**Perfect Squares and Patterns**If you have been following my last blog, you may find this new post helpful for you in solving questions of Pythagorean Theorem.

Firstly, I am going to talk about perfect squares.

The following are some examples and non-examples of perfect squares from the online resource:

http://www.mathwarehouse.com/arithmetic/numbers/what-is-a-perfect-square.php.

### Examples of perfect squares

- 9
- 9 is a perfect square because it can be expressed as 3 * 3 (the product of two equal integers)

- 16
- 16 is a perfect square because it can be expressed as 4 * 4 (the product of two equal integers)

- 25
- 25 is a perfect square because it can be expressed as 5 * 5 (the product of two equal integers)

###
__Non__ examples of perfect squares

- 8
- 8 is a not perfect square because you cannot express it as the product of two equal integers

- 5
- 5 is a not perfect square because it cannot be expressed as the product of two equal integers

- 7
- 7 is a not perfect square because you cannot express it as the product of two equal
- integers

Hopefully, these above examples has helped you understand more about perfect squares.

Secondly, I would like to introduce some easier and faster ways for students to calculate their questions as I did in my last post.

Usually, students would just know the square of one digit number which is from 1

^{2}up to 9^{2}. However, it would be even better if students can memorize up to 19^{2}. They could either memorize them just by memorizing or they could memorize them using a pattern.**1) Perfect Squares - Pattern 1**

#’s Squares Difference between squares Difference increases by

11

^{2}= 121 [+23] +2
12

^{2 }= 144 [+25] +2
13

^{2 }= 169 [+27] +2
14

^{2 }= 196 [+29] +2
15

^{2 }= 225 [+31] +2
16

^{2 }= 256 [+33] +2
17

^{2 }= 289 [+35] +2
18

^{2 }= 324 [+37] +2
19

^{2 }= 361
Pattern: the difference between 11

^{2}and 12^{2}is +23, and then the difference increases by +2 as the number increases by 1.**2) Perfect Squares - Pattern 2**

#’s Squares Difference between squares Difference increases by

152 = 225 +400 +200

252 = 625 +600 +200

352 = 1225 +800 +200

452 = 2025 +1000 +200

552 = 3025 +1200 +200

652 = 4225 +1400 +200

752 = 5625 +1600 +200

852 = 7225 +1800 +200

952 = 9025

Pattern: the difference between 15

^{2}and 25^{2}is +400, and then the difference increases by +200 as the number increases by 10.
By knowing and memorizing these patterns, students can calculate squares faster.

Hope this post helps you in calculating squares as well.