108
/www/nrich/html/content/02/05/bbprob1/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p>In Teddy Town, teddies are either red or yellow and they live in
red or yellow houses. There are 4 teddies - 2 red and 2 yellow, and
4 houses - 2 red and 2 yellow.</p>
<p>Can you match each teddy to a house so that the four pairs are
all different from each other?</p>
<p><mdo:flash height="300" width="300"><param value="teddy2.swf" name="movie" ></param><param value="high" name="quality" ></param><param value="#FFFFCC" name="bgcolor" ></param></mdo:flash></p>
<p style="text-align: center;"> </p>
<p style="text-align: center;">Imagine now that there are three
different colours of teddies and houses. red, yellow and blue. In
Teddy Town now there are 9 teddies and 9 houses:</p>
<p><mdo:flash height="300" width="560"><param value="teddy3.swf" name="movie" ></param><param value="high" name="quality" ></param><param value="#FFFFCC" name="bgcolor" ></param></mdo:flash></p>
<p>What are the nine different combinations of teddies and
houses?</p>
<p>Here is a map showing Teddy Town:</p>
<p><mdo:image alt="Map of Teddy Town." src="map.gif"></mdo:image></p>
<p>The streets are very special. If you walk along a street from
east to west, or west to east, all the houses are a different
colour and the teddies living in the houses are a different colour
too. The same is true if you walk along the streets in a
north-south or south-north direction.<br></br>
In other words, looking at the map grid, each row and column must
have different coloured houses and different coloured
teddies.<br></br>
Can you arrange the nine different combinations you've found on the
map grid?</p>
<p><mdo:flash height="400" width="450"><param value="teddy3g.swf" name="movie" ></param><param value="high" name="quality" ></param><param value="#FFFFCC" name="bgcolor" ></param></mdo:flash></p>
<p>Teddy Town is expanding rapidly as green teddies move to the
area and green houses are built. Now there are 16 teddies and 16
houses:</p>
<p><mdo:flash height="550" width="560"><param value="teddy4.swf" name="movie" ></param><param value="high" name="quality" ></param><param value="#FFFFCC" name="bgcolor" ></param></mdo:flash></p>
<p>Find the sixteen different ways to combine the teddies and
houses now.</p>
<p>How could these sixteen households be organised on the map now?
Remember that in each row and column there must be both different
coloured houses and teddies.</p>
<p><mdo:flash height="500" width="450"><param value="teddy4g.swf" name="movie" ></param><param value="high" name="quality" ></param><param value="#FFFFCC" name="bgcolor" ></param></mdo:flash></p>
<p>Now ... yes, you've guessed it. Another colour teddy bear has
moved to Teddy Town. As well as red, yellow, blue and green teddies
there are now purple teddies. Of course, this means that purple
houses will have to be built. So, now in Teddy Town there are 5 of
each colour bear, making 25 teddies in all, and also 25 houses,
again 5 of each colour. Can you make the 25 different combinations
of teddy and house now?</p>
<p><mdo:flash height="550" width="560"><param value="teddy5.swf" name="movie" ></param><param value="high" name="quality" ></param><param value="#FFFFCC" name="bgcolor" ></param></mdo:flash></p>
<p>Arrange these on the street map below in the same way as
before:</p>
<mdo:flash height="600" width="550"><param value="teddy5g.swf" name="movie" ></param><param value="high" name="quality" ></param><param value="#FFFFCC" name="bgcolor" ></param></mdo:flash>
<p>Teddy Town is becoming very overcrowded! However, there is just
enough room for some black teddies to join. Living there now are 36
teddy bears: 6 red, 6 yellow, 6 blue, 6 green, 6 purple and 6
black. There are 36 houses for them to live in: 6 red, 6 yellow, 6
blue, 6 green, 6 purple and 6 black. Make the 36 combinations of
teddies and houses.</p>
<p><mdo:flash height="550" width="560"><param value="teddy6.swf" name="movie" ></param><param value="high" name="quality" ></param><param value="#FFFFCC" name="bgcolor" ></param></mdo:flash></p>
<p>Do you think it will be possible to put these 36 combinations in
the street grid? May be it's not. Have a go!</p>
<p><mdo:flash height="600" width="550"><param value="teddy6g.swf" name="movie" ></param><param value="high" name="quality" ></param><param value="#FFFFCC" name="bgcolor" ></param></mdo:flash></p>
<p>Now, look back at what you have done and ask yourself some of
these questions:</p>
<ul>
<li>Was it easier to arrange the combinations in some of the grid
sizes compared with others?</li>
<li>Why do you think this is?</li>
<li>What was your strategy for solving the arrangement puzzle each
time?</li>
<li>What would happen if the two diagonals on the map also had to
have different coloured houses and different coloured teddies? Can
you solve the problem for each street plan now?</li>
</ul>
<h3>Acknowledgements.</h3>
<p>This activity is based on a Bernard's Bag problem from December
1997 called Tea Cups. The idea for the teddies came from Andrew
Massey who is an Advisor for Worcestershire County Council. Thank
you! Many thanks also to <a href="http://www.learningresources.com/">learningresources.com</a> for
the use of the bear images.</p>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<span class="editorial">Tim has been very busy arranging teddies.
