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2011-02-01T00:00:01
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<p>These are the 'double-3 down' dominoes.</p>
<div style="text-align: center;"><mdo:image height="199" width="305" alt="" src="domino5.gif"></mdo:image></div>
<br></br>
<p>Use these dominoes to make this square so that each side has
eight dots.</p>
<div style="text-align: center;"><mdo:image alt="example square" src="domino2.gif"></mdo:image></div>
<br></br>
<br></br>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=6833&part=">
Click here for a poster of this
problem</a>. </div>
<br></br>
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<p><span class="editorial">Andrew and Michael from Old Earth
Primary, and Sam, Harry, Molly and Callum from St Nicolas C of E
Junior School, Newbury started this challenge in the same way. Here
is what Sam and Harry said:</span></p>
Sam and I, Harry, counted all the dots on the double-3 down
dominoes, so we got 30. But we needed 8 dots on each row and 8 x 4
= 32.<br></br>
So we knew we had to put 2 dots in the corners because the dots in
the corners count double because they are in two rows. <br></br>
<br></br>
<p><span class="editorial">Andy and Michael expanded a bit on
this:</span></p>
All the corners must add up to 2, which is either 0, 0, 0, 2 or 0,
0, 1, 1.<br></br>
<br></br>
<p><span class="editorial">In other words, the dots in the four
corners must make a total of 2 dots. The only ways to have a total
of 2 dots is by having one corner with 2 dots and the others all
blank, or two corners blank and two with one dot.</span></p>
<p class="editorial">Sam and Harry found this solution which had 2
dots in one corner and the other corners blank:</p>
<br></br>
<br></br>
<mdo:image height="324" width="314" src="dominosol1.gif" alt=""></mdo:image><br></br>
<br></br>
<p><span class="editorial">Ellie and Kerry from Stoke Mandeville
Combined School also sent a picture of the same solution. Thank
you! Andrew and Michael found a different solution, using 2 dots in
one corner:</span></p>
<br></br>
<br></br>
<mdo:image height="237" width="243" src="dominosol3.gif" alt=""></mdo:image> <br></br>
<br></br>
<p><span class="editorial">Rather than draw dominoes, they have
chosen to write digits to represent the spots, which might be a bit
quicker!</span></p>
<p class="editorial">Livvy and Connor from Cutthorpe got out their
dominoes and experimented. They also found a solution with just 2
dots in one corner:</p>
<br></br>
<br></br>
<mdo:image height="237" width="243" alt="" src="dominosol4.gif"></mdo:image><br></br>
<br></br>
<p><span class="editorial">Abbas, Alfin, Yasemin and Godwin from
Archbishop Lanfranc also found the solution above. Lucy from
Middlefield School said:</span></p>
I cut out all the dominoes on the sheet. I then experimented by
sorting out one side, then the next and so on until I did it.<br></br>
<br></br>
<p><span class="editorial">Lucy discovered another way to arrange
the dominoes with 2 dots in one of the corners:</span></p>
<br></br>
<br></br>
<mdo:image height="284" width="278" alt="" src="dominoesol5.gif"></mdo:image><br></br>
<br></br>
<p><span class="editorial">Tom and Ted from Ysgol Abergynolwyn
thought that blanks worked best in the corners and they found yet
another way using a 2 in one corner:</span></p>
<br></br>
<br></br>
<mdo:image height="351" width="475" src="dominosol6.gif" alt=""></mdo:image><br></br>
<br></br>
<p><span class="editorial">Are there any more ways to arrange the
dominoes with a 2 in one corner and the other corners blank? Let us
know if you find any.</span></p>
<br></br>
<p><span class="editorial">Molly and Callum from St Nicolas C of E
Junior School, Newbury found a solution
which has two blanks and two ones in the corners:</span></p>
<br></br>
<br></br>
<mdo:image height="324" width="313" src="dominosol2.gif" alt=""></mdo:image><br></br>
<br></br>
<p><span class="editorial">We also had three other solutions sent
in which had two blanks and two ones in the corners. Here is Nick's
(who goes to the English College in Dubai):</span></p>
<br></br>
<br></br>
<mdo:image height="200" width="216" alt="seventh solution" src="dominosol7.gif"></mdo:image><br></br>
<br></br>
<p><span class="editorial">Rebecca and Amy from Sawtry CC sent this
one, saying that they used trial and error to work this out,
starting with the double 3 on the top line and working
through:</span></p>
<br></br>
<br></br>
<mdo:image height="200" width="216" src="dominosol8.gif" alt="eighth solution"></mdo:image><br></br>
<br></br>
<p><span class="editorial">And Rachael from Old Earth Primary sent
this:</span></p>
<br></br>
<br></br>
<mdo:image height="200" width="216" alt="ninth solution" src="dominosol9.gif"></mdo:image><br></br>
