How Learning Patterns Leads to Brighter Students?

Do you know we could make optimal use of the things that we have been told they exist since we were primary school students such as the table that contains the very common info, like the table of squares?

$x$         $x^2$

$1$          $1$
$2$          $4$
$3$          $9$
$4$        $16$

What can we do about it, you might wonder. It isn't like we could change the values of the squares, could we? Nope, of course not! But we could let our students to study the patterns of the table of squares, little do they know we could observe many interesting facts from the table.

First of all, let us look at the first square numbers as follows:

$1,\,4,\,9,\,16,\,25,\,36,\,49,\,64,\,81,\,100$

Note that

[MATH]\color{black}1+\color{yellow}\bbox[5px,purple]{3}\color{black}=4[/MATH]

[MATH]\color{black}4+\color{yellow}\bbox[5px,purple]{5}\color{black}=9[/MATH]

[MATH]\color{black}9+\color{yellow}\bbox[5px,purple]{7}\color{black}=16[/MATH]

[MATH]\color{black}16+\color{yellow}\bbox[5px,purple]{9}\color{black}=25[/MATH]

[MATH]\color{black}25+\color{yellow}\bbox[5px,purple]{11}\color{black}=36[/MATH]

[MATH]\color{black}36+\color{yellow}\bbox[5px,purple]{13}\color{black}=49[/MATH]

[MATH]\color{black}49+\color{yellow}\bbox[5px,purple]{15}\color{black}=64[/MATH]

[MATH]\color{black}64+\color{yellow}\bbox[5px,purple]{17}\color{black}=81[/MATH]

[MATH]\color{black}81+\color{yellow}\bbox[5px,purple]{19}\color{black}=100[/MATH]

Note that the addends to the squares to form the next square is actually an arithmetic sequence, with the first term as [MATH]\color{yellow}\bbox[5px,purple]{3}[/MATH] and the common difference as [MATH]\color{yellow}\bbox[5px,purple]{2}[/MATH].

That means, if two squares have the difference of [MATH]\color{yellow}\bbox[5px,green]{19}[/MATH], that means the two squares must come from the pair $81$ and $100$ and no other pairs else.

This might not immediately "mean" something to the students, but this observation could help us to solve for one really hard mathematics competition problem, a problem that the math educator could use in classroom to promote creativity and critical thinking in my next blog post.