A. INTRODUCTION

Kids,do you know that besides adding, subtracting, multiplying and dividing numbers, there are two other very interesting math operations that are also very useful in many activities? If you wish to know the area of a square or the measures of the sides of a right triangle, two special math operations - getting the square values or getting the square roots of numbers, are some of the more advanced math operations that you must learn.

They are advanced because unlike ordinary multiplication and division, their operations (methods of getting the answers), are indeed, much more tricky and difficult.

SQUARE OF A NUMBER

Getting the ‘square value’ of a number is like doing a special kind of multiplying a number, in which both the multiplicand and the multiplier are equally the same values. Sometimes, it is described as product of a number multiplied by itself.

Examples:

2 x 2 = 4
27 x 27 = 729
146 x 146 = 21,316

The value 4 is sometimes called the ‘square value’ of 2, or simply, “square of 2”. The same way, 729 is the “square of 27” and 21,316, the “square of 146”. Sometimes, instead of writing 2x2, 27x27 or 146x146, “a small number 2 in upper right side” of a given number is used as a symbol, telling you to multiply that number by itself. So, instead of 2x2, we write 2² = 4 and 27x27 as 27² = 729, while 146x146 as 146² = 21,316.

Maybe, you are wondering why it is called ‘square’. Probably, early mathematicians noticed that the measure of the area of a square is always equal to a certain ‘number multiplied by itself’, so they named it, that way.

SQUARE ROOT

On the other hand, getting the square root of a number, needs a very different way, of dividing a number. Unlike in ordinary division at which you need to mention the value of the divisor, in getting the square root of a number, both the divisor and the quotient are unknown and the difficult thing is, both divisor and the quotient must be equally in the same values. 

Examples:  

36 ÷ 3 = 12               36 ÷ 4 = 9           36 ÷ 6 = 6

In the above examples, 36 can be divided by 3 or 4 but the quotient would not be equal or the same with the divisor. Dividing 36 by 6, we can get a quotient equal to 6, which is the same exact value as to the divisor. In this situation, we can say then, that, 6 is a square root of 36.

In doing this special kind of division, the symbol √ is used before a given number (example √144, read as, “the square root of one hundred forty-four’), to tell you to look for a divisor that will give a quotient, equal to that divisor. Dividing 144 by 12, we come up with a quotient equal to 12 (144 ÷ 12 = 12). Showing equal values for both divisor and quotient we can say then, that √144 = 12.

But there are occasions that the given numbers are in large values (example, √139,876). Getting the square root of such large valued numbers requires a very tedious and tricky method called ‘long hand division’. But as a practice, small valued numbers are introduced for grade school children, to make them easier to memorize.

TABLE OF SQUARE ROOTS

√1 = 1
√4 = 2 
√9 = 3
√16 = 4
√25 = 5
√36 = 6
√49 = 7
√81 = 9 
√100 = 10

PERFECT SQUARES

Not all numbers from 1 to 100 give a square root in whole exact values. Most of them are in decimal values. Below is a list of examples of numbers, having no whole exact square root values:

√ 2, √3, √10, √99 , √28 , √50

Counting from 1 to 100, there are only ten numbers having square roots in ‘exact whole values’ and they are called perfect squares (or simply call them “PERKS”).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

(Using a calculator, find the square roots of each numbers from 1 to 100 and write down which numbers have an exact whole numbers)
SET OF “PERFECT SQUARES” = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}
 
For fun way of naming things, let's call them Perkies, pertaining to "squares of whole numbers".


TABLE OF SQUARES

1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81 
10² = 100

D. Multiple-Digit Squaring (Rules to Follow)

SSQ is applicable to any number. But SSQ become a little bit complicated when the digits of a number increases. The two-digit SSQ method is the easiest and the very fundamental of the principles behind SSQ. But when the digits of numbers increases, such as in squaring 54,675², it will involve many 'sub-products' and many 'partial sums' and the process of multiplying and adding the set of digits will also become very complex.

But don't worry. As long as you already have the idea of squaring two-digit numbers and learn to adjust and become familiar in squaring three-digit numbers using SSQ, squaring multiple-digit numbers will be just another way of squaring number in a more challenging way. 


