— it is not a rough migration of “the whole number being counted as a decimal point”, but rather using the algorithm as a bottom logic, using structural consistency to bridge the cognitive barriers of integers and decimals。
Real decimals are easy to calculate, not by the means of memory skills, but by the means of:
“the law remains the same, the position is the same, the point is the form of the number, the same is the calculated skeleton.”
The following are strictly observed:
Each algorithm is accompanied by a four-step closed loop of "integer-numbered prototype decimal migration and error-prone resolution of the true-protocol simulation "
(b) the dangerously misleading rejection of "first integer and then decimal" (this is bound to collapse in a chain, hybrid operation)
All cases are taken from the final/preliminary question of mathematics at the end of primary school in the last five years, with reference to the number of the points of reference (e. G., 6 in human education 1. 3)
#i, plus-equivalent & combining law
Integer prototype:
28 + 73 + 72 = (28 + 72) + 73 = 100 + 73 = 173
Core: reconstruction order for a full 10-hundred-hundred pair。
Decimal migration:
Key cognitive upgrading:
The word “integer” for decimals is not “no decimal part”, but is used to refer to the decimals which add up to 0. 10, 0. 20... Or 1. 00 whole per cent。
For example: 0. 27 + 0. 73 = 1. 00 (wind 1), 1. 48 + 0. 52 = 2. 00 (wind 2)。
Error resolution:
Error: "0. 3 + 0. 7 = 1 so 0. 03 + 0. 07 = 1" forget decimal places, actual = 0. 10
Correct: at the end of the alignment — 0. 03 and 0. 07, both percentages, 3+7 = 10 = 0. 10 per cent。
The real story quick-track demonstration (2023 shenzhen fukuda at the end of the sixth round):
Calculate: 4. 65 + 2. 38 + 5. 35 + 7. 62
Thinking chain:
Observation decimal component: 0. 65 + 0. 35 = 1. 00, 0. 38, + 0. 62 = 1. 00 pair
- reorganization (4. 65 + 5. 35) + (2. 38 + 7. 62) = 10. 00 + 10. 00 = 20. 00
Point: when the decimals are aligned, the last two add up to 100, i. E., one in percent。
#ii, multiplication exchange & combining method # the core of decimal is "accumulation."
Integer prototype:
25 x 13 x 4 = (25 x 4) x 13 = 100 x 13 = 1300
Core: 10/100 full pair (25 x 4 = 100)。
Decimal migration:
Key cognitive upgrading:
The essence of the decimal multiplication is the conversion of decimals into " integers with decimals " using the interaction of decimals with integers。
For example: 0. 25 × 4 = 1 (due to 0. 25 = 1/4), 1. 25 × 8 = 10 (due to 1. 25 = 5/4, 5/4 × 8 = 10)。
- misdiagnostic:
Error: "0. 125 x 8 = 1, so 0. 0125 x 8 = 1" → actual = 0. 1 (one decimal to the left)
Correct: first calculates the fraction of the integer and then determines the number of decimal places in the accumulation by the sum of the median decimal places of the factor。
0. 0125 (4 decimals) x 8 decimals; 125 x 8 = 1000 minus 10,000? Wrong
→ correct: 0. 0125 x 8 = 0. 1000 = 0. 1 (end 0 omitted)。
Realistic quick-track demonstration (2024 nanjing drum building mini-up, compute volume):
Calculate: 0. 125 x 3. 7 x 8 x 0. 4
Thinking chain:
- looking for “gold combination”: 0. 125 x 8 = 1 (core anchor)
- residual: 3. 7 x 0. 4 = ? ? 37 × 4 = 148, due to 3. 7 (1), 0. 4 (1) 2 decimals → 1. 48
Final: 1 x 1. 48 = 1. 48
Point: in the multiplication method, priority is given to the combination of factors that remove decimals (e. G. 0. 125 x 8, 2. 5 x 4) and the residuals are treated。
The essence of multiplication is preservation
Integer prototype:
102 x 37 = (100 + 2) x 37 = 100 x 37 + 2 x 37 = 3700 + 74 = 3774
Decimal migration:
Key cognitive upgrading:
Decimal fractions shall be kept to the original number accuracy and shall be broken to " integer part + decimal part " instead of being randomly severed。
> for example: 4. 8 should be reduced to 4 + 0. 8 (not 4 + 0. 80, equal value but emphasis on bet awareness)
12. 05 disassembly to 12 + 0. 05 (non-disable 10 + 2. 05, damage to decimal structures)。
- misdiagnostic:
Error: 9. 9 x 4. 5 = (10 - 0. 1 x 4. 5 = 10 x 4. 5 - 0. 1 x 4. 5 = 45 - 0. 45 = 44. 55 (correct!)
