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## 'Perimeter Expressions' printed from http://nrich.maths.org/

Charlie took a sheet of paper and cut it in half.

Then he cut one of those pieces in half, and repeated until he had
five pieces altogether.

He labelled the sides of the smallest rectangle, $a$ for the
shorter side and $b$ for the longer side.

Here is a shape that Charlie made by combining the largest and
smallest rectangles:

Check you agree that the perimeter is $10a + 4b$.

Alison combined the largest and smallest rectangles in a different
way. Her shape had perimeter $8a + 6b$. Can you find how she
might have done it?

Charlie and Alison made sure their rectangles always met along an
edge, with corners touching.

Can you combine the largest and smallest rectangles in this way to
create other perimeters?

Create some other shapes by combining two or more rectangles,
making sure they meet edge to edge and corner to corner. What can
you say about the areas and perimeters of the shapes you can
make?

If you have a friend to work
with, you could each create a shape and work out the area and
perimeter. Can you recreate each other's arrangement if you only
know the area and perimeter?
Here are some questions to
consider:

What's the largest perimeter you can make using ALL the
pieces?

Can you make two different shapes which have the same area

and perimeter as each
other?

Can you make two different shapes which have the same perimeter but
different areas?

How do you combine any set of rectangles to create the largest
possible perimeter?

Charlie thinks he has found a shape with the perimeter $7a + 4b$.
Can you find his shape?

What can you say about the perimeters it is possible to make, if
$a$ and $b$ are the dimensions of one of the other
rectangles?

This problem is based on an idea
shared by Sue Southward.