54
/www/nrich/html/content/98/11/bbprob1/
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2011-02-01T00:00:01
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<p>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.</p>
<p>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:-</p>
<p style="text-align: center;"><mdo:image height="373" width="370" alt="pic3" src="Picture%203.jpg"></mdo:image></p>
<p style="text-align: center;">or as a diagram:-</p>
<p style="text-align: center;"> </p>
<div style="text-align: center;"><mdo:image src="Grid1.gif" alt=""></mdo:image></div>
<br></br>
<p>The spaces in between the blocks are the roads running from
North to South and from East to West.</p>
<p>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.:-</p>
<mdo:image height="373" width="372" src="Picture%205.jpg" alt="pic5"></mdo:image><mdo:image src="Grid1a.gif" alt=""></mdo:image> <br></br>
<p>And another way of putting the fence would be like this:-</p>
<mdo:image height="369" width="367" src="Picture%208.jpg" alt="pic8"></mdo:image><mdo:image src="Grid1b.gif" alt=""></mdo:image> <br></br>
<p>But, so that not one block is separated from the rest of their
8, so you CANNOT have something like:-</p>
<mdo:image height="370" width="369" src="Picture%206.jpg" alt="pic6"></mdo:image><mdo:image src="Grid1Not.gif" alt=""></mdo:image> <br></br>
<p>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.</p>
<p>So, the next one is counted as different from the first route
shown above:-</p>
<mdo:image height="371" width="366" src="Picture%207.jpg" alt="pic7"></mdo:image><mdo:image src="Grid1c.gif" alt=""></mdo:image> <br></br>
<p>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!</p>
<p>Find some good interesting ways of recording the different
fence-routes you find.</p>
<br></br>
<p>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!]</p>
<br></br>
<p>It's probably time to ask the usual question:- "I wonder what
would happen if ...?''</p>
<p>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.</p>
<p>b) I could also suggest, "What would happen if the
roads were laid out in triangular arrangements?''</p>
<p style="text-align: center;"><mdo:image height="571" width="490" alt="Mant Tri" src="Many%20Tri%20Roofs.jpg"></mdo:image></p>
<div style="text-align: center;"><mdo:image src="Grid2.gif" alt=""></mdo:image></div>
<br></br>
<p>This could lead to one solution such as:-</p>
<div style="text-align: center;"><mdo:image src="Grid2a.gif" alt=""></mdo:image></div>
<br></br>
<p>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?''</p>
<br></br></mdoxml>
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<p class="editorial">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.</p>
<hr width="50%"></hr>
<strong>Solution 1, Joanne, West Flegg Middle School</strong>
<p>I first thought about the simple routes, for example straight
down. That can be changed by going horizontally across the
middle.</p>
<p>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.</p>
<p>In finding this out, I have found out the number of times each
route can be used.</p>
<p>The set of road patterns are as follows:<br></br>
(each map shows one route and all its starting points)</p>
<table>
<tbody>
<tr>
<td><mdo:image src="t-joanne_1.gif" alt=""></mdo:image></td>
<td><mdo:image src="t-joanne_2.gif" alt=""></mdo:image></td>
<td><mdo:image src="t-joanne_3.gif" alt=""></mdo:image></td>
</tr>
<tr>
<td><mdo:image src="t-joanne_4.gif" alt=""></mdo:image></td>
<td><mdo:image src="t-joanne_5.gif" alt=""></mdo:image></td>
<td><mdo:image src="t-joanne_6.gif" alt=""></mdo:image></td>
</tr>
<tr>
<td><mdo:image src="t-joanne_7.gif" alt=""></mdo:image></td>
<td><mdo:image src="t-joanne_8.gif" alt=""></mdo:image></td>
<td><mdo:image src="t-joanne_9.gif" alt=""></mdo:image></td>
</tr>
</tbody>
</table>
<hr width="50%"></hr>
<strong>Solution 2, Natasha, West Flegg Middle School</strong>
<p>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.</p>