He sent us pictures of his arrangements. Well done Tim!</span>
<br></br>
<br></br>
<mdo:image width="302" height="302" src="teddytown1.gif" alt="Red and yellow teddies and their houses"></mdo:image><mdo:image width="548" height="300" src="teddytown2.gif" alt="Red, yellow and blue teddies and their houses"></mdo:image><mdo:image width="439" height="395" src="teddytown3.gif" alt="Red, yellow and blue teddies and houses arranged on the map"></mdo:image><mdo:image width="553" height="389" src="teddytown4.gif" alt="Red, yellow, blue and green teddies and their houses"></mdo:image><mdo:image width="339" height="432" src="teddytown5.gif" alt="Red, yellow, blue and green teddies and their houses arranged on the map"></mdo:image>
<mdo:image width="547" height="431" src="teddytown6.gif" alt="Red, yellow, blue, green and purple teddies and their houses"></mdo:image><mdo:image width="378" height="520" src="teddytown7.gif" alt="Red, yellow, blue, green and purple teddies with their houses arranged on the map"></mdo:image>
<br></br>
<mdo:image width="555" height="434" src="teddytown8.gif" alt="Red, yellow, blue, green, purple and black teddies and their houses"></mdo:image>
<br></br>
<br></br>
Tim said that he couldn't arrange the houses and teddies on the map
when there were six colours, but he wasn't sure why. In fact, this
is quite a famous problem. You can read about it on the <a href="http://nrich.maths.org/public/viewer.php?obj_id=1453&part=index">NRICH</a> site or at <a href="http://www.cut-the-knot.org/arithmetic/latin3.shtml" onclick="mediaSave()">www.cut-the-knot.org</a> , for example (our teddy
bear problem is equivalent to finding two orthogonal Latin
squares of order 6). <br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><div class="embed">
<h2>Teddy Town</h2>
<br></br>
<p>In Teddy Town, teddies are either red or yellow and they live in red or yellow houses. There are 4 teddies - 2 red and 2 yellow, and 4 houses - 2 red and 2 yellow.</p>
<p>Can you match each teddy to a house so that the four pairs are all different from each other?</p>
<p><mdo:flash height="300" id="/content/02/05/bbprob1/teddy2.swf" width="300"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param>
<param name="movie" value="teddy2.swf"></param>
<param name="quality" value="high"></param>
<param name="bgcolor" value="#FFFFCC"></param>
</mdo:flash></p>
<p style="text-align: center;"> </p>
<p style="text-align: center;">Imagine now that there are three different colours of teddies and houses. red, yellow and blue. In Teddy Town now there are 9 teddies and 9 houses:</p>
<p><mdo:flash height="300" id="/content/02/05/bbprob1/teddy3.swf" width="560"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param>
<param name="movie" value="teddy3.swf"></param>
<param name="quality" value="high"></param>
<param name="bgcolor" value="#FFFFCC"></param>
</mdo:flash></p>
<p>What are the nine different combinations of teddies and houses?</p>
<p>Here is a map showing Teddy Town:</p>
<p><mdo:image alt="Map of Teddy Town." src="map.gif"></mdo:image></p>
<p>The streets are very special. If you walk along a street from east to west, or west to east, all the houses are a different colour and the teddies living in the houses are a different colour too. The same is true if you walk along the streets in a north-south or south-north direction.<br></br>
In other words, looking at the map grid, each row and column must have different coloured houses and different coloured teddies.<br></br>
Can you arrange the nine different combinations you've found on the map grid?</p>
<p><mdo:flash height="400" id="/content/02/05/bbprob1/teddy3g.swf" width="450"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param>
<param name="movie" value="teddy3g.swf"></param>
<param name="quality" value="high"></param>
<param name="bgcolor" value="#FFFFCC"></param>
</mdo:flash></p>
<p>Teddy Town is expanding rapidly as green teddies move to the area and green houses are built. Now there are 16 teddies and 16 houses:</p>
<p><mdo:flash height="550" id="/content/02/05/bbprob1/teddy4.swf" width="560"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param>
<param name="movie" value="teddy4.swf"></param>
<param name="quality" value="high"></param>
<param name="bgcolor" value="#FFFFCC"></param>
</mdo:flash></p>
<p>Find the sixteen different ways to combine the teddies and houses now.</p>
<p>How could these sixteen households be organised on the map now? Remember that in each row and column there must be both different coloured houses and teddies.</p>
<p><mdo:flash height="500" id="/content/02/05/bbprob1/teddy4g.swf" width="450"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param>
<param name="movie" value="teddy4g.swf"></param>
