1, decimal multiplication integer (p2,3): meaning-a simple operation to find the sum of several identical addends.
For example, 1.5×3 indicates how many times 1.5 is or the sum of three 1.5.
Calculation method: first expand the decimal into an integer; Calculate the product according to the law of integer multiplication; Look at a factor * * *, how many decimal places there are, and count the decimal points from the right side of the product.
2. Decimal times decimal (P4, 5): that is, what is the score of this number.
For example, 1.5×0.8 is to find what is eight tenths of 1.5.
How much is 1.5× 1.8? It is 1.8 times 1.5.
Calculation method: first expand the decimal into an integer; Calculate the product according to the law of integer multiplication; Look at the number of decimal places in factor one * * *, and count the decimal points from the right side of the product.
Note: In the calculation results, the 0 at the end of the decimal part should be removed to simplify the decimal; When the number of decimal places is not enough, use 0 to occupy the place.
3. Rule (1)(P9): the product of a number (except 0) multiplied by a number greater than 1 is greater than the original number;
A number (except 0) is multiplied by a number less than 1, and the product is less than the original number.
4. There are generally three methods to find the divisor: (P 10)
(1) rounding method; (2) into law; (3) Tailing method
5. Calculate the amount of money and keep two decimal places, indicating that the calculation is completed. Keep one decimal place, indicating that the angle has been calculated.
6. The operation of (p11) four decimal places is the same as that of an integer.
7, operation law and nature:
Addition: additive commutative law: a+b=b+a Addition Law: (a+b)+c=a+(b+c).
Subtraction: Subtraction property: A-B-C = A-(B+C) A-(B-C) = A-B+C.
Multiplication: multiplication commutative law: a× b = b× a.
Law of multiplicative association: (a×b)×c=a×(b×c)
Multiplication and distribution law: (a+b) × c = a× c+b× c (a-b) × c = a× c-b× c.
Division: nature of division: a÷b÷c=a÷(b×c)
For exercises:
1, column vertical calculation.
27×0.430.86× 1.2 1.2× 1.4
(Calculation check) (two decimal places are reserved) (accurate to ten places)
2. Calculate the following problems, which can be simple to simple.
7.06×2.4-5.72.33×0.5×40.65× 105
3.76×0.25+25.84.8×0.25 1.2×2.5+0.8×2.5
Decimal division
1, the meaning of fractional division: know the product of two factors and one of them, and find the operation of the other factor.
For example, 0.6÷0.3 means an operation to find another factor by knowing the product of two factors 0.6 and one factor 0.3.
2. Calculation method of decimal divided by integer (P 16): decimal divided by integer and then divided by integer. The decimal point of quotient should be aligned with the decimal point of dividend. The integer part is not divided enough, quotient 0, decimal point. If there is a remainder, add 0 and divide it.
3.(P2 1) Calculation method of division with divisor as decimal: first expand the divisor and dividend by the same multiple to make the divisor become an integer, and then calculate according to the rule that the divisor is an integer.
Note: If there are not enough digits in the dividend, make up the dividend with 0 at the end.
4.(P23) In practical application, the quotient obtained by fractional division can also be obtained by "rounding" to keep a certain number of decimal places as needed.
5. (P24,25) Variation law in division: ① quotient invariance: divisor and divisor expand or shrink by the same multiple (except 0) at the same time, and the quotient remains unchanged. (2) The divisor remains the same, the dividend expands, and the quotient expands. The dividend is constant, the divisor decreases and the quotient expands. ③ The dividend is constant, the divisor decreases and the quotient expands.
6.(P28) Cyclic decimal: the decimal part of a number. Starting from a certain number, one number or several numbers appear repeatedly in turn. Such decimals are called cyclic decimals.
Circular part: the decimal part of a circular decimal, which is a number that appears repeatedly in turn. For example, the periodic part ......................... of 6.3232 is 32.
7. The number of digits in the decimal part is a finite decimal, which is called a finite decimal. The number of digits in the decimal part is infinite decimal, which is called infinite decimal.
Observe an object
1, correctly identify the shape of the object viewed from the top, front and left.
2, there is a trick to observe the object, first count a few faces, and then look at its arrangement, pay attention when drawing graphics, only draw up and down.
3. When you observe the same object from different positions, you may see the same or different graphics.
4. Observing different objects from the same position, you may see the same or different graphics.
Only by observing from different positions can we know an object more comprehensively.
Simple equation
1, (P45) In a formula containing letters, the multiplication sign in the middle of the letters can be recorded as ""or omitted.
The plus sign, minus sign, division sign and multiplication sign between numbers cannot be omitted.
