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 shown above:-
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 spelling!]
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 trial routes.
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?''