<p class="editorial">So many solutions! But are there any others
with two ones and two blanks in the corners? Do let us know.</p>
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<div class="embed">
<h2>Domino Squares</h2>
<br></br>
<p>These are the 'double-3 down' dominoes.</p>
<div style="text-align: center;"><mdo:image alt="" height="199" src="domino5.gif" width="305"></mdo:image></div>
<br></br>
<p>Use these dominoes to make this square so that each side has eight dots.</p>
<div style="text-align: center;"><mdo:image alt="example square" src="domino2.gif"></mdo:image></div>
<br></br>
<br></br>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=6833&part=">Click here for a poster of this problem</a>.</div>
<br></br>
</div>
<h3> </h3>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
Dominoes are a great resource and <a href="http://nrich.maths.org/public/viewer.php?obj_id=146&part=index">this problem</a> is an intriguing way to use them. Not only does this activity require logical thinking but it is also an interesting way of practising addition and subtraction.<br></br>
<br></br>
<h3>Possible approach</h3>
<div>If you have an interactive whiteboard, you may find our <a href="http://nrich.maths.org/public/viewer.php?obj_id=6361&part=index">Dominoes Environment</a> useful for this problem.</div>
<br></br>
<div>You could introduce this problem by laying out ten large dominoes in a square on the floor (it does not matter which dominoes go where). Ask the class to gather round and ask a few questions about the sum of dots on each side so that learners understand how the corner spots are counted in both the horizontal side and the vertical side.</div>
<br></br>
<div>Introduce the problem itself and ask pairs of children to talk for a minute or two about how they might tackle the problem. Share some of their suggestions among the whole group before giving them time to work in their pairs with dominoes. Using real dominoes whenever possible would be advantageous, but you can use <a href="/content/99/03/letme1/6SpotDoms.pdf">this sheet</a> of a standard
set of dominoes to be cut out. Squared paper would also be useful for jottings and recording.</div>
<br></br>
<div>As well as talking about the solutions in the plenary, you could focus on how children recorded their solutions. Some may well have just used the dominoes and moved them around as they went but how did they keep track of what they had tried? Some may have jotted down pictures of different arrangements. It would be useful to have a conversation about what ways of recording are most useful in
this context.</div>
<br></br>
<h3>Key questions</h3>
<div>What do the numbers on this side add to?</div>
<div>What do you need to make eight?</div>
What could you try instead?<br></br>
<br></br>
<h3>Possible extension</h3>
<div>Use the 'double 4 down' dominoes to make a rectangle with equal numbers of dots on each side. Repeat with 'double 5 down' etc.</div>
<br></br>
<div>What numbers of dominoes can be made into a true square? Explore the numbers that emerge and explain why certain numbers of dominoes cannot be made into a square.</div>
<br></br>
<h3>Possible support</h3>
<div>Use real dominoes and sort out the '3 spot down' ones and use them to make a square. Then count the dots on the sides and work on the problem in a 'trial and improve' basis.</div>
<br></br>
<div>You could start with the 'double 2 down' dominoes making each side add to 16 and using a square like this:</div>
<br></br>
<div style="text-align: center;"><mdo:image alt="" height="154" src="domino4.gif" width="155"></mdo:image></div>
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<mdoxml version="1.0">Which dominoes could go on the same side as the double-3
domino?<br></br>
It would be very useful to have some dominoes to move around, or
you could print off and cut out them out from <a href="/content/99/03/letme1/6SpotDoms.pdf">this sheet</a>.<br></br></mdoxml>
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Nick (from Dubai) sent us this solution: <br></br>
<br></br>
<mdo:image height="252" width="423" alt="" src="domino3.gif"></mdo:image><br></br>
<br></br>
Can you find any more arrangements?<br></br>
<br></br>
Another -3-3, 1-1 along top<br></br>
0-3, 1-3, 1-0 along left (top to bottom) <br></br>
0-2, 2-2 along bottom<br></br>
0-0, 3-2, 1-2 along right (top to bottom)<br></br>
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Domino Square
Use the 'double-3 down' dominoes to make a square so that each side
has eight dots.
Calculations and Numerical Methods
Addition & subtraction
Using, Applying and Reasoning about Mathematics
Trial and improvement
Mathematics Tools
Dominoes