## Thursday, July 30, 2015

### Perfect Squares and Patterns

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 12 up to 92. However, it would be even better if students can memorize up to 192. 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
112   = 121                 [+23]                                                        +2
122   = 144                 [+25]                                                        +2
132   = 169                 [+27]                                                        +2
142   = 196                 [+29]                                                        +2
152   = 225                 [+31]                                                        +2
162   = 256                 [+33]                                                        +2
172   = 289                 [+35]                                                        +2
182   = 324                 [+37]                                                        +2
192   = 361

Pattern: the difference between 112 and 122 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 152 and 252 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.