A Trick for Calculating Squares

Here's a nice trick to figure out the square of a number next to one you know.

If you know that 12²=144 and you want to know what 13² is all you need to do is add 12 and 13 to 144 so 

13² = 144 + 12 + 13 

    = 169.

If you know that 20²=400, 

then 21²= 400 + 20 + 21 = 441. 

This is a nice little trick to make it look like you're really good at knowing perfect squares.  To be even more impressive, you can use a similar trick even if the numbers are more then 1 apart.

You know that 20²=400 and you want to know what 23² is.  

20 and 23 are 3 apart.  

And 20 + 23 = 43.

So: 

23² = 400 + 3 x 43 

        = 400 + 129 

        = 529.  

Oh, and you can also do it backwards!  If you want to know what 18² is, start with 20².

20² = 400

20 and 18 are 2 apart.

20 + 18 = 38

So:

18² = 400 - 2 x 38

        = 400 - 76

        = 324

Note here you're subtracting instead of adding, since you're going down from 20 to 18.

And, a challenge for grade 11 and 12 students:  figure out the math behind this trick.   Hint:  try rewriting the problem (in this case calculating 23²) as 23² - 20²  and using a difference of squares factoring!