## Squares ...5

Squares:  of numbers ending in a five
Vocabulary: a square is a number multiplied by itself.  In the junior part of school there is an implied assumption that the number to be squared is an integer. That may be because you don’t like decimlas and fractions, but it could be the opther way around, that you don’t ‘like’ fractions becasue you haven’t used them (enough to like them).

Learning: the squares of 1 to 12 are assumed to be known before joining Y8. Learn 13 to 30 squared (after doing this section, because this should help you make some sense of the numbers !).

Look at this table: when you have had a good look, copy some columns of it into your book.

 Number Front end factors of the front bit Number Front end ...............................Ideas.............................. 05 00 25 0 x 1 25 000 625 15 01 25 125 25 06 25 2 x 3 225 050 625 35 12 25 325 45 20 25 425 180 625 55 30 25 525 65 42 25 625 390 625 75 56 25 85 72 25 95 90 25 105 110 25 1025 115 132 25 125 156 25 1225 135 182 25 145 210 25 155 240 25 1525

Can you see that all the squares end in 25?

Can you see any connections for the front few digits ?  I have separated them out to help.

Look at the factors of the front digits.             Can you find a pattern that predicts these?

Predict some of the factors; test your ideas to see if you have found a rule of some sort.

Try to write your rule down.

What would happen if you wrote down the squares of numbers ending in a half?

Try writing the corresponding table for squares of numbers ending in a half. You will find the process easier if you use fractions rather than decimals (honest!). Hopefully you think the connection with the table above is pretty obvious; Is the rule for predicting these squares a bit easier to write down? Most people find it easier, but are reluctant to work with fractions…….

I’ve added some more columns on the right for those who thought the page too easy - let us call this extension work.

Other pages on Squares are:
The Next Square,
Squares - Theory,
Pythagoras,
Arithmetic Skills

Theory:
(10x+5)² = 100x²+100x+25 = 100x(x+1) + 25. the x00 makes space for the 25 to fit.
(100x+25)² = 10000x²+5000x+625 = 10000x(x+½) +625=1000x(10x+5) +625.
the x000 makes space for the 625 to fit.

Email: David@Scoins.net      © David Scoins 2018