SQUARING NUMBERS ENDING IN FIVE

[Sirjon]

Since this is math Tutorial section, let me share something that I wish, could be a big help for kids who's really having hard time understanding (or I may say, following) the traditional method of finding the square root of a number (as I do, until now). but before we go to that, let me first share you the following:

1) SQUARING TWO-DIGIT NUMBERS ENDING IN FIVE

                5^2 =      25                         

              15^2 =    225

              25^2 =    625 

              35^2 = 1,225

             45^2 = 2,025

             55^2 = 3,025

             65^2 = 4,225

             75^2 = 5,625

             85^2 = 7,225

             95^2 = 9,025

What do you notice?

First, when you square any number ending in five, you will always get an answer always ending in 25

Second, the first digit before the 5 (as in 15^2), as long as the given number is a two-digit number, always follow the rule below:

               /N//5/^2 = /Nx(N+1)//25/

Example:

                15^2 => /1//5/^2 => /1x(1+1)//25/ => /1x2//25/ => 2'25 = 225

2) SQUARING NUMBERS MORE THAN TWO DIGITS ENDING IN FIVE USING THE COPY AND PASTE RULE

 I'm sure most of us, know it well, mentally that 12^2 =144, What then if make it into 125^2? 

             125^2 =>  /12//5/^2 => /144//25/

                            +    C.A.P.    => /  12/   |               (the symbol  "|" means to bring down)                            

                                                     /156//25/ => 1'56'25 = 15,625

The C.A.P stands for copy and paste, meaning, as we already know the the square of 12, which is 144, then we automatically include 25 at the end as to the rule that the square of any number ending in 5 always ends in 25 and simply copy the number (12) and then paste it down and add to the known square of 12 (which is 144).  This rule will always work as long as we already know the square of the digits on the left of 5. 

3) AVERAGING AND FLOATING SIX RULE

Now, let's consider this, let's say that we have numbers, 10 and 15. As we can do it mentally, 10^2 = 100 and 15^2, applying the technique i presented above, will be 225. We want to know the square of 12.5, which is actually, in between the squares of  10 and 15.

                        15^^2 = 225

                   +   10^2  = 100  

                                     325 / 2 = 162.5   << This value is not accurate

If we actually  try to get the square of 12.5 , which is the in-between of 10 and 15, we got 156.25

To settle this problem, as a rule - we must omit the last 5 of the quotient (162.5 to 162) then subtract by 'floating 6' then include 25 at the end

                                     162.5 => /162./ =>/162. - 6 /<->/25/ = /156.//25/ => 156.25 = 156.25

I call it 'floating 6' because it does not follow the real math rule that involves decimal places.

So as a general rule, for this particular section,  we must omit the last digit 5 (as it should always be), then subtract by floating 6 and include the

 25 at the end part.

Now how is this relevant to finding the square root of a number? Well let me introduce first, to you another thing I call it 'String Squaring'

Let's start with                 

               12^2 = 144

Now let's include 3 next to 12 ...

               123^2  = /144/09/ (fundamental squares)

            + 12x6    = /     7  2|/  (sub-product)   

                                  15129   (total)

This time, make it 1234^2 ......

              1234^2 = 1'51'29'16

        +     123x8  =        '98'4|

                               1'52'27'56

I called it 'string squaring' as if you include another digit to the right of a number,  we can easily get the square of the new number, as long as you already know or had the square of the previous number. Again, this is important to ESR.

Now let's try to apply those rule in finding the square root of 2.00

Without specifically presenting all the details, just to give you an idea how ESR works, let's start with 1 as the first digit of our answers. Now, for most kids, the next digit after 1 is really somewhat a big question... you have nine possible guess (1.1, 1.2, 1.3, ....up 1.9) and only one is eligible .

To make it easier, we need to know the middle square value between 2 and 1 (since the 2.00 is between the squares of 2 and 1), we will get 1.5

Using the first rule, finding the square of two-digit numbers ending in five, we have now an idea that 1.5^2 = 2.25 which is way above 2.00.
So a kid will now have 4 choices instead of nine, (4,3,2,1) as the next digit after 1. (one then the decimal point).  

Then apply the 'averaging', add the square of 1.5 to 1.00, and divide by 2 the sum we get 1.625, or 1.5625 to be exact, which is way below 2.00. 