Ultimate Rules in SSQ

Now that we are dealing to do the squaring of numbers in four, five or six-digit numbers, it is important that you must know and follow these 'golden rules':


First Golden Rule:

If you're going to square a number, make sure to count the digits. 

 "The count of digits doubles as you square a number."

35² = 1,225

(Take note, a from two-digit number it becomes a four-digit number)

But in certain cases, this rule seems not been followed...

12² = 144

But in SSQ, it is a 'strict rule' that you must follow this golden rule. 12² = 144 will then, become...

12² = 01'44


Second Golden Rule:

The number of sub-products increases as the digit of numbers increases. But the number of sub-products depends on the count of digits. 
"The sub-products are always less than one to the count of digits of the given number".
 
You might noticed that in two-digit SSQ, there is only one sub-product (SP1), while doing the three-digit SSQ, there appears a second sub-product. It is easy to follow the pattern - that in a "four-digit.SSQ", you will then need three sub-products (SP1, SP2 and SP3), to complete the process. So in five digits, there will be four SP's. Six... five, and so on.

Third Golden Rule:
 
As a initial rule, always remember that the count of partial sums increases as the counts of digits of the given number increases. But to be specific, keep it in your mind that:
"The partial-sums are always one less than to the count of sub-products in a multiple-digit number SSQ"

In a way, the partial-sum, when squaring a number having more than two digits, is simply a temporary result of adding the sub-product to the PSL .


Take note, in a two-digit SSQ, there is only one sub-product and "no partial-sum" at all

 (Example 1)

 23²       = 04’09 ← PSL
 +2x3x2 =   1'2   ← SP1  (2 of 12 is aligned to second 0 of 04'09)

................05’29 ← T-Sum 

As a simple way of explaining things, we may consider that a 'two-digit' SSQ is somehow, not part of the multiple-digit SSQ at all - in a sense that it doesn't include any partial sum. In a two-digit SSQ, the sub-product (SP) is directly added to the partial squares line (PSL) to get the 'total sum' (T-Sum).  

Look below. Take note that there are now  two sub-products (SP1 and SP2) and the T-Sum in the above (example1), become the 'partial-sum' (in yellow shades).   

(Example 2)
23..²      = 04’09.. ← PSL 1
+2x3x2 =   1'2    . ← SP1  (2 of 12 is aligned to second 0 of 04'09)
238²   = 05’29'64   ← PSL 2  (the new PSL)
23x16 = .   36'8  .   ← SP 2  (the new SP) (8 of 36'8 aligned to 6 of 05'29'64)
.............05'66'44    ← T-Sum
  

Fourth Golden Rule:
 "The partial sum will become part of the new PSL along with the next index square"  

In example 2, the 05'29, which is the actual square of 23, become part of the new PSL. We can say that that 05'29 is the "index square" of 23, if we are dealing with multiple-digit SSQ such the example above. 

Fifth Golden Rule:
 "The total sum in a multiple-digit number SSQ will reflect the true square value of the given number". 

 

B. Squaring Two-Digit Numbers

Common Way of Multiplying Numbers

Squaring a number is the same as multiplying two numbers having identical values.

Example:

Square the number 743 = 743x 743.

1) Multiply 743 by 3. Put the carries above 743 and the partial product = 2229

2) Then multiply 743 by 4. The partial product = 2972. Put the last digit 2 on the tens decimal place.

3) Lastly, multiply 743 by 7. the partial product = 5201. Put the last digit 1 on the hundreds decimal place.

4) Add the partial products and what we get is = 552049 or 552,049

That is how we commonly get the square of a number.

Systematic Squaring Method (SSQ)

This time, I’ll teach a new way of getting the squares of numbers in an easier and orderly manner. But first, you must also know some new things.

Digit Number

A digit (what I’m talking about here is the numeric digit), is either any of the following;

0, 1, 2, 3, 4, 5, 6, 7, 8 or 9

A number such as 743 has three digits, 7, 4 and 3. Sometimes it is called a three-digit number. All you have to do is to count the digits. Counting the digits of 4,569,742, we can then, name that number, as a seven-digit number. In SSQ, the “count of digits of a number is important”. Later, you will realize the reason why it is important. But for now, giving you the idea of what a digit of a number is all about, would be enough.