2.a×a can be written as A or A, A is read as the square of A, and 2a stands for A+A..
3. Equation: An equation with an unknown number is called an equation.
The value of the unknown that makes the left and right sides of the equation equal is called the solution of the equation.
The process of solving an equation is called solving an equation.
4. Principle of solving equations: balance.
The equation still holds when the left and right sides of the equation add, subtract, multiply and divide the same number (except 0) at the same time. 、
5. A quantitative relationship: addition: sum = addend+addend; One addend = sum-another addend.
Subtraction: difference = minuend-meimei = difference+meimei = meimei-difference.
Multiplication: product = factor × factor One factor = product ÷ another factor.
Division: quotient = dividend/divisor = quotient × divisor = dividend/quotient
6. All equations are equations, but not all equations.
7. Equation test process: left side of equation = ...
8. The solution of the equation is a number;
Calculation process of solving equations. = Right side of the equation
So, X=… is the solution of the equation.
For practice
1. Judge whether the following statement is correct.
(1) equations are all equations, but they are not necessarily equations. ()
(2) An equation with an unknown number is called an equation. ()
(3) The solution of the equation is the same as the solution of the equation. ()
(4) 10=4x-8 is not an equation. ()
(5)x=0 is the solution of equation 5x=5. ()
9.3- 1.3 = 10-2 is an equation. ()
2. Solve the equation.
x+53= 102x- 17=54
x-0.9= 1.2x+3 10=690
8.5+x = 10.2 x-0.74 = 1.5
Area of polygon
1, formula: rectangle: perimeter = (length+width) ×2- length = perimeter ÷2- width; Width = perimeter ÷2- Long letter formula: C=(a+b)×2
Area = area = length × width Letter formula: S=ab
Square: perimeter = side length ×4 letters formula: C=4a
Area of parallelogram = base × high letter formula: S=ah
Area of triangle = base × height ÷2- base = area × 2 height; Height = area ×2÷ bottom letter formula: S=ah÷2.
Trapezoidal area = (upper bottom+lower bottom) × height ÷2 letter formula: S=(a+b)h÷2.
Upper bottom = area ×2÷ height-lower bottom, lower bottom = area ×2÷ height-upper bottom; Height = area ×2 (upper bottom+lower bottom)
2. Derivation of parallelogram area formula: shearing and translation.
3. Derivation of triangle area formula: rotation
Parallelogram can be changed into rectangle;
Two identical triangles can be combined into a parallelogram,
The length of a rectangle is equivalent to the base of a parallelogram;
The base of parallelogram is equivalent to the base of triangle;
The width of the rectangle is equivalent to the height of the parallelogram;
The height of parallelogram is equivalent to the height of triangle;
The area of a rectangle is equal to the area of a parallelogram,
The area of parallelogram is equal to twice the area of triangle,
Because rectangular area = length x width, parallelogram area = bottom x height.
Because parallelogram area = because parallelogram area = base x height, triangle area = base x height ÷2.
4. Derivation of trapezoid area formula: rotation
5. The second derivation method of triangle and trapezoid was taught by the teacher, and I read the book by myself.
Two identical trapezoids can be combined into a parallelogram, as long as you know.
The base of parallelogram is equivalent to the sum of the upper and lower bases of trapezoid;
The height of parallelogram is equivalent to the height of trapezoid;
The area of that parallelogram is equal to twice that of the trapezoid,
Because parallelogram area = bottom × height, trapezoid area = (upper bottom+lower bottom) × height ÷2.
6. Parallelograms with equal bases and equal heights have equal areas;
Triangles with equal bases and equal heights have equal areas;
The area of a parallelogram with equal base and equal height is twice that of a triangle.
7. The rectangular frame is drawn as a parallelogram with constant perimeter and smaller area.
8. Combination diagram: convert it into a simple diagram that has been learned and calculate it through addition and subtraction.
Statistics and possibilities
I. Classification and main points of statistical charts (1) Bar statistical charts: A bar statistical chart represents a certain quantity by unit length, draws straight lines with different lengths according to the quantity, and then arranges these straight lines in a certain order.
Function: It is easy to see the figures of various quantities from the bar chart.
(2) statistical chart of broken line: statistical chart of broken line represents a certain quantity with unit length, then draws points according to the quantity, and then connects the points with line segments in turn.
Function: The broken line statistical chart can not only show the quantity, but also clearly show the change of the quantity.
(3) Sector statistics chart: Sector statistics chart represents the total with the whole circle, and the size of each sector in the circle represents the percentage of each part in the total.
