Sydney worked hard at this problem. He wrote:

I tried various combinations of numbers of dots to make rectangles. I discovered by factoring each number of dots I could figure out how many rectangles I could make out of each number of dots.

That is very well expressed. In other words, by finding pairs of numbers that multiply together to make each number of dots, you can find out how many rectangles there are. Sydney continued:

For example:
So, all numbers make a skinny rectangle.
$6$ also makes a $2\times3$
$12$ dots make three rectangles: $1\times12$, $2\times6$, and $3\times4$.
$8$ a $2\times4$
$10$ a $5\times2$
$12$ a $2\times6$ and $3\times4$
$14$ a $2\times7$
$16$ a $2\times8$
$18$ a $2\times9$ and $3\times6$.

Well done. I think by a "skinny rectangle" you mean a rectangle which is just one row of counters?

Cong of St Peter's RC Primary School, Aberdeen, points out that:
If a square can be thought as a special rectangle, then square numbers $1$, $4$, $9$, $16$ can also make special rectangles as $1 = 1 \times 1$; $4 = 2 \times 2$; $9 = 3 \times 3$; $16 = 4 \times 4$
He also spotted that
$15$ counters can make a rectangle as $15 = 3 \times 5$ (three rows of $5$ counters).