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?
It is wonderful to receive solutions that show
so much thought and effort. Joanne has a really neat way of
thinking about patterns with start points for the routes. (Well
done!) Natasha has tried to report her work in an organised way and
come up with some good "What if...'' questions. Jenny has a nice
way of explaining her thinking. Jill tries to look at the
investigation in some different ways.
I first thought about the simple routes, for example straight
down. That can be changed by going horizontally across the
Then I worked on harder routes. I fund that all regular routes
had 4 starting points around the road pattern (without going back
over the same route). The irregular pattern of routes had 8
starting points around the road pattern.
In finding this out, I have found out the number of times each
route can be used.
The set of road patterns are as follows:
(each map shows one route and all its starting points)
I am finding the routes by picking a square and then just
doodling until I find a route that is split into halves (8). Also,
by doing the opposite to the one I had done before. I am also
drawing a very faint line horizontally - and vertically in pencil,
so that I can see roughly where I have to split the street into
two. So really what I had to do was divide 16 by 2 = 8.
I had this sudden thought that what if I worked out how many
ways I could divide 16 and then I thought maybe I could then prove
that I had found them all. But as you can see below, that was
definitely not going to work because I only found 2 ways.
My suggestions for "What if''
What would happen if two houses didn't approve of a street
party, therefore you wouldn't be able to split the houses (build a
What would happen if someone from the street were on holiday and
they were coming back on the night of the party and you didn't want
to build a fence because you didn't know how they would react.
What would happen if you had 49 houses to try and split
What you would have to do is split the 49 houses into 2 24s that
would make 48, then you would have to share the house left over so
that you could have 6 hours there each (half of 12 which is the
night), or if you were counting it by the whole day (24 hours) 1
street party would use that house 1/4 of the way through the day
and have 6 hours and the other party would start 3/4 of the way