However, if it is written (10-0. 10) x 4. 5 zirconium, the same result reveals a vagueness of “0. 1 and 0. 1 but different bits” - the examination may deduct step points。
Corrected: when split, the decimal part number is strictly consistent with the original number (9. 9 is 1 decimal to 10 - 0. 1, not 10 - 0. 1)。
The real story quick-run demonstration (end of beijing haidian zone 2023):
Calculation: 3. 6 x 99. 8
Thinking chain:
- consider 99. 8 as 100 - 0. 2 (1 decimal matching)
3. 6 x (100 - 0. 2) = 3. 6 x 100 - 3. 6 x 0. 2
3. 6 x 100 = 360 (two decimal places to the right)
3. 6x0. 2: 36x2 = 72 first, due to 3. 6 (1), 0. 2 (1) → 2 decimals → 0. 72
360 - 0. 72 = 359. 28
Point: minus minus zero alignment (360. 00 - 0. 72) - it's the iron rule of small numbers, not to be omitted
# # # # # iv, deductive & separatist # unitization policy
Integer prototype:
125 - 37 - 63 = 125 - (37 + 63) = 125 - 100 = 25
360 ÷ 12 ÷ 3 = 360 ÷ (12 x 3) = 360 ÷ 36 = 10
Decimal migration:
Key cognitive upgrading:
Company reductions/companys are to be combined provided that the units are consistent (same in dollar, square and kilograms) and the decimal numbers are consistent。
> if you have a different number of digits, you must add zero uniform precision and then run it。
> for example: 5. 2 − 1. 37 − 0. 63 → with 5. 20 − 1. 37 − 0. 63, combined minus 1. 37 + 0. 63 = 2. 00 = 5. 20 − 2. 00 = 3. 20
Error resolution:
Error: 8. 4 ÷ 0. 2 ÷ 0. 7 = 8. 4 ÷ (0. 2 × 0. 7) = 8. 4 ÷ 0. 14 → calculated for 8. 4 ÷ 0. 14 =, not converted to integer division (840 ÷ 14 = 60), with a direct mouth error。
Correct: decimal chain division, prioritized into a single division, with the addition of a “commercial non-change” to widen the division by the same number as the division:
8. 4 ÷ 0. 14 = 840 ÷ 14 = 60 (same x 100)。
Realistic quick-run demonstration (2024 shanghai zone simulation six):
Calculation: 7. 5 - 2. 86 - 1. 14 + 0. 5
Thinking chain:
Combined minus 2. 86 + 1. 14 = 4. 00 (6 + 4 = 10 in percentage, plus 1; 8 + 1 = 10, plus 1; 2 + 1 = 3 + 1 = 4)
- prototype 7. 5 - 4. 00 + 0. 5
-systems: 7. 50 - 4. 00 + 0. 50 = (7. 50 + 0. 50) - 4. 00 = 8. 00 - 4. 00 = 4. 00
Point: when adding or subtracting a mixture, the word “plus” is more stable than “left-to-right” — since the law of exchange is established without conditions, the reduction requires caution。
# # final toolbox: a simple three-order capability map of decimals spectrum
The level of competence, criteria, typical performance, teacher tips
L1: a bit sensitive person can accurately say any decimal number of names and units (e. G. 3. 05 "0" in deciles, 0. 1) and see 0. 25 react "1/4" and 0. 125 react "1/8" with "1/8" base and all speeds are in the sky
L2: sensors are free to discover reorganizable structures in decimals (e. G. 0. 73 and 0. 27, 1. 25 and 8) and to proactively call the corresponding algorithms without relying on the title hint, “a simple method”, to decide which numbers to count first
L3: the handler of the movement has precise control of the decimal point position in the multiplication method (multiplication view of the total bit, division of the non-change properties) and automatically adds zero alignment in the hybrid operation. Gap
I'll give you a final test:
“simply calculated `simplicity', not with few steps but with clear logic
It's poop, not fast, but right in the head."
When you stop asking, "how can this be simple?"
But ask:
"what two numbers add up/ multiply/decreasing here to make the decimal zero?"
- congratulations. You've got the key to the world。
