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2011-02-01T00:00:01
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<p>Take three different colour blocks, maybe red, yellow and blue.<br></br>
Make a tower using one of each colour.<br></br>
Here's one with red on top, blue in the middle and yellow on the bottom.</p>
<br></br>
<div style="text-align: center;"><mdo:image alt="" height="139" src="blocks2.gif" width="60"></mdo:image></div>
<p>Now make another tower with a different colour on top.</p>
<p>How many different towers can you make?</p>
<p>When you are sure you have found them all, try it with four colours.</p>
<p>You may like to print off one of these sheets for recording three-block towers: <a href="/content/98/10/letme2/ThreeBlocks.doc">Word document</a> or <a href="/content/98/10/letme2/ThreeBlocks.pdf">pdf</a> . Here are sheets for recording four-block towers <a href="/content/98/10/letme2/FourBlocks.doc">Word document</a> or <a href="/content/98/10/letme2/4Blocks.pdf">pdf.</a></p>
<br></br>
<a href="http://nrich.maths.org/7196">Click here for a poster of this problem</a>. </mdoxml>
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<span class="editorial">Jemima sent us her work on this problem.
Well done, Jemima!</span> <br></br>
<br></br>
I used multilink cubes to make the towers. First of all, I looked
for towers that were red at the top. Then I looked for towers that
were yellow at the top. Then I looked for towers that were blue at
the top. These are the ones I found.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image height="133" width="62" src="tower1.gif" alt="Red, blue, yellow"></mdo:image><mdo:image height="133" width="62" src="tower2.gif" alt="Red, yellow, blue"></mdo:image><mdo:image height="133" width="62" src="tower3.gif" alt="Yellow, red, blue"></mdo:image><mdo:image height="133" width="62" src="tower4.gif" alt="Yellow, blue, red"></mdo:image><mdo:image height="133" width="62" src="tower5.gif" alt="Blue, red, yellow"></mdo:image><mdo:image height="133" width="62" src="tower6.gif" alt="Blue, yellow, red"></mdo:image></div>
<br></br>
<span class="editorial">Jack from Gosforth Central Middle School
explained why only six ways are possible:</span> <br></br>
<br></br>
With three different colour blocks there are six possible ways of
building the tower. This is because for each colour on top there
are two options for the other blocks. Therefore as there are three
colours to go on top there are $3$ x $2$ options in total, that
means six. <br></br>
<br></br>
<span class="editorial">David tried the problem with four blocks.
Here are the towers he found:</span> <br></br>
<br></br>
<br></br>
<mdo:image height="154" width="46" src="talltower1.gif" alt="Red, yellow, blue, green"></mdo:image><mdo:image height="154" width="46" src="talltower2.gif" alt="Red, yellow, green, blue"></mdo:image><mdo:image height="154" width="46" src="talltower3.gif" alt="Red, blue, yellow, green"></mdo:image><mdo:image height="154" width="46" src="talltower4.gif" alt="Red, blue, green, yellow"></mdo:image><mdo:image height="154" width="46" src="talltower5.gif" alt="Red, green, yellow, blue"></mdo:image><mdo:image height="154" width="46" src="talltower6.gif" alt="Red, green, blue, yellow"></mdo:image><mdo:image height="154" width="46" src="talltower7.gif" alt="Yellow, red, blue, green"></mdo:image><mdo:image height="154" width="46" src="talltower8.gif" alt="Yellow, red, green, blue"></mdo:image><mdo:image height="154" width="46" src="talltower9.gif" alt="Yellow, blue, red, green"></mdo:image><mdo:image height="154" width="46" src="talltower10.gif" alt="Yellow, blue, green, red"></mdo:image><mdo:image height="154" width="46" src="talltower11.gif" alt="Yellow, green, red, blue"></mdo:image><mdo:image height="154" width="46" src="talltower12.gif" alt="Yellow, green, blue, red"></mdo:image><mdo:image height="154" width="46" src="talltower13.gif" alt="Blue, red, yellow, green"></mdo:image><mdo:image height="154" width="46" src="talltower14.gif" alt="Blue, red, green, yellow"></mdo:image><mdo:image height="154" width="46" src="talltower15.gif" alt="Blue, yellow, red, green"></mdo:image><mdo:image height="154" width="46" src="talltower16.gif" alt="Blue, yellow, green, red"></mdo:image><mdo:image height="154" width="46" src="talltower17.gif" alt="Blue, green, red, yellow"></mdo:image><mdo:image