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.