Function: The relationship between the number of each part and the total number can be clearly expressed through the fan-shaped statistical chart.
The broken-line statistical chart can not only reflect the data quantity (quantity), but also reflect the change of data (quantity) of a project in a certain period of time.
Second, the average, mode, median comparison
similar
The similarities of the three statistics are as follows: (1) they are all statistics describing the trend of data concentration; Can be used to reflect the general level of data; Can be used as a representative of a set of data.
discrepancy
The difference between the two is mainly manifested in the following aspects.
1, different definitions
Average value: the quotient obtained by dividing the sum of a set of data by the number of the set of data is called the average value of the set of data.
Median: arrange a set of data in order of size, and a number in the middle is called the median of this set of data.
Mode: The number with the highest frequency in a set of data is called the mode of this set of data.
2. Different solutions
Average value: the sum of all data divided by the number of data, which can only be found after calculation.
Median: arrange the data from small to large or from large to small. If the number of data is odd, the number in the middle position is the median of this set of data; If the number of data is even, the average of the middle two data is the median of this set of data. Its solution needs no or only simple calculation.
Mode: The number with the highest frequency in a set of data can be found without calculation.
3. The quantity is different
In a set of data, the mean and median are unique, but the mode is sometimes not unique. In a set of data, there may or may not be multiple patterns.
4, present different
Average value: it is an imaginary number, which is calculated, not the original data in the data.
Median: It is an incomplete "imaginary" number. When a group of data is odd, it is the middle data after the sorting of the group of data, and it is a real data in the group of data; But when the number of data is even, the median is the average of the middle two data, which is not necessarily equal to some data in this group of data. At this point, the median is a virtual number.
Pattern: The original data in a set of data is real.
5, on behalf of different
Average: It reflects the average size of a group of data, and is usually used to represent the overall "average level" of data.
Median: Like a dividing line, the data is divided into the first half and the second half, so it is used to represent the "medium level" of a group of data.
Mode: It reflects the most frequent data and is used to represent the "majority level" of a set of data.
Although these three statistics reflect different things, they can all represent the concentration trend of data and the general level of data.
6. Different characteristics
Average value: it is related to every data, and the change of any data will cause the corresponding change of average value. The main disadvantage is that it is easily influenced by extreme values. Extreme values here refer to numbers that are too big or too small. When the number is too large, the average value will be increased, and when the number is too small, the average value will be decreased.
Median: it is related to the arrangement position of data, and some data changes have no effect on it; It is the representative value of the middle position of a group of data and is not affected by the extreme value of the data.
Mode: It is related to the number of data occurrences, focusing on the frequency of each data occurrence. Its size is only related to some data in this set of data and is not affected by extreme values. Its disadvantage is not unique. There may be one pattern, multiple patterns or none in a set of data.
7. Different functions
Average value: it is the most commonly used data representative value in statistics, which is relatively reliable and stable, because it is related to every data and reflects the most sufficient information. The average value can not only describe the overall average value of a group of data itself, but also be used as a standard for comparing different groups of data. So it is widely used in life, such as average score, average height, average weight and so on.
Median: As a representative of a set of data, the reliability is poor because only part of the data is used. When the individual data of a set of data is too large or too small, it is more appropriate to use the median to describe the concentration trend of this set of data.
Mode: As a representative of a set of data, its reliability is also poor, because it only uses part of the data. . In a group of data, if individual data changes greatly and a certain data appears the most times, it is more appropriate to use this data (that is, the mode) to represent the "concentration trend" of this group of data.
The connection and difference between average, median and mode;
Average is widely used. As a representative of a set of data, it is relatively stable and reliable. However, the average value is related to all the data in a set of data and is easily influenced by extreme data; Simply put, it is the average of this set of data. Median is in the middle position in the numerical ranking of a group of data, and people can make general judgment and control on things through median. Although it is not affected by extreme data, its reliability is poor. So the median only represents the general situation of this set of data. This mode focuses on the frequency of a set of data. As a representative of a set of data, it is not affected by extreme data, and its size is related to some data in a set of data. When the individual data in a set of data changes greatly and a certain data appears frequently, it is more appropriate to use the mode to express the concentration trend of this set of data, which reflects the concentration degree of the whole data.
Average, median and mode all have their own advantages and disadvantages:
Average: (1) All data of the whole group need to be calculated;
(2) It is susceptible to extreme values in data.
Median value: (1) can only be determined by arranging the data in sequence;
(2) Not easily influenced by extreme values in data.
Mode:
Get by counting (1);
(2) Not easily influenced by extreme values in data.
Third, the possibility of size.
Possibility is related to the number of objects and may be expressed by a score.