<param name="quality" value="high"></param>
<param name="bgcolor" value="#FFFFCC"></param>
</mdo:flash></p>
<p>Now ... yes, you've guessed it. Another colour teddy bear has moved to Teddy Town. As well as red, yellow, blue and green teddies there are now purple teddies. Of course, this means that purple houses will have to be built. So, now in Teddy Town there are 5 of each colour bear, making 25 teddies in all, and also 25 houses, again 5 of each colour. Can you make the 25 different combinations of
teddy and house now?</p>
<p><mdo:flash height="550" id="/content/02/05/bbprob1/teddy5.swf" width="560"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param>
<param name="movie" value="teddy5.swf"></param>
<param name="quality" value="high"></param>
<param name="bgcolor" value="#FFFFCC"></param>
</mdo:flash></p>
<p>Arrange these on the street map below in the same way as before:</p>
<mdo:flash height="600" id="/content/02/05/bbprob1/teddy5g.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param>
<param name="movie" value="teddy5g.swf"></param>
<param name="quality" value="high"></param>
<param name="bgcolor" value="#FFFFCC"></param>
</mdo:flash>
<p>Teddy Town is becoming very overcrowded! However, there is just enough room for some black teddies to join. Living there now are 36 teddy bears: 6 red, 6 yellow, 6 blue, 6 green, 6 purple and 6 black. There are 36 houses for them to live in: 6 red, 6 yellow, 6 blue, 6 green, 6 purple and 6 black. Make the 36 combinations of teddies and houses.</p>
<p><mdo:flash height="550" id="/content/02/05/bbprob1/teddy6.swf" width="560"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param>
<param name="movie" value="teddy6.swf"></param>
<param name="quality" value="high"></param>
<param name="bgcolor" value="#FFFFCC"></param>
</mdo:flash></p>
<p>Do you think it will be possible to put these 36 combinations in the street grid? May be it's not. Have a go!</p>
<p><mdo:flash height="600" id="/content/02/05/bbprob1/teddy6g.swf" width="550"><param name="allowfullscreen" value="true"></param><param name="allowfullscreen" value="true"></param>
<param name="movie" value="teddy6g.swf"></param>
<param name="quality" value="high"></param>
<param name="bgcolor" value="#FFFFCC"></param>
</mdo:flash></p>
<p>Now, look back at what you have done and ask yourself some of these questions:</p>
<ul>
<li>Was it easier to arrange the combinations in some of the grid sizes compared with others?</li>
<li>Why do you think this is?</li>
<li>What was your strategy for solving the arrangement puzzle each time?</li>
<li>What would happen if the two diagonals on the map also had to have different coloured houses and different coloured teddies? Can you solve the problem for each street plan now?</li>
</ul>
<h3>Acknowledgements.</h3>
<p>This activity is based on a Bernard's Bag problem from December 1997 called Tea Cups. The idea for the teddies came from Andrew Massey who is an Advisor for Worcestershire County Council. Thank you! Many thanks also to <a href="http://www.learningresources.com/">learningresources.com</a> for the use of the bear images.</p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>The original version of this problem uses only a 4x4 grid, but reducing the size makes this <a href="http://nrich.maths.org/public/viewer.php?obj_id=108&part=">investigation</a> accessible to younger children too. The learning objectives covered are numerous and cover the entire KS1/KS2 age range:</div>
<div>Sorting, classifying and organising</div>
<div>Choosing and using appropriate strategies</div>
<div>Explaining methods of reasoning Understanding and using vocabulary related to position</div>
<div>Recognising, explaining, generalising and predicting patterns</div>
<br></br>
<h3>Possible approach</h3>
<div>No matter how old the children, it would be advisable to have objects to represent the teddies and houses as an introduction to the activity. These could be, for example, coloured counters and coloured squares if the real thing weren't to hand. Coloured magnets would be ideal for use on a white board as a demonstration. If you prefer, click on the following links to download word documents
of the different coloured houses which you could print, laminate and cut out: <a href="/content/02/05/bbprob1/yellow.doc">yellow</a>, <a href="/content/02/05/bbprob1/red.doc">red</a>, <a href="/content/02/05/bbprob1/blue.doc">blue</a>, <a href="/content/02/05/bbprob1/green.doc">green</a>, <a href="/content/02/05/bbprob1/orange.doc">orange</a>, <a href="/content/02/05/bbprob1/purple.doc">purple</a>.</div>
<br></br>
<div>It would be worth clarifying the very first instruction. Work out the four different combinations together with the children, using teddies and houses of two different colours.</div>
