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Why do this problem?
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.
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
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
Identify a central place where students can post their conjectures
or other observations and encourage students to check their
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
, to support your presentation/discussion of the
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
If $(i)$ is fixed and $(p)$ increases by 1, what is the effect on
Does the same relationship hold when shapes are drawn on isometric
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