<p>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.</p>
<mdo:image width="250" height="250" src="t-natasha_idea.gif" alt=""></mdo:image> <br></br>
<table>
<tbody>
<tr>
<td><mdo:image src="t-natasha_1.gif" alt=""></mdo:image></td>
<td><mdo:image src="t-natasha_2.gif" alt=""></mdo:image></td>
</tr>
<tr>
<td><mdo:image src="t-natasha_3.gif" alt=""></mdo:image></td>
<td><mdo:image src="t-natasha_4.gif" alt=""></mdo:image></td>
</tr>
<tr>
<td><mdo:image src="t-natasha_5.gif" alt=""></mdo:image></td>
<td><mdo:image src="t-natasha_6.gif" alt=""></mdo:image></td>
</tr>
<tr>
<td><mdo:image src="t-natasha_7.gif" alt=""></mdo:image></td>
<td><mdo:image src="t-natasha_8.gif" alt=""></mdo:image></td>
</tr>
<tr>
<td></td>
</tr>
<tr>
<td><mdo:image src="t-natasha_9.gif" alt=""></mdo:image></td>
<td><mdo:image src="t-natasha_10.gif" alt=""></mdo:image></td>
</tr>
<tr>
<td></td>
<td>This is as far as I got until I moved on</td>
</tr>
<tr>
<td><mdo:image src="t-natasha_11.gif" alt=""></mdo:image></td>
<td><mdo:image src="t-natasha_12.gif" alt=""></mdo:image></td>
</tr>
</tbody>
</table>
<p><strong>My suggestions for "What if''</strong></p>
<p>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?)</p>
<table>
<tbody>
<tr>
<td><mdo:image src="t-natasha_f1.gif" alt=""></mdo:image></td>
<td><mdo:image src="t-natasha_f2.gif" alt=""></mdo:image></td>
</tr>
<tr>
<td><mdo:image src="t-natasha_f3.gif" alt=""></mdo:image></td>
<td><mdo:image src="t-natasha_f4.gif" alt=""></mdo:image></td>
</tr>
</tbody>
</table>
<p>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.</p>
<table>
<tbody>
<tr>
<td><mdo:image src="t-natasha_h1.gif" alt=""></mdo:image></td>
<td><mdo:image src="t-natasha_h2.gif" alt=""></mdo:image></td>
</tr>
<tr>
<td><mdo:image src="t-natasha_h3.gif" alt=""></mdo:image></td>
<td><mdo:image src="t-natasha_h4.gif" alt=""></mdo:image></td>
</tr>
</tbody>
</table>
<p>What would happen if you had 49 houses to try and split
equally?</p>
<p><mdo:image src="t-natasha_49.gif" alt=""></mdo:image></p>
<p>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.</p>
<br></br>
<br></br>
<br></br>
<br></br></mdoxml>
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<mdoxml version="1.0"><div class="embed">
<h2>Street Party</h2>
<br></br>
<p>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.</p>
<p>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:-</p>
<p style="text-align: center;"><mdo:image alt="pic3" height="373" src="Picture%203.jpg" width="370"></mdo:image></p>
<p style="text-align: center;">or as a diagram:-</p>
<p style="text-align: center;"> </p>
<div style="text-align: center;"><mdo:image alt="" src="Grid1.gif"></mdo:image></div>
<br></br>
<p>The spaces in between the blocks are the roads running from North to South and from East to West.</p>
<p>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.:-</p>
<mdo:image alt="pic5" height="373" src="Picture%205.jpg" width="372"></mdo:image><mdo:image alt="" src="Grid1a.gif"></mdo:image><br></br>
<p>And another way of putting the fence would be like this:-</p>
<mdo:image alt="pic8" height="369" src="Picture%208.jpg" width="367"></mdo:image><mdo:image alt="" src="Grid1b.gif"></mdo:image><br></br>
<p>But, so that not one block is separated from the rest of their 8, so you CANNOT have something like:-</p>
<mdo:image alt="pic6" height="370" src="Picture%206.jpg" width="369"></mdo:image><mdo:image alt="" src="Grid1Not.gif"></mdo:image><br></br>
<p>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.</p>
<p>So, the next one is counted as different from the first route shown above:-</p>
<mdo:image alt="pic7" height="371" src="Picture%207.jpg" width="366"></mdo:image><mdo:image alt="" src="Grid1c.gif"></mdo:image><br></br>
<p>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!</p>
<p>Find some good interesting ways of recording the different fence-routes you find.</p>
<br></br>
<p>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!]</p>
