decimeter and cubic meter.
1 m3 = 1000 cubic decimeter
1 cubic decimeter = 1000 cubic centimeter
1 m3 = 1000000 cubic centimeters
1 1.
The volume of the object that a container can hold is called the volume of the container. The commonly used unit of volume are liters and milliliters.
1L = 1000ml
1 l = 1 cubic decimeter
1 ml = 1 cm3
12.
Interaction between adjacent volume units
Low-level unit
Advanced unit
13.
Unit of volume is used to calculate the volume of objects, and unit of volume is generally used to calculate the volume of liquids and gases.
14.
Cuboid volume = length × width × height
15.
Volume of cube = side length × side length × side length
16.
Volume of cuboid (cube) = bottom area × height
Four. per cent
1.
Percent indicates the percentage of one number to another. Percentage is also called percentage, percentage.
Writing 22%, reading 22%
2.
Exchange of percentages and decimals:
(1) Decimal percentage: move the decimal point to the right by two places and add a percentage sign. For example, 0.5=50%
(2)
Decimal Percentage: Delete the percent sign and move the decimal point before the percent sign to the left by two places.
For example: 234%=2.34
3.
Conversion between percentages and fractions:
( 1)
Fractional percentage: divide the numerator by the denominator, and then divide the quotient by the percentage. Or the decimal part is a fraction with the letter 100, and then rewritten as a percentage.
For example:
(2) Percentage score: write the percentage as a score, and the denominator is 100, and the quotation that can be reduced is divided into the simplest score.
For example:
4.
Excellent rate = excellent number ÷ total number.
Pass rate = number of people who passed/total number of people.
Qualified rate = number of qualified products ÷ total number of products.
Attendance = Attendance ÷ Total number of people
Hit ratio = number of hits ÷ total times
Germination rate = number of germinated seeds/total number of seeds.
Survival rate = number of trees survived/total number of trees planted.
Flour yield = flour weight/wheat weight.
Oil yield = weight of pressed oil-weight of peanut kernel.
Verb (abbreviation for verb) statistics
1.
Bar charts can clearly show the quantity of various kinds of land for comparison.
2.
The fan-shaped statistical chart can intuitively show how much each quantity accounts for the total amount.
3.
The broken-line statistical chart can display the change of quantity intuitively.
4.
Average value = total quantity/total number of copies
5.
Arrange a set of data from small to large (or from large to small), and the middle number is called the median of this set of data.
6.
The number that appears most frequently in a set of data is called the mode of this set of data.