The first unit equation of the important knowledge points of mathematics in the second volume of the fifth grade
1, the expression of the equation is called equality.
2. An equation with an unknown number is an equation.
3. Equation must be an equation; An equation is not necessarily an equation. Equation > equation
4. Adding or subtracting the same number on both sides of the equation at the same time results in an equation. This is the essence of the equation.
When both sides of an equation are multiplied or divided by the same number that is not equal to 0, the result is still an equation. This is also the nature of the equation.
5. The process of finding the unknown quantity in the equation is called solving the equation.
Common relations in solving equations:
One addend = sum-another addend = minuend-difference = minuend+difference.
One factor = product/divisor = dividend/quotient dividend = quotient × divisor
Note: After solving the equation, you should form a good habit of testing.
6. The sum of five consecutive natural numbers (or consecutive odd numbers and consecutive even numbers) is equal to five times the median number. The sum of odd consecutive natural numbers (or even consecutive odd numbers) is equal to the middle number.
7. The sum of four consecutive natural numbers (or consecutive odd numbers or even numbers) is equal to the sum of the middle two numbers or the first two numbers × number ÷2 (Gaussian sum formula).
8. enumerate the ideas of solving application problems by equations: a. examine the problems and understand the known conditions and problems of the problems. B, clear the equal relationship of the topic. C, set an unknown number, and the number that is generally found is represented by X. D, list equation E, solve equation F, test G and answer according to the equivalence relation.
The second unit determines the position
1. When determining the position, the vertical rows are called columns and the horizontal rows are called rows. Determine which column is generally counted from left to right and which row is generally counted from front to back.
2. The number pair (x), 1 number indicates which column (x), and the second number indicates which row (). When writing a number pair, write the number of columns first, then the number of rows.
3. Seen from the globe, it is the meridian connecting the North Pole and the South Pole, and the ring perpendicular to the meridian is the latitude. Longitude and latitude are arranged in a certain order, indicating "longitude" and "latitude" respectively, and both "longitude" and "latitude" are expressed in degrees (), minutes (') and seconds (").
4. Move a point to the left and right by several squares, except that the number on column (x) changes, decreases to the left and increases to the right, while the number on row () remains unchanged. For example, the position of point (6, 3) is (8, 3) after being shifted to the right by 2 units, and column 6+2 = 8; The position of the point (6,3) is (4,3) after being shifted to the left by 2 units, and the column 6-2=4.
5. Move a point up and down a few squares, except that the number on the row () has changed, and the number on the column (x) has not changed. For example, the position of point (6, 3) is (6, 5) after moving up by 2 units, and line 3+2 = 5; When the point (6,3) moves down by 2 units, the position is (6, 1), and the column 3-2= 1.
Unit 3 Common Multiples and Common Factors
1, the smallest factor is 1, the largest factor is itself, and the number of a number factor is limited.
The minimum multiple of a number is itself, and there is no maximum multiple. The multiple of a number is infinite.
The largest factor of a number is equal to the smallest multiple of this number.
2. The common multiple of several numbers is called the common multiple of these numbers, and the smallest is called the least common multiple of these numbers, with the symbol =40,(5,8)= 1;
The greatest common factor of two adjacent numbers is 1, and the least common multiple is their product. =72,(9,8)= 1;
Numbers with special relationships (both are composite numbers, one is odd and the other is even, but there is only one common factor between them 1), such as 4 and 9, 4 and 15, 10 and 2 1, and the greatest common factor is 1.
Extended reading: fifth grade, book 1, mathematical knowledge points, unit 1, decimal multiplication
1, decimal times integer: meaning-a simple operation to find the sum of several identical addends.
For example, 1.5×3 indicates how many times 1.5 or how many three 1.5 are.
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 multiplied by decimal: meaning-that is, what is the fraction of this number.
For example, 1.5×0.8 (the integer part is 0) is to find eight tenths of 1.5.
1.5× 1.8 (integer part is not 0) is 1.8 times of 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.
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: a number (except 0) is multiplied by a number greater than 1, and the product 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 usually three ways to find the approximate value:
(1) rounding method; (2) into law; (3) Tailing method
5. Calculate the amount of money, and keep two decimal places, indicating that the calculation has reached the point. Keep one decimal place, indicating that the angle has been calculated.
6. The order of four decimal operations is the same as that of integers.
7, operation law and nature:
Add:
Additive commutative law: A+B = B+A.
Additive associative law: (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 or a×c+b×c=(a+b)×c(b= 1, b) omitted.
