5467
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2011-02-01T00:00:01
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You will need to download and print out <a href="/content/id/5467/Fractions%20Jigsaw%20%28word%29.doc">this
jigsaw</a> (or draw the pieces on squared paper), and cut up the
pieces into squares (don't cut along the diagonal lines!).<br></br>
<br></br>
<div style="text-align: center;"><mdo:image width="137" height="135" alt="wonky tile 1" src="wonky1.gif"></mdo:image><mdo:image width="137" height="137" alt="wonky tile 2" src="wonky2.gif"></mdo:image><mdo:image width="135" height="135" src="wonky3.gif" alt="wonky tile 3"></mdo:image></div>
<br></br>
<br></br>
Now try to put the square pieces together without rotating any of
them (so that in the finished jigsaw all the numbers are the right
way up). <br></br>
<br></br>
Two pieces may only go next to each other if the edges that touch
contain fractions that are equivalent.<br></br></mdoxml>
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<p><span class="editorial">A Maths Focus Group from Reigate Priory School wrote up the strategy they used to complete the jigsaw:</span></p>
<p>First we found the corners, which were pretty easy as they had to have $2$ black triangles.<br></br>
Then we did the edges which have $1$ black triangle but we found that we had $4$ spare, $1$ from each edge.<br></br>
We realised that these probably had to go together in the middle of the jigsaw, ie: with black triangles touching.<br></br>
We then worked out the 'inner' corners, ie: the squares that had to have $2$ triangles matching, and then we worked out the squares that had to join these up.<br></br>
After that there was only $1$ piece left to fit in the middle!!</p>
<p class="editorial">Jamie from Great Sankey High School described his strategy:</p>
I looked at matching pairs first, and then tried to match them up into a group.<br></br>
I then saw a pattern and worked from that to complete the jigsaw.<br></br>
<br></br>
<span class="editorial">George and Sophie from Hoyle Court Primary in Baildon, Shipley, West Yorkshire started off finding the more obvious equivalent fractions, and formed small sections of the jig-saw. Then they realised they could start to piece sections together. Eventually they recognised the pattern of the triangles around the edge. Their solution is below. Well done to both of
them.</span><br></br>
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<mdo:image src="Jigsaw.jpg"></mdo:image><br></br>
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<h3>Why do this problem:</h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=5467">This
problem</a> provides an opportunity to find equivalent fractions
and carry out some simple additions and subtractions of fractions
in a context that may challenge and motivate students.</div>
<br></br>
<div>For some students this will also invite questions like:</div>
<div>How has this puzzle been created, and how much freedom is
there in this structure?</div>
<h3>Possible approach :</h3>
<div>Give the jigsaw to pairs of students to complete, being ready
for discussion that may follow about fractions or puzzles of this
type.</div>
<br></br>
<div><a href="/content/id/5467/Fractions%20Jigsaw%20--%20outline.gif">Here</a>
is a blank outline of the jigsaw for students to create their own,
harder/easier versions of the puzzle.</div>
You can create, print out, save and exchange customised jigsaws,
domino activities and a variety of rectangular card sort activities
using "Formulator Tarsia", free software available from the
<a href="http://www.mmlsoft.com/index.php?option=com_content&task=view&id=9&Itemid=10">
Hermitech Laboratory website</a> .<br></br>
<h3>Key questions :</h3>
<ul>
<li>How did you start with this puzzle?</li>
<li>At what stage did it get hard?</li>
<li>How did you get through that block?</li>
<li>How could it be made harder?</li>
</ul>
<ul class="noindent"></ul>
<ul class="noindent"></ul>
<h3>Possible extension :</h3>
<div>Would it have been harder if the numbers did not have to be
"the right way up"? What if the same answer occured more than once?
What if there were calculations on the outside edges, rather than
grey? Can you make a harder (but still possible) puzzle?</div>
<br></br>
<div>Or consider the structure that makes this puzzle possible.
Matching here is specified to be by equivalent fractions but the
means of matching is unimportant. If we use a letter to stand for
each 'match' : a to a, b to b etc., and if we release the
constraint that 'numbers are the right way up', how easy is it to
arrive at the arrangement which is the solution?</div>
<ul>
<li>
<p>Can a set of pieces solve in more than one way ?</p>
</li>
<li>
<p>How does our answer to these questions change for different size
jigsaws.</p>
</li>
</ul>
<h3>Possible support :</h3>
<div>Use the blank template and create a jigsaw using only simple
fractions. Give some of the piece positions at the start. Get the
group creating appropriate jigsaws for each other to try. There may
be good discussions in what makes one puzzle harder than
another.</div>
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Count the number of pieces in the puzzle.<br></br>
What will the dimensions of the finished jigsaw be?<br></br>
<br></br>
Some people find it easiest to complete the edges first.<br></br></mdoxml>
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Here is one possible solution to the puzzle.<br></br>
<div style="text-align: center;"><mdo:image width="600" height="600" alt="solution" src="Fractions%20Jigsaw%20--%20solution.gif"></mdo:image></div>
<div>Here is a blank outline for Charlie to put in the Notes when
the solution has been published:</div>
<br></br>
<div style="text-align: center;"><mdo:image width="600" height="600" src="Fractions%20Jigsaw%20--%20outline.gif" alt="outline"></mdo:image></div>
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