Parents are saying that the subject of getting the square and square roots of numbers are very difficult tasks for grade school children to understand. What more if it is in large values? Not anymore. INTRODUCING, SQUARE EDGING - the fun, easy way of learning the methods of getting the square roots of numbers that children will enjoy to work with.
2) Had they ever tried taking the square root of an eight-digit number such as 38,775,529 without the aid of a calculator?
3) Are they familiar with the “long hand division method”?
If they answer “yes” to all my questions, maybe some how, they will agree with me, along with the other parents and teachers that such a kind of mathematics, is a “real burden” for grade school kids like you.
But I devised a much easier method that I called Square Edging (MSM-1 Format), designed specially, for kids like you.
WELCOME TO THE WORLD OF SQUARE EDGING!
Notes from the Author:
For parents and Math teachers who have the ideas about Squaring and Taking the Square Roots of Numbers, start with SE Telegram and if you're interested to know how it works, start reading the topic "New Method of Squaring" to fully understand what "Square-Edging MSM-1 Format" is all about
Now that you’d learned all the details in square edging a five or six digits perfect square number, it would be easier for you to understand this fast-track version of square edging.
Given problem
√ 62,001
Step 1:
Follow the instructions in 3D/4D.SE
√ 6’20’01
2 N 1
2 N 9
Take Note: As long as you become familiar with the rules in 5D/6D.SE, I will allow you not to include a zero before 6 in √ 06’20’01, as a strict rule in SSQ
Step 2:
Create a Parameter Checker
As a first rule, always write this phrase “ Let P = “ and then copy the first two groups of digits of the given problem, followed by two dots
Let P = 6’20..
Helpful Tips:
Fast Way of Creating a Parameter Checker
1) Check the first digit of any of the two possible square roots. Use it as the reference digit.
The first digits of the possible square roots (2N1and 2N9) are the same. They are all equal to 2.
reference digit = 2
2) Find a number, next to the reference digit, higher by 1. Put a zero after it together with the ‘two dots’ notation.
Next to 2 is 3. Writing 0 and .. , they will appear as this:
30..
3) On the next line, write down the reference digit and follow it with a digit ‘5’ and the ‘two dots’ notation.
30..
25..
4) On the third line, just copy the reference digit and follow it with a ‘0’ digit and the ‘two dots’ notation.
30..
25..
20..
5) Square them all. Include the ‘two dots’ notations at the end of each line.
30.. 2 = 9’00..
25.. 2 = 6’25..
20.. 2 = 4’00..
Below will be the ‘general form’ that you must always follow in creating a P-Checker
P-Chk
Let P = 6’20..
30.. 2 = 9’00..
25.. 2 = 6’25..
20.. 2 = 4’00..
Using the value of P, find out where it is located by looking at the right side values of the P-Chk.
If you will notice P = 6’20.. is above 4’00.. but below 6’25..
25.. 2 = 6’25..
…………… } P = 6’20..
20.. 2 = 4’00..
Determine if N is 5↑ or 4↓
Five Up and Four Down
25..2 is also known as the middle-half square of 20..2 and 30..2. The 5 in 25..2 actually represent the “next digit after the first digit”. We need to know if the next possible digit belongs to numbers from 5, 6, 7, 8 or 9 (five up) or belongs to 4, 3, 2, 1 or 0 (four down). So the Mid-half is also a reference, in knowing if the square root is in the HLA (higher limit area), or in LLA (Lower Limit Area).
…. 30.. 2 = 9’00..
…{ Higher Limit Area
…. 25.. 2 = 6’25..
4↓{ Lower Limit Area
…. 20.. 2 = 4’00..
Write down below, “N = 4, 3, 2, 1, 0”
…. 30.. 2 = 9’00..
…. 25.. 2 = 6’25..
4↓{ …………… } P = 6’20..
…. 20.. 2 = 4’00..
N = 4, 3, 2, 1, 0
Step 3:
Find out the Missing Digits
2N12 = ..nn’01 ← PSL
...Nx1x2 = ..0 ←SP1 (Add this SP1 to PSL)
....................01 ← T Sum
Nx2 = ..0
N = 0, 5
201
Take note: No need to write down 251 because the P-Chk gave as a hint that the true square root is at 4↓
2N92 = ..nn’81 ← PSL
...Nx9x2 = ..2 ←SP1 (Add SP1 toPSL)
....................01 ← TSum (provided that TSum aligned to PSL)
Nx18 = Nx8 = ..2
N = 4, 9
249
Take Note: With the use of parameter checker, weminimized at once the possible square roots from four into only two possible square roots.