height="154" width="46" src="talltower18.gif" alt="Blue, green, yellow, red"></mdo:image><mdo:image height="154" width="46" src="talltower19.gif" alt="Green, red, yellow, blue"></mdo:image><mdo:image height="154" width="46" src="talltower20.gif" alt="Green, red, blue, yellow"></mdo:image><mdo:image height="154" width="46" src="talltower21.gif" alt="Green, yellow, red, blue"></mdo:image><mdo:image height="154" width="46" src="talltower22.gif" alt="Green, yellow, blue, red"></mdo:image><mdo:image height="154" width="46" src="talltower23.gif" alt="Green, blue, red, yellow"></mdo:image><mdo:image height="154" width="46" src="talltower24.gif" alt="Green, blue, yellow, red"></mdo:image><br></br></mdoxml>
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<h3><span style="font-weight: bold;">Why do this
problem?</span></h3>
<div>There are two parts to <a href="http://nrich.maths.org/public/viewer.php?obj_id=137&part=index">
this problem</a> - firstly making as many different towers as you
can think of and then being sure that all possible ones have been
made. The first is easy, the second requires some systematic
working and logical thinking.</div>
<br></br>
<h3>Possible approach</h3>
<div>You could use the three block problem as a whole class
introduction to the four block one, which children could then go
off and try on their own.</div>
<br></br>
<div>Start with a whole class activity in which you invite children
to come to the front and make a three-block tower that's different
from the one made by the person before. Check in each case why
theirs is different, encouraging the children to do the explaining.
Keep going until you have six different ones, and then ask another
child or two to come and make another and different tower. They
will of course find it impossible, which gives you a nice way in to
ask "Have we got them all? How can we be sure?".</div>
<br></br>
<div>Allow some time for different children to suggest ideas.
Encourage any sort of rearranging which helps the pattern to be
clear. If no-one suggests it, point out that we can group them in
different ways - with red at the top, blue at the top, etc. and
that this helps us to see if there are any missing.</div>
<br></br>
<div>Give each pair three different colours of blocks to replicate
the activity and convince you that they have found all possible
towers.</div>
<br></br>
<div>When they have done that to your, and their own, satisfaction,
suggest that they work in groups of four to produce as many
four-block towers as possible, arranging them as they work so that
they can be pretty sure that they have found them all. The
recording sheet (<a href="/content/98/10/letme2/FourBlocks.doc">Word document</a> or
<a href="/content/98/10/letme2/4Blocks.pdf">pdf</a>) may be useful
if you are likely to run out of multilink. Cutting out and
rearranging is a powerful way of organising the thinking.</div>
<br></br>
<h3>Key questions</h3>
<div>Have we got them all? How can we be sure?</div>
<div>How could we rearrange them to help?</div>
<br></br>
<h3>Possible extension</h3>
<div>Working with four blocks is an extension to this problem
anyway, but as an added extension you could again ask if the
children are sure that they haven't missed any out, and how they
know. Look for different ways of arranging and listen for the
explanations that go with them.</div>
<br></br>
<h3>Possible support</h3>
<div>Children who find this way of working difficult could be
encouraged to colour their solutions on the three-block sheet
provided (<a href="/content/98/10/letme2/ThreeBlocks.doc">Word
document</a> or <a href="/content/98/10/letme2/ThreeBlocks.pdf">pdf</a> ), cut each out and
rearrange them into a pattern. Look for other similar problems
(such as dressing teddy in red green or yellow hats, t-shirts and
shorts) which help to reinforce the importance and use of working
systematically.</div>
<br></br>
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How many towers will have a red block at the top?<br></br>
How many will have yellow at the top? What about
blue?<br></br>
How will you know when you have got them all? Perhaps you could
rearrange them to help?<br></br>
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<span class="editorial">Jemima sent us her work on this problem.
