At this stage, I will admit that I'm not sure if during my grade school days that any of my teachers in Math (Grade IV, Grade V or even in Grade VI) discussed in their classes 'anything' about SQUARE ROOTS. I just remember that, during my Junior High, my Math teacher gave us an assignment about "taking the square roots of numbers". I even asked my elder sisters and brothers about it (including my neighbors) but 'little' they had an idea about it. So I went to the library and searched on it. In my dismayed, I didn't truly understand the explanation or the procedure about it. The next day, my Math teacher explained it for us, so, at least I had a better grasp on it, but still, not completely clear. After that, she gave a quiz and it took the whole session of the class, just for us, to try to answer that single given question. To tell you the truth, I didn't even finished my own computation.
Okay, let me give you a similar quiz::
QUESTION: What is the square root of 471,969
..................
√471,969 = ?
Using the LONG HAND DIVISION METHOD...
STAGE 1
Step 1: Group the digits by two's starting from the last digit
..................
√47'19'69
Step 2: Find a number when multiplied by itself will bring out a product near but less than 47
....6............
√47'19'69 ( Choose 6 since 6 x 6 = 36)
Step 3: Subtract 36 from 47
....6............
√47'19'69
- 36 ,.
..11'19 (Bring down 19)
Step 4: Find a number when multiplied by blah blah blah... (Please read the INTRODUCTION)
6 x 20 x ? + (? x ?) = equal or less than 1119
6 x 20 = 120 (This is our basis to find out which among from 1 to 9 is the possible next digit)
STOP HERE!
In Introduction, I gave you an 'alternative approach' to make it easier, which next number from 1 to 9 is the possible candidate by 'rounding off the numbers' and then try to divide...
But how can we find an easy way of relating 1119 to 120?
Possibly by rounding off 1119 into 1000 and 120 into 100...
1000 ÷ 100 = 10
This is one of the 'complication' of this method. We cannot use 10 because the only set of numbers, possible to choose are from 1 to 9.
Okay, let's have a wild guess - a number below 10
STAGE 2
Sample 1: Let's try 9...
6 x 20 x 9 + (9 x 9) = equal or less than 1119
120 x 9 = 1080 .... 9 x 9 =81
Seems 1080 is less than 1119. But the thing is, we must still add 81
1080 + 81 = 1161
1161 is way 'above' or over than 1119.
So, we 'discard' 9
Sample 2: Let's try 8...
6 x 20 x 8 + (8x8) = equal or less than 1119
120 x 8 = 960 .... 8 x 8 = 64
960 + 64 = 1024
1024 is less than 1119. We 'accept' 8 as our next digit...
....6...8........
√47'19'69
- 36 ,.
..11'19
-.10'24 .
......95'69 (Bring down 69)
STAGE 3
68 x 20 x ? + (? x ?) = equal or less than 94'69
68 x 20 = 1360
Again, finding a way to relate 9469 to 1369
Let's round off 9469 into 9000 and 1369 into 1000
9000 ÷ 1000 = 9
It appears that 9 is the better guess, so far.
SAMPLE 1: Let's try 9...
68 x 20 x 9 + (9 x 9) = equal or less than 95'69
1320 x 9 = 12240 ... seems 'above' 9569
So, we discard 9...
SAMPLE 2: Let's try 8...
68 x 20 x 8 + (8 x 8) = equal or less than 95'69
1320 x 8 = 10560 ... still, 'above' 9569
So, we must also discard 8...
SAMPLE 3: Let's try 7...
68 x 20 x 7 + (7 x 7) = equal or less than 95'69
1320 x 7 = 9520, less than 9569 ... seems we have the right 'candidate' this time
... just continue
7 x 7 = 49
9520 + 49 = 9569
The result, 9569 is in fact equal to 9569, so we accept 7 as the remaining digit
....6...8..7...
√47'19'69
- 36 ,.
..11'19
-.10'24 .
......95'69
- ....95'69 .
........ < 0 > (Zero Remainder)
WHAT'S WRONG WITH THE LONG-HAND DIVISION
The long-hand division is based on the "process of reduction ", meaning, we try to subtract values until it become zero or null. In the above example, 687 is the exact square root of 471,969 because at the end of the process, there is no more remainder left. Comparing it to how we got the value of 1.4142 for the √2, if we continue doing the same process, we will become 'frustrated', in a sense, that there's no end in trying to zero-in the remainder ( reducing the remainder until it become zero). The reason for this is that √2 is an 'irrational' number - meaning, how much you try to reduce the remainder, it will not end up to zero(SEE THIS).
In my own opinion, there are at least 'four' reasons why the long-hand division is not so popular to grade school kids (or even for most adults).
1) Kids hate to subtract numbers
2) Kids also hate to divide numbers, or even trying to round off numbers to make the process of division much easier (Making the Divisor divisible to the Dividend).
3) Kids have difficulty in choosing among which numbers from 1 to 9 is fit enough to be the next digit as part of the answer (in getting the square root of a certain number).
4) The trial and error method of 'sampling' those numbers, one at a time, is a long, slow process for them.
But DON"T WORRY. I found a better method that will make children become more interested in 'taking the square root of a number' without the painstaking and tedious way of doing it, the old fashion way.
WELCOME TO "EASY SQUARE ROOTING"...
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