Meaning of SSQ

SSQ stands for Systematic Squaring. It is based on a popular algebraic equation, (X + Y)². It is much different from the common method of multiplying two identical numbers.

SSQ has only three main parts, namely:

1) PSL (Partial Squares Line)

2) Sub-product

3) Total Sum (TSum)

Index Squares

Always remember that there are only ten basic digits (numeric digits) and these are;

0, 1, 2, 3, 4, 5, 6, 7, 8 and 9

An index square is a product of a ‘basic digit’ multiplied by itself:

0x0 = 0 / 1x1 = 1 / 2x2 = 4 / 3x3 = 9 / 4x4 = 16

5x5 = 25 / 6x6 = 36 / 7x7 = 49 / 8x8 = 64 / 9x9 = 81

It is safe to call 0, 1, 4, 9, 16, 25, 36 , 49, 64 and 81 as index squares but in SSQ, an index square must be expressed as “two-digit square”. So the proper way of writing them are as follows:

Table of Index Squares

0² = 00

1² = 01

2² = 04

3² = 09

4² = 16

5² = 25

6² = 36

7² = 49

8² = 64

9² = 81

Two-Digit SSQ

Let's start by squaring a two-digit number, using the SSQ method

Question: What is the square of 23?

23² = ?

Step 1: Create a PSL

Partial Squares Line (PSL)

The PSL is simply, the “two-digit squares” representation of each, individual digits of a certain number. In 23², the two-digit squares representation of the digits, 2 and 3 are 04 and 09, respectively. So we simply write it this way:

23² = 04’09 ← PSL

But don’t forget to also include this sign - ’ (a special character called single close quote). It will easily give us a clue of how many index squares are there in a PSL.

Step 2: Solve the sub-product

Sub-Product  (SP)

Don’t think that the value we’d taken from the PSL is already the correct answer. The value 04’09 is still incomplete. We must add a sub-product to come up with the ‘true’ square value of 23. But to get the sub-product of 23, we must multiply the digits 2 and 3 in a special kind of pattern.

General Rules in Dealing with the Sub-product

Rule 1: Look for the last digit of the given number

Rule 2: Double its value, meaning, multiply it by 2

Rule 3: Then multiply that value to the remaining digits on its left.

 

DOU-LAL Multiplication Pattern

Dou-LAL stands for "Double the Last Digit and All to It's Left". It is simply an easy to memorize acronym which is kind of multiplication pattern that is, effective in getting the sub-product. How it works?

HOW DOU-LAL WORKS?

In the given number 23, you may notice that 3 is the last digit, and on the left of 3 is 2.

Activity 1: Simply double the value of the last digit  (which in this case, is 3)

 3 x 2 = 6

Activity 2: Now, the "all to the left" of the last digit is in this case, is a single digit which is 2.

The next thing to do is to multiply 2 by 6, which happens to be the 'double value of the last digit' 3 of the given value 23

2 x 6 = 12

Officially, 12 will be the value of our Sub-product


Step 3: Find the 'total sum' by simply adding the Sub-product to the PSL with some little adjustment needed to perform.

 Adjust SP One Digit to The Left

In strict ruling in Math, the sub-product of 23 should be:

20 x ( 2 x 3)  = 120

But it is a 'short cut' way of writing number (for the value of SP in a two-digit squaring), if we try to omit the zero at the end of the SP

Instead of 120, we should simply write 12,  by which the last digit of SP, must be aligned to the next digit to the left of the last digit of the PSL.

 23²       = 04’09 ← PSL
 +2x3x2 =   1'2   ← SP1  (2 of 12 is aligned to second 0 of 04'09)

................05’29 ← T-Sum

Total Sum

In the above example, the value 05'29  will be our T-Sum, which is simply the result of adding the SP to the PSL. T-Sum reflects the true value of 23²  or the square value of 23.

But in 'reality', we can't simply say that the square of 23 is 05'29. Instead, we must omit the zero (0) before the 5 and erase the ' (  single close quote). 

So, the final answer will be:

23² = 529