Step 4:
To know it quickly which among the two possible square roots is the true square root, use the Square root locator. But this time, instead of writing down too many numbers, simply write down H, M and L and follow these mathematical operations;
1) Add H and L
2) Divide by 2. ( I represented it as _____/2 as to directly divide 10’25 by 2)
3) This is important. Always subtract the quotient by 6. It is much nearer to the “true square value” of the middle number of 20.. and 25.. ( which is 22.5..), than simply averaging the values of 6’25.. and 4’00..
H = 6’25..
M = ?
L = 4’00..
….10’25 / 2
….. 5’12 - 6
M = 5’06
1) Add H and L
2) Divide by 2. ( I represented it as _____/2) as to directly divide 10’25 by 2
3) This is important. Always subtract the quotient by 6. It is much nearer to the “true square value” of the middle number of 20.. and 25.. ( which is 22.5..), than simply averaging the values of 6’25.. and 4’00..
Our Square Root Locator will appear this way;
Sq. Rt. Loc
H = 6’25..
M = 5’06
L = 4’00..
4) Arrange the two possible square roots. On the left, is the lower square root with a down arrow (↓) and on the right, the higher square root with an up arrow (↑). Locate P (remember, P = 6’20..), by using the above Sq. Rt. Locator
↓201 ……………. 249↑
… H = 6’25..
↑ { ……….. } P = 6’20..
… M = 5’06
… L = 4’00..
The ‘up arrow’ shows that the true square root is in the upper portion of the HLA, so we choose 249 with an up arrow, as our answer.
After you become familiar with the basic rules and procedures of Square Edging, you can use the SE Telegram, to ease up the computation. It is a combination of doing some parts mentally and writing some other parts directly without the formal representations, such as the use of square sign, equal sign and omitting some letters and others, not so important.
Example:
√38,775,529
Step 1: Determine first, the two IPS (initial possible square roots). There is no need to write down the middle letters M and N
*At Left Column, 1st Row
√38,775,529
... 6 ......... 3
... 6 ......... 7
Step 2: Create a modified P CHK.
1) On the first line, write down the notation “ P: ” followed by the first 4 digits of the given problem.
2) On the second line, write down the middle numbers, separated by a colon “ : ” and no square sign. On its right side, draw an arrow, either an up ↑ or down ↓ arrow, based on the following conditions:
Condition 1: If P is less than the middle square value, draw a down ↓ arrow
Condition 2: If P is greater than the middle square value, draw an up ↑ arrow
3)On the third line, write down M, followed by an either a 5↑ or a 4↓, based on the following conditions:
Condition 4: If the arrow on the second line is ↑, write down 5↑, then on its right side, write the notation /_ 750/
Condition 5: If the arrow on the second line is ↓, write down 4↓, then on its right side, write the notation /_ 250/
*At Right Column, 1st Row
.....P: 38’77
. 65 : 42’25 ↓
....M :4↓ / 6250/
Step 3: Start computing the SE data.
Let’s begin with 6MN3
1) Looking for the ‘N’ digits
* At Left Column, 2nd Row
N3 : 09
(Blank)
....... 2 ... ← (take note, the 2 here is thesecond to the last digit of the given problem)
The letter N is included because we’re looking for that missing digit. Leave the second line blank and ask your self this question:
Q: What number needed to add to 0 to have a sum of 2?
A:2
Write down on the second line “ Nx6 ” (6 is the double value of 3), followed by a colon :, then the answer “2”. On their right, decide which “pair of multiplicands” is the correct combination:
..... N3 : 09
... Nx6 : 2 . ..... 2, 7
............. 2
2) Looking for the M digits
Since we’re doing some parts mentally, directly write down the two combinations from the digits we gathered:
* At Left Column, 3rd Row
..... 23 : 4’09
... 2x6 : 1’2 .
............ 5’29
. Mx6 :(Blank)
............ 5 ← (this is the third to the last digit of the given problem)
Leaving some space in the second line blank, give you time to think first of what digit to write, by looking for a number needed to add to come up with the correct sum
..... 23 : 4’09
... 2x6 : 1’2 .
............ 5’29
. Mx6 : 0 . ... 0, 5
............ 5
6023
Underline the 0 since in P Chk,4↓ indicates that the digits for M is 4 below.
Write down on the sixth line, the first complete digits of the first possible square root
Helpful Tip: Write down first 0, then 23 and then look for the first digit 6, so you would not be confused in writing down 6023
Do the Same procedures for the combination 73
* At Left Column, 4th Row
..... 73 : 49’09
... 7x6 :4’2 .
