At its simplest level this problem allows students to consolidate their understanding of how to calculate the area of irregular shapes. The extra mathematical demand comes from requiring students to identify the relationship between three variables.
This printable worksheet may be useful: Pick's Theorem.
You may be interested in our collection Dotty Grids - an Opportunity for Exploration, which offers a variety of starting points that can lead to geometric insights.
Draw a polygon on a square dotty grid on the board. Clarify that we shall be interested in three variables: the number of dots on the perimeter, $(p)$, the number of dots in the inside, $(i)$, and the area $(A)$. Ask students to work out the $(p)$, $(i)$ and $(A)$ of the shape that you have drawn.
Display the five shapes from the problem. For each, ask students to work in pairs and agree on the values of $(p)$, $(i)$ and $(A)$.
Draw attention to the two shapes that have an area of 1. What do they notice about their $(p)$ and $(i)$? Is this true for all shapes that have an area of 1? Allow the students some time to draw and share results. Confirm that there are an infinite number of possibilities of shapes which satisfy these conditions.
"The size of the shape will determine the area, so $(p)$ and $(i)$ may well determine the area. Your challenge is to draw some more shapes and find out if there is a relationship between these three variables."
Identify a central place where students can post their conjectures or other observations and encourage students to check their validity.
At an appropriate time bring the students back together to discuss the relationships they have discovered. Use this also as an opportunity to discuss effective strategies for identifying relationships, eg keeping one variable fixed.
You may wish to use the interactive pinboard, found here , to support your presentation/discussion of the problem.
Here is an account of one teacher's approach to using this problem.
If $(p)$ is fixed and $(i)$ increases by 1, what is the effect on the area?
If $(i)$ is fixed and $(p)$ increases by 1, what is the effect on the area?
Suggest that students start with shapes with small areas. How many different shapes can they draw for an area of 2? What possible values of $(p)$ and $(i)$ can they find? What about an area of 3..4..5...?
Does the same relationship hold when shapes are drawn on isometric dotty paper?