This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
This activity investigates how you might make squares and pentominoes from Polydron.
If you had 36 cubes, what different cuboids could you make?
I am told that in the United States, where perhaps some of you
might live, or might have been, there are some cities and towns
where the streets are arranged in very straight lines and cross
each other at right angles. There are many mathematics problems and
challenges which use this idea and today we are going to think
about having a street party or street parade or festival.
Let's imagine that there is a small town made up of 16 blocks of
flats, so a view from above might look a bit like this:-
or as a diagram:-
The spaces in between the blocks are the roads running from
North to South and from East to West.
In this town there are going to be two different parties on the
same day which they want to keep separate. So they decide that they
will put a fence down the middle of some roads to divide the town
up into two equal parts. Because there are 16 block of buildings
altogether we'll have to have 8 blocks in each half. Each of the
two 8 blocks need to be kept together with no block being separated
from the rest of that group. So this is O.K.:-
And another way of putting the fence would be like this:-
But, so that not one block is separated from the rest of their
8, so you CANNOT have something like:-
You know, sometimes when you are doing these kinds of challenges
you have to make decisions about whether two answers are the same
if they look the same but just happen to be in a different place.
In this challenge they are different, because a route going from
North to South is different from a route going from West to East,
since you pass different people's houses.
So, the next one is counted as different from the first route
Well, you can probably guess that the challenge is to find as
many routes as you can for the fence to go so that the town is
divided up into to halves, each with 8 blocks of course!
Find some good interesting ways of recording the different
fence-routes you find.
When you've done a few there may be some things that you want to
say about how you are finding the different routes and you may be
able to prove that you've found them all! It's good to write such
ideas down and when you send in your results make sure that you
include the writing. [Don't worry about wonderful sentences or
It's probably time to ask the usual question:- "I wonder what
would happen if ...?''
a) One child, Michael, recently suggested that it would be
interesting to find out what would happen if there was a road
blockage somewhere which meant you could not lay a fence down that
bit of the road. It lead to some interesting thoughts and some
b) I could also suggest, "What would happen if the
roads were laid out in triangular arrangements?''
This could lead to one solution such as:-
c) A very obvious question would be; "I wonder what would happen
if there were more blocks of houses in the town so that the grid
was 6 by 4, maybe, or a bigger square that would be 6 by 6?''