............ 3’29
. Mx6 : 2 . ... 2, 7
............ 5
6273
On the right column, below the P- Chk, do the same procedures starting from Step 3, to complete the data for the second IPS ( 6MN7)
*At Right Column, 2nd Row
..... N7 : 49
... Nx4 : 8 . ..... 2, 7
............. 2
*At Right Column, 3rd Row
....... 27 : 4’49
... 2x14 : 2’8 .
.............. 7’29
... Mx4 : 8 . ... 2, 7
.............. 5
6227
*At Right Column, 4th Row
....... 77 : 49’49
..... 7x14 : 9’8 .
................ 9’29
..... Mx4 : 6 . ... 4, 9
................ 5
6477
Step 4: Arrange the 4 possible square roots from the lowest value to the highest by inserting the letters A, B, C, D on their right sides to avoid writing them again.
A - the lowest possible square root
B - 2nd to the lowest
C - 2nd to the highest
D – the highest of all
At Left Column ..... At Right Column
...... 6023 (A) ....................6227 (B)
...... 6273 (C) ..................... 6477 (D)
Step 5: Use the Square Root locator to determine which of the 4 possible square roots will remain
42’25
(Blank)
36’00 .
78’25 / 2
39’12
Leave the second line (M) blank. Add 42’25 (H) and 36’00 (L).
We come up with a sum of 78’25. Divide by 2
Doing it mentally, always subtract the quotient by 6.
* At Left Column, 5th Row
42’25
39’06 \ ↓(Using this middle value as reference, the P: 38’77 is lesser than)
36’00 /
78’25 / 2
39’12
The ↓ arrow indicates that the two higher values (C and D) are eliminated. A and B remained
Step 6: Use the 2nd Square Root Locator to determine which of the two remaining possible square roots, is the true square root of 38,775,529
39’06
(Blank)
36’00
75’06/2
37’53
Complete the data and determine where P is located
In the above example, there are two possible square roots that ‘automatically eliminated’ due to the reason that the sub-products tend to end in odd digit, which violate the general rule of SSQ that all sub-products must be in even numbers.
Take note that there is always a pattern when a “not a square root of P” notation appears on the left column or right column.
If it is in left column3rd row, the next≠ √Pis on the right column, 2nd row or vice versa.
If it is in the left column 2nd row, the next≠ √P is on the right column, 3rd row and vice versa.
The notations /_750/ at 5↑ and /_250/ at 4↓ (in P Chk), are only reminders that in arranging the possible square roots, make sure thatthere should be two values less than the indicated notation and two other possible square roots greater than it, or else, there could be an error in the process.
FINAL WORDS:
I believe that some people would agree with me that this kind of technique in taking the square root of numbers is much easier. But I don’t intend to replace the traditional method of taking the square root of numbers (long hand division).
I have good reasons why this method will benefit grade school children
1) It will sharpen their skills in adding and multiplying numbers.
2) There is little division and subtraction. I do believe that children hate to divide or subtract large numbers.
3) It will introduce them to the idea of what squares and square roots of numbers are, which they will learn soon in trigonometry (Pythagoreans Theorem), and higher mathematics.
4) The introduction of letters (M, N, P etc), might be strange to them but at that early age, SE prepares them in some basic ideas of algebra. It is up to the parents or teachers’ imaginations. It could be this way “imagine M as a box, where we don’t know what digit is inside that box. “
5) It gives them ideas of the relationships of numbers, such as, which numbers have the same last digits or what number should be multiplied by such number to get a product ending with this and that.
6) SE is Three-in-One (like a 3-1 coffee?). They will learn to know the basic squares of numbers and at the same time, sharpen their skills in addition and multiplication (repeated addition and multiplication of numbers), all in a one package.
7) It is more like a guessing game, a fun way of doing arithmetic. Knowing the missing digits challenges them to work it out than giving them a task of multiplying two numbers or adding four-digit numbers where they don’t have a clue, what the answers should look like.
I am advocating this method, to become part of the curriculum in elementary schools. I am looking for people who would agree with me with this idea and help me spread this message.
As a TOKEN OF APPRECIATION, I wish people will call this method, Square Edging MSM-1 (as part of my name is included in it).
Why MSM-1? The reason is that there are two other methods:
MSM- 2 Square Edging Numbers ending in 25, where MSM-1 failed
MSM-3 Universal Square Edging, a true way of taking the square root of any number, perfect square or not that is much easier to use than the long-hand division method.
