Making Shapes

Arrange any number of counters from these 18 on the grid to make a rectangle. What numbers of counters make rectangles? How many different rectangles can you make with each number of counters?
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
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Problem



I have three counters.  I arrange them on a grid to make a rectangle like this:



Image
Making Shapes
 
I wonder if there are any other rectangles I could make with just three counters?

If I had six counters, I could make a rectangle like this:

Image
Making Shapes


Are there any other rectangles that I could make with six counters?

Imagine you have $18$ counters to put on a grid.

Arrange any number of counters on the grid to make a rectangle (not just its outline).

How many different rectangles can you make with each number of counters?