Squares: theory

Theory for handling Squares


Here is one of my favourite haiku, developed from a year 8 homework:


This + That all squared 
is This squared plus That squared plus
two times This times That


We can write this in a sort of algebra as 
               
(This + That)² = This² + That² + 2 x This x That
or, if you replace This with x and That with y. it reads more like   
                            (x + y)
² = x² + y² + 2 x y

which is usually written  in order of the x terms as (x + y)² = x² + 2xy + y²


When the ‘That’ is 1, (y = 1) then we have  (x + 1)² = x² + 2x + 1 and we might recognise that  2x + 1 = x + x+1; this looks unnecessarily silly at first but this is x plus x+1, so:

Example::  if x is ten then x plus x+1 means ten plus eleven….. so eleven² is ten² plus ten plus eleven      which is 100+10+11 = 121.
In the same way,  if I tell you that 31
² is 961 you can work out 32² = 31² + 31 + 32 = 961 + 31+ 32 = 1024.
Exercises:  13², 21², 14², 31², 15², 33², 34², 35², 21², 51². Check the last digit makes sense.
Similarly 19
², 99², 101², 102², 103², 1013², using previous answers perhaps


A special case:

Using the expansion for a square we can see that (x+½)²  = x² + x + 1/4 . 
If x is a whole number then x
² is; so is x² + x; clearly (well,  I think it is clear) x is a factor of  both x and of x² , so it must be a factor of x² + x. The other factor must be x+1. If you think about (x+1), it is the next whole number after the one you’ve read on the left hand side, the bit before the “and a half”, so (x+½)² = x (x+1) + ¼
Example:  5½ ² = 5x6 + ¼ = 30¼

Exercises:  (these should take little or no calculation,  but show detail for the first few)

1  Write down a) 7½ ² b) 8 ½ ² c) 9½ ²   d) 20½ ²   e)  15½ ²  

2  By adding the right numbers of zeros, use the last answers to write down 
a) 75²  b) 85²,  c) 95²,  d) 155²  e) 750²  f)  0.75²  g) 0.155² h) 0.25²  i) 0.085².
3Find
a) 10½ ²  b) 30½ ²  c) 99½ ²   d) 200½ ²   e)  44½ ².


No answers given, instant calculation for marker.

© David Scoins 2017