### Geoboards

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

### Polydron

This activity investigates how you might make squares and pentominoes from Polydron.

If you had 36 cubes, what different cuboids could you make?

# Street Party

##### Stage: 2 Challenge Level:

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.

Solution 1, Joanne, West Flegg Middle School

I first thought about the simple routes, for example straight down. That can be changed by going horizontally across the middle.

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)

Solution 2, Natasha, West Flegg Middle School

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.

 This is as far as I got until I moved on

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 fence there?)

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 equally?

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 through.