<br></br>
<p>It's probably time to ask the usual question:- "I wonder what would happen if ...?''</p>
<p>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.</p>
<p>b) I could also suggest, "What would happen if the roads were laid out in triangular arrangements?''</p>
<p style="text-align: center;"><mdo:image alt="Mant Tri" height="571" src="Many%20Tri%20Roofs.jpg" width="490"></mdo:image></p>
<div style="text-align: center;"><mdo:image alt="" src="Grid2.gif"></mdo:image></div>
<br></br>
<p>This could lead to one solution such as:-</p>
<div style="text-align: center;"><mdo:image alt="" src="Grid2a.gif"></mdo:image></div>
<br></br>
<p>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?''</p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>This is a good <a href="http://nrich.maths.org/public/viewer.php?obj_id=54&part=">activity</a> for getting good language flowing when encouraging the children to explain their thinking.</div>
<br></br>
<h3>Possible approach</h3>
<div>It is probably best to start with the pupils sitting round in a circle with some blocks in the middle that represent the blocks of flats. Then, introduce the challenge and get some ideas, showing them physically where the fence could go with string or ribbon.</div>
<br></br>
<div>Then the pupils can have time for exploring the ideas on their own or in small groups.</div>
<br></br>
<h3>Key questions</h3>
<div>How did you get this idea?</div>
<div>Are any of yours the same as any others?</div>
<div>How do you make sure that you have done the halving of the 16 blocks?</div>
<br></br>
<h3>Possible extension</h3>
<div>As suggested in the problem, some children will enjoy looking at other shapes and extending the number of blocks in either a square or rectangular shape.</div>
<br></br>
<h3>Possible support</h3>
<div>Some blocks or something else for pupils to use individually to represent the blocks of flats will be useful.</div>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">One way to begin this challenge might be to see how
many fence routes you can find by always starting in the same
place.<br></br>
You could use some blocks or cardboard boxes to represent the
buildings, and lay them on the floor or on a table. Then you could
use string or ribbon to show the route of the fence.
You could draw your different arrangements, or perhaps take some
photos. <br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<a href="http://nrich.maths.org/public/viewer.php?obj_id=54&part="></a>This
isa good little challenge. Many of the children found,
last time I used it, the number of repeats of the same shaped fence
that were able to be used. I found it valuable for getting good
language flowing when encouraging the children to explain why there
were 2 of one shape and 4 of another shape of route. <br></br>
<br></br>
For higher-attaining pupils I would suggest that the triangular
variety should also be approached. I've only played around with the
ideas with another friend and it seems to have quite a potential
for developing good Mathematics thoughts and proofs.<br></br>
<br></br>
If your pupils are not so good with fine motor skills and you want
them to concentrate on the mathematics rather than the drawing, do
print out a hard copy of the challenge and photocopy the empty
grids, or create some of your own. I suggest this as you can
appreciate that many children will find it hard to get the gaps
between the block suitable for them to use in order to show the
route of the fences.<br></br></mdoxml>
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Street Party
The challenge here is to find as many routes as you can for a fence
to go so that this town is divided up into two halves, each with 8
blocks.
Using, Applying and Reasoning about Mathematics
Investigations
Decision Mathematics and Combinatorics
Combinations
Using, Applying and Reasoning about Mathematics
Working systematically