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 31² 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 19², 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 x² 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.
