Why do this problem?
helps pupils explore the area of rectangular shapes and see the effects of changing the length of their sides. There are obvious implications for multiplication facts too. The activity also opens out the possibility of pupils asking “I wonder what would happen if . . .?” If and when they do this
their curiosity may be easily aroused and they'll pursue things further.
It would be good to introduce this activity with a practical demonstration using some blocks like Cuisenaire. It would also be good to have some discussion about the kinds of corners you can have and those that are not acceptable.
Some children get hooked on the idea of creating the nearest things to a square each time so as to get the largest area.
Others tend to just lengthen the rectangle each time and therefore lose out when they come to using eight blocks to surround the largest area - this time in the shape of a square.
If you have a large group of children doing this activity then it is very useful to have them working with different lengths of rods. This allows an interesting comparison between those who used [for example] the $4$ rods with those who used the $5$ rods and the $6$ rods. A large table of results leads usually to interesting discussions.
How are you working out what the area is that you have got your fence around?
Have you got a special way of doing these so that you know you're getting the largest area possible?
Encourage pupils to explain in words how you make sure you have the shape with the largest area each time.
For the exceptionally mathematically able
Extend the whole idea by producing $3$D cuboid cages using blocks like above and trting to get the largest volume with eeach number that you work up to. To start how about $12$ $4$'s producing a cage that is $6$x$5$x$5$:-
Keeping to blocks of four and adding $1$ more at a time, discover the largest colume of cage that can be produced. After several steps some generealizations should be looked for.
It may be necessary to help some pupils with strategies for working out the area.