1, algorithm
Add:
Additive commutative law a+b=b+a
Additive associative law (a+b)+c=a+(b+c)
Multiplication:
Multiplicative commutative law a×b=b×a
Multiplicative associative law (a×b)×c=a×(b×c)
Multiplication and distribution law (a+b)×c=a×c+b×c
Subtraction:
The nature of subtraction a-b-c=a-(b+c)
Department:
The nature of division a÷b÷c=a÷(b×c)
2. Add (delete) brackets
Parentheses are preceded by+,× and invariant symbols; The brackets are preceded by-,,and the symbol should be changed.
Sign and change rules:+change-,-change+; Change, change.
Step 3 move the position
Move with the number: when moving, move with the symbol in front of the number.
Second, problem-solving skills.
Some students, you test his arithmetic rules, and he knows it backwards, but when they encounter specific problems, like a tiger biting a hedgehog, they don't know where to start. In the final analysis, I still don't know all kinds of simple calculation methods and specific application scenarios.
Next, I will specifically talk about what kind of simple calculation method is used under what circumstances.
First of all, we need to know two concepts: peer operation and two-level operation.
Addition and subtraction are first-order operations, and multiplication and division are second-order operations. If an expression only contains addition, subtraction or multiplication and division, we say it is the same level operation; An expression, if it includes both addition and subtraction and multiplication and division (usually multiplication and addition or multiplication and subtraction), is called secondary operation.
Ⅰ. Two-stage operation
You can only use the multiplication distribution law!
Example 1, 25×(4+8)
=25×4+25×8
= 100+200
=300
Parentheses, multiply separately, and then add.
Example 2, 17×23-23×7
=23×( 17-7)
=23× 10
=230
No brackets, seek the same number.
Give the same number and write the rest in brackets. If the middle is+,write it as+,if the middle is-,write it as-.
Example 3, 99×38+38
=38×99+38× 1
=38×(99+ 1)
=38× 100
=3800
Example 4, 88×20 1-88
=88×20 1-88× 1
=88×(20 1- 1)
=88×200
= 17600
It is a two-stage operation, but it is not a standard form, and can be transformed into a standard form through appropriate deformation. The first step can be omitted after proficiency.
Ⅱ. Operation at the same level
1, including addition only.
Additive commutative law and the law of association are used comprehensively to piece together a piece that can be rounded and put it in brackets.
Example 5,5+137+45+63+50
=(5+45+50)+( 137+63)
= 100+200
=300
2. Only multiplication is included.
Using multiplicative commutative law and associative law synthetically, a piece that can be rounded up is pieced together and enclosed in brackets.
Example 6,8× 25×125× 4
=( 125×8)×(25×4)
= 1000× 100
= 100000
3. Continuous reduction
The essence of subtraction
Example 7,347-148-52
=347-( 148+52)
=347-200
= 147
4. Even division
The essence of division
Example 8, 16000÷ 125÷8
= 16000÷( 125×8)
= 16000÷ 1000
= 16
Step 5 have brackets
Remove brackets
Example 9,740 ℉ (37× 4)
=740÷37÷4
=20÷4
=5
Pay attention to changing the number.
6. Same mantissa
shift position
Example10,445+87-45
=445-45+87
=400+87
=487
III. Multiply two numbers and then divide them.
Multiplication of two numbers is only directly applicable to the multiplication and exchange law, which can not make the calculation simple, and needs to be converted into the same-order operation or the second-order operation through division.
1, there is a number close to the whole hundred (the whole ten is almost the same as the whole thousand).
Break a number close to the whole hundred into "whole hundred+several" or "whole hundred-several".
Example 1 1, 87×99
=87×( 100- 1)
=87× 100-87× 1
=8700-87
=86 13
For example, 12,103x12.
=( 100+3)× 12
= 100× 12+3× 12
= 1200+36
= 1236
2. A number is 25 or 125.
25 in the case of 4, 125 in the case of 8.
Example13,25× 28
=25×(4×7)
=25×4×7
= 100×7
=700
For example, 14 and 125×72.
= 125×(8×9)
= 125×8×9
= 1000×9
=9000
Can also be divided into two levels of operation.
125×72
= 125×(80-8)
= 125×80- 125×8
= 10000- 1000
=9000
Third, error-prone analysis.
1, the multiplication and distribution law only multiplies the first number.
Examples: 15, 125×(80+8)
Wrong solution:
125×(80+8)
= 125×80+8
= 10000+8
= 10008
Positive solution:
125×(80+8)
= 125×80+ 125×8
= 10000+ 1000
= 1 1000
2. Operations at the same level are divided into two levels.
Example16,25× 32
Wrong solution:
25×32
=25×(4×8)
=25×4+25×8
= 100+200
=300
Positive solution:
25×32
=25×(4×8)
=25×4×8
= 100×8
=800
3. I forgot to bring my number to move.
Example17,253-87+53
Wrong solution:
253-87+53
=253-53+87
=200+87
=287
Positive solution: just calculate in the order of operation.
4. Add (delete) brackets,-,and forget to change the symbol.
Example18,3700 ÷ 25× 4
Wrong solution:
3700÷25×4
=3700÷(25×4)
=3700÷ 100
=37
Positive solution: just calculate in the order of operation.
5. An error occurred while disassembling the project.
Example19,37× 99
Wrong solution:
37×99
=37×(99+ 1)
=37× 100
=3700
Positive solution:
37×99
=37×( 100- 1)
=37× 100-37× 1
=3700-37
=3663
Fourth, expand and upgrade.
Two-level operation, no brackets, no same sign.
Example 20,46× 32+27× 64
=46×32+54×32
=32×(46+54)
=32× 100
=3200
Find the multiple, and transform it into the standard form of the multiplication and distribution law by using the changing law of the product.