Variants: (a-b)×c=a×c-b×c or a× c-b× c = (a-b )× c.
Subtraction: Subtraction property: a-b-c=a-(b+c)
Division: nature of division: a÷b÷c=a÷(b×c)
Second unit position
8. In order to determine the position of an object, you need to use several pairs (first column: vertical, then horizontal). Two problems can be solved by using number pairs: first, given a number pair, the point where the object is located is marked on the way of coordinates. The second is to give a point in the coordinates, which can be expressed by several pairs.
Unit 3 Decimal Division
10, 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 that the product of two known factors is 0.6, one factor is 0.3, and what is the other factor?
1 1. Calculation method of decimal divided by integer: decimal divided by integer, divided by integer to divide. 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.
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Note: If there are not enough digits in the dividend, make up the dividend with 0 at the end.
12. In practical application, the quotient obtained by fractional division can also be rounded to a certain number of decimal places as needed to obtain the approximate number of quotients.
13, the change law of division: ① quotient invariance: the dividend and divisor are expanded or reduced by the same multiple at the same time (except 0), and the quotient remains unchanged. (2) The divisor is constant, the dividend is enlarged (reduced), and the quotient is enlarged (reduced). (3) The divisor is constant, the divisor decreases, but the quotient expands; The dividend is constant, the divisor is enlarged, but the quotient is reduced.
14, 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. Abbreviation 6.32.
15, 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. Decimals are divided into finite decimal and infinite decimal.
Unit 4 Possibility
16. There are three situations when the incident occurs: possible, impossible and certain.
17, possible events, probability. By adding the number of copies of several possible situations as the denominator and taking a single possibility as the numerator, the possibility of the corresponding event can be obtained.
Unit 5 Simple Equation
18. In the 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.
19, a×a can be written as a or a, and a is pronounced as the square 2a of a, which means a+a.
Especially here1a = a: "1"we don't write.
20. Equation: An equation with an unknown number is called an equation (★ Conditions that the equation must meet: there must be an unknown number, and both are indispensable). 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.
2 1, principle of solving equation: 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.
22. 10 quantitative relationship: addition: sum = addend+addend; One addend = and-the other 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
All equations are equations, but all equations are not necessarily equations.
24. Equation test process: left side of equation = ...
25. The solution of the equation is a number; Calculation process of solving equations. = the right side of the equation, so X=… is the solution of the equation.
Unit 6 Area of Polygon
26. Formula:
Square:
Area of a square = side length x side length s positive = AXA = A2
Known: the area of a square, find the side length;
Rectangular:
Area of rectangle = length × width;
S length =aXb
Known: rectangular area and length, width;
Parallelogram:
Area of parallelogram = base x height;
Class s =aXh
The known height h=S is obtained from the area and bottom of parallelogram;
Triangle:
Area of triangle = base x width and height ÷ 2;
Three =aXh÷2
It is known that the area and bottom of a triangle are determined by its height;
H=S three X2÷a
Trapezoid:
Trapezoidal area = (upper bottom+lower bottom) x height ÷2
Stitch =(a+b)X2
It is known that the sum of trapezoidal area and upper and lower bottoms is the height.
Height = area ×2 (upper bottom+lower bottom)
Upper bottom = area ×2÷ height-lower bottom
Combined graphics:
When the combined figure is convex, the areas of two or three simple figures are added and calculated.
When the combined pattern is concave, the largest simple pattern area is used to subtract the smaller simple pattern area for calculation.
27. Derivation of parallelogram area formula: shear and translation.
Parallelogram can be changed into rectangle; The length of a rectangle is equivalent to the base of a parallelogram; The width of the rectangle is equivalent to the height of the parallelogram; The area of rectangle is equal to the area of parallelogram, because the area of rectangle = length × width, so the area of parallelogram = bottom × height.
28. Derivation of triangle area formula: rotation
Two identical triangles can be combined into a parallelogram, and the base of the parallelogram is equivalent to the base of the triangle; The height of parallelogram is equivalent to the height of triangle;
The area of parallelogram is equal to twice the area of triangle, because the area of parallelogram = base× height, so the area of triangle = base× height ÷ 2;
29. Derivation of trapezoidal area formula: rotation
30. Two identical trapezoids can be combined into a parallelogram. 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 parallelogram is equal to twice the area of trapezoid, because the area of parallelogram is equal to bottom × height, so the area of trapezoid is equal to (upper bottom+lower bottom) × height ÷2.