<br></br>
<div>Throughout all of this investigation, encourage the children to explain their thinking orally. This may be to each other, or to the class as a whole. Either way, it is vital in allowing them to clarify their own ideas, reflect critically on their work and so move themselves forward.</div>
<br></br>
<h3>Key questions</h3>
<div>How can we make sure they are all different?</div>
<div>Is there a way to go about making the combinations so we don't leave any out?</div>
<div>Talk about being methodical and systematic i.e. planning and checking</div>
<br></br>
<h3>Possible extension</h3>
<div>Try <a href="http://nrich.maths.org/public/viewer.php?obj_id=32&part=">Tea Cups</a>.</div>
<br></br>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
You could use counters or pieces of coloured card if you don't have
teddies.<br></br>
Have you checked your combinations are all different?<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
The original version of this problem uses only a 4x4 grid, but
reducing the size makes this investigation accessible to younger
children too.<br></br>
The learning objectives covered are numerous and cover the entire
KS1/KS2 age range:<br></br>
<div style="margin-left: 40px;">Sorting, classifying and
organising</div>
<div style="margin-left: 40px;">Choosing and using appropriate
strategies</div>
<div style="margin-left: 40px;">Explaining methods of
reasoning</div>
<div style="margin-left: 40px;">Understanding and using vocabulary
related to position</div>
<div style="margin-left: 40px;">Recognising, explaining,
generalising and predicting patterns</div>
<br></br>
No matter how old the children, it would be advisable to have
objects to represent the teddies and houses as an introduction to
the activity. These could be, for example, coloured counters and
coloured squares if the real thing weren't to hand. Coloured
magnets would be ideal for use on a white board as a demonstration.
If you prefer, click on the following links to download word
documents of the different coloured houses which you could print,
laminate and cut out: yellow , red , blue , green , orange , purple
.<br></br>
<br></br>
It would be worth clarifying the very first instruction. Work out
the four different combinations together with the children, using
teddies and houses of two different colours:<br></br>
<br></br>
<mdo:image width="627" height="319" src="ted%26houses.jpg" alt="ted&houses"></mdo:image><br></br>
<br></br>
Ask them:<br></br>
<div style="margin-left: 40px;">How can we make sure they are all
different?</div>
<div style="margin-left: 40px;">Is there a way to go about making
the combinations so we don't leave any out?</div>
<div style="margin-left: 40px;">Talk about being methodical and
systematic i.e. planning and checking</div>
<br></br>
<div>It is vital that the children understand how each of the above
is different from the rest and, in addition, that they realise you
can put a teddy in a house of the same colour (this has caused
confusion in past experience).</div>
<div>Discuss strategies for working out the arrangements of
combinations on the streets. Try inviting your pupils to
suggest:</div>
<div style="margin-left: 40px;">Starting points -- what might we
put in the grid first and where?</div>
<div style="margin-left: 40px;">Logical approaches to filling in
the grid</div>
<div style="margin-left: 40px;"></div>
<div>When a particular grid is complete, look for patterns and find
out whether these can be applied to the next grid size up. This can
also entail different recording/representation methods and allows
you to explore vocabulary of position.</div>
<br></br>
<div>Throughout all of this investigation, encourage the children
to explain their thinking orally. This may be to each other, or to
the class as a whole. Either way, it is vital in allowing them to
clarify their own ideas, reflect critically on their work and so
move themselves forward.</div>
<br></br>
<div>To find out about the history and theory of this problem, look
at the article <a href="http://nrich.maths.org/public/viewer.php?obj_id=1453&part=">Latin
Squares</a> on the NRICH website or <a href="http://www.cut-the-knot.org/Curriculum/Algebra/Latin.shtml">www.cut-the-knot.org</a></div>
<br></br>
<br></br></mdoxml>
2
4
1
1
0
0
0
Teddy Town
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
Using, Applying and Reasoning about Mathematics
Investigations
Mathematics Tools
Compare bears
Using, Applying and Reasoning about Mathematics
Working systematically
Decision Mathematics and Combinatorics
Combinations
Information and Communications Technology
Interactivities