So we come up with two numbers to guess - 4 or 3. Applying the 'string squaring'...
1.4^2 = 1.16 (FS)
+1x8 =     8 |  (SP)

              1.96  Acceptable

Then for the next digit after 1.4, apply the copy and paste...

1.45^2 = /1.96/25

+CAP  =       14 |  

                 2.10'25
Way above 2.00, so again, the averaging, we come up with only two guesses, 2 and 1, use the 'string squaring'.

Repeat the process to get the other next digits.

I hope my presentation is clear to you and I hope it will help your kids, as I'm sure as you will agree, kids hate to subtract huge number as well as dividing numbers, except dividing numbers by 2, which is an easier thing to do. 

 

 

 

Thanks moderator,  for putting it on the proper section. Again, still not familiar with other sections...

 

Edited March 28, 2019 by Sirjon

[Sirjon]

Okay, since I'm still waiting for the food to cook, let me add this thing,for others who are a bit confused with squaring a number using 'string squaring'...

Always remember that in squaring a single digit number (from 1 to 9), the square value should always be in two digits (or two decimal places). So let me give you the list

                                0^2 = 00                                      5^2 = 25  
                                1^2 = 01                                      9^2 = 81

                                2^2 = 04                                      8^2 = 64

                                3^2 = 09                                      7^2 = 49

                                4^2 = 16                                      6^2 = 36

You will also notice that both squares of 1 and 9 ends in 1 (1,81). So do.to the others, with similar pattern except 0 and 5. I call them 'ten-square co-related'. This is useful when dealing with perfect square numbers. (i will show how it works, some other time).

Now, let's apply this to 'String Squaring',  this time to determine if the square root of 2, which is 1.4142,  is approximately correct. 

                      1.4142^2 = ?
                             14^2 = 01'16    <<<      Fundamental squares as equal to /x^2//y^2/ in (x+y)^2 = x^2+2xy+y^2  
                              1x8   =      8 |     <<<      Sub-product as equal to 2xy 
                          1.41^2 = 01.96'01
                           14x2   =         2'8 |  
                      1.414^2   = 01.98'81'16
                         141x8   =         1'12'8 |  
                    1.4142^2   =  01.99'93'96'04
                        " X 4       =               5'65'6 |       The symbol " just to save time in repeating to write numbers
                                            01.99'99'61'64
                     Therefore 1.4142^2 = 1.99'99'61'64 = approx. 2.00

Edited March 29, 2019 by Sirjon

[Taniya]

For the squaring numbers rule of any number of digits I found this pattern- exclude 5, multiply the remaining number with its successor and write 25 in the end of the product.

Eg  sq of 125 = (12*13)25 = 15625

      Sq of 25 = (2*3)25 = 625

     Sq of 1435 = (143*144)25 = 2059225

[Sirjon]

Yes, Tanya, that is also a good way of squaring numbers (of any number of digits) ending in five. Thank you.

Now, when we deal with perfect square numbers less than 10000, that is in four-digits, we can also do a short cut method
Check the list below:

1^2 = 1         9^2 = 81  both ends in 1
2^2 = 4         8^2 = 64  both ends in 4
3^2 = 9         7^2 = 49  both ends in 9
4^2 = 16       6^2 = 36  both ends in 6
0^2 = 0         5^2 = 25  ( no co-related pair)

Although not all numbers ending in 1, 4, 6, or 9 are perfect square numbers, it's a smart practice to always make an 'assumption' that if the last digit falls in either numbers, then it could possibly be a perfect square.

   _______
\/ 1,296
Step 1: Group the number into two digits starting from the decimal point
   ______
\/ 12'96.

Step 2: Find a single number which square value equal or nearest but less than to 12
Answer: 3 (the square of 3, which is 9)

Since the given number end in 6, we can make assumption that last digit of your possible answers could either be 4 or 6
Possible Answers:   34, 36

Step 3: To get the correct answer, multiply the first digit of your known possible square roots  by a number next to it, higher by 1. 
Next to 3 is 4, so 3 x 4 =12 (to get the estimate middle square value) 

Rule 1: If the first two digits of the actual given value (12'96) is higher, or equal to the estimate middle square value (which is12), choose the higher possible answer
Rule 2: If it is lower to the estimate middle square value, choose the lower possible answer

Answer : 36
Check: Square of 34 = 1,156  ( in 11'56, 11 is less than 12)