Well done, Jemima!</span> <br></br>
<br></br>
I used multilink cubes to make the towers. First of all, I looked
for towers that were red at the top. Then I looked for towers that
were yellow at the top. Then I looked for towers that were blue at
the top. These are the ones I found.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image width="62" height="133" src="tower1.gif" alt="Red, blue, yellow"></mdo:image><mdo:image width="62" height="133" src="tower2.gif" alt="Red, yellow, blue"></mdo:image><mdo:image width="62" height="133" src="tower3.gif" alt="Yellow, red, blue"></mdo:image><mdo:image width="62" height="133" src="tower4.gif" alt="Yellow, blue, red"></mdo:image><mdo:image width="62" height="133" src="tower5.gif" alt="Blue, red, yellow"></mdo:image><mdo:image width="62" height="133" src="tower6.gif" alt="Blue, yellow, red"></mdo:image></div>
<br></br>
<span class="editorial">David tried the problem with four blocks.
Here are the towers he found:</span> <br></br>
<br></br>
<mdo:image width="46" height="154" src="talltower1.gif" alt="Red, yellow, blue, green"></mdo:image><mdo:image width="46" height="154" src="talltower2.gif" alt="Red, yellow, green, blue"></mdo:image><mdo:image width="46" height="154" src="talltower3.gif" alt="Red, blue, yellow, green"></mdo:image><mdo:image width="46" height="154" src="talltower4.gif" alt="Red, blue, green, yellow"></mdo:image><mdo:image width="46" height="154" src="talltower5.gif" alt="Red, green, yellow, blue"></mdo:image><mdo:image width="46" height="154" src="talltower6.gif" alt="Red, green, blue, yellow"></mdo:image><mdo:image width="46" height="154" src="talltower7.gif" alt="Yellow, red, blue, green"></mdo:image><mdo:image width="46" height="154" src="talltower8.gif" alt="Yellow, red, green, blue"></mdo:image><mdo:image width="46" height="154" src="talltower9.gif" alt="Yellow, blue, red, green"></mdo:image><mdo:image width="46" height="154" src="talltower10.gif" alt="Yellow, blue, green, red"></mdo:image><mdo:image width="46" height="154" src="talltower11.gif" alt="Yellow, green, red, blue"></mdo:image><mdo:image width="46" height="154" src="talltower12.gif" alt="Yellow, green, blue, red"></mdo:image><mdo:image width="46" height="154" src="talltower13.gif" alt="Blue, red, yellow, green"></mdo:image><mdo:image width="46" height="154" src="talltower14.gif" alt="Blue, red, green, yellow"></mdo:image><mdo:image width="46" height="154" src="talltower15.gif" alt="Blue, yellow, red, green"></mdo:image><mdo:image width="46" height="154" src="talltower16.gif" alt="Blue, yellow, green, red"></mdo:image><mdo:image width="46" height="154" src="talltower17.gif" alt="Blue, green, red, yellow"></mdo:image><mdo:image width="46" height="154" src="talltower18.gif" alt="Blue, green, yellow, red"></mdo:image><mdo:image width="46" height="154" src="talltower19.gif" alt="Green, red, yellow, blue"></mdo:image><mdo:image width="46" height="154" src="talltower20.gif" alt="Green, red, blue, yellow"></mdo:image><mdo:image width="46" height="154" src="talltower21.gif" alt="Green, yellow, red, blue"></mdo:image><mdo:image width="46" height="154" src="talltower22.gif" alt="Green, yellow, blue, red"></mdo:image><mdo:image width="46" height="154" src="talltower23.gif" alt="Green, blue, red, yellow"></mdo:image><mdo:image width="46" height="154" src="talltower24.gif" alt="Green, blue, yellow, red"></mdo:image><br></br></mdoxml>
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3 blocks towers
Take three differently coloured blocks - maybe red, yellow and
blue. Make a tower using one of each colour. How many different
towers can you make?
Decision Mathematics and Combinatorics
Combinations
Using, Applying and Reasoning about Mathematics
Working systematically
Mathematics Tools
Interlocking cubes