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Floors and the ways that tiles are arranged have fascinated me for
a long time. In different countries, cities and even small towns
you find both walls and floors with interesting patterns.
<p>I don't know about you but I find myself counting the tiles in
particular ways. Maybe even your much younger brother or sister
tends to count things that are around them. Keep your eyes open for
patterns in tiles and you could discover a lot!</p>
<p>I was in an old church in Cambridge, where I live, and I looked
at the floor.</p>
<p>This is what I saw:-</p>
<p style="text-align: center;"><mdo:image alt="" src="F3.jpg"></mdo:image></p>
<p>What a design!!</p>
<p>There were these things to count:-<br></br>
. . . the tiles making up the three squares<br></br>
. . . the dark green tiles that made a border for the squares<br></br>
. . . the pale yellow tiles that filled in the gaps into the long
rectangle.</p>
<p>When I got home I drew this design out (I had made some notes in
the church, which is a good idea as it's sometimes hard to remember
exactly how it was).</p>
<p>I then drew a 2 square version:-</p>
<p style="text-align: center;"><mdo:image alt="" src="F2.jpg"></mdo:image></p>
<p>and a 1 square!!</p>
<p style="text-align: center;"><mdo:image alt="" src="F1.jpg"></mdo:image></p>
<p>Well, it's really up to you to count and compare the numbers of
particular tiles in one or all of these three designs.</p>
<p>See what number patterns you can come up with. I guess there are
very many.</p>
<p>Now I wonder what would happen if we had four squares? I suppose
we could put two lots of 2 together:-</p>
<p style="text-align: center;"><mdo:image alt="" src="F4a.jpg"></mdo:image></p>
<p>I was not altogether happy with this, but maybe you are, and
that's O.K.</p>
<p>I decided to remove that centre line and come up with:-</p>
<p style="text-align: center;"><mdo:image alt="" src="F4.jpg"></mdo:image></p>
<p>Mind you, you could just continue with things in a straight
line. What do you think?</p>
<p style="text-align: center;"><mdo:image alt="" src="F4b.jpg"></mdo:image></p>
<p>Try this out for yourself and maybe extend the pattern for even
greater numbers of squares.</p>
<p>Someone might be asking "What about smaller sized squares?" You
then might have something like :-</p>
<p style="text-align: center;"><mdo:image alt="" src="F1Sm.jpg"></mdo:image></p>
<p>You could try some more like these.<br></br>
What rules do you decide on, so that the pattern works out in the
same way?</p>
<p>Perhaps something rather different appears with different sized
squares and you could follow that line!!</p>
<p>And for those who really are fed up with squares why not look at
the same idea with triangles!!</p>
<p>Yes, here we are :-</p>
<p style="text-align: center;"><mdo:image alt="" src="tri1.gif"></mdo:image></p>
<p>for the small one and my next size up is like this :-</p>
<p style="text-align: center;"><mdo:image alt="" src="tri4.gif"></mdo:image></p>
<p>But what happens when we want more of the central triangles?</p>
<p>I wonder??</p>
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<p><span class="editorial">There are lots of answers to this
problem, depending on what questions you choose to ask.</span></p>
<br></br>
<br></br>
Have a go yourself, and if you discover anything interesting,
<a href="mailto:%20nrich@damtp.cam.ac.uk">e-mail</a> us to tell us
what you've done! Please don't worry that your solution is not
"complete" - we'd like to hear about anything you have tried.
Teachers - you might like to send in a summary of your children's
work. <br></br></mdoxml>
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<h2>The Great Tiling Count</h2>
<br></br>
Floors and the ways that tiles are arranged have fascinated me for a long time. In different countries, cities and even small towns you find both walls and floors with interesting patterns.
<p>I don't know about you but I find myself counting the tiles in particular ways. Maybe even your much younger brother or sister tends to count things that are around them. Keep your eyes open for patterns in tiles and you could discover a lot!</p>
<p>I was in an old church in Cambridge, where I live, and I looked at the floor.</p>
<p>This is what I saw:-</p>
<p style="text-align: center;"><mdo:image alt="" src="F3.jpg"></mdo:image></p>
<p>What a design!!</p>
<p>There were these things to count:-<br></br>
. . . the tiles making up the three squares<br></br>
. . . the dark green tiles that made a border for the squares<br></br>
. . . the pale yellow tiles that filled in the gaps into the long rectangle.</p>
<p>When I got home I drew this design out (I had made some notes in the church, which is a good idea as it's sometimes hard to remember exactly how it was).</p>
<p>I then drew a 2 square version:-</p>
<p style="text-align: center;"><mdo:image alt="" src="F2.jpg"></mdo:image></p>
<p>and a 1 square!!</p>
<p style="text-align: center;"><mdo:image alt="" src="F1.jpg"></mdo:image></p>
<p>Well, it's really up to you to count and compare the numbers of particular tiles in one or all of these three designs.</p>
<p>See what number patterns you can come up with. I guess there are very many.</p>
<p>Now I wonder what would happen if we had four squares? I suppose we could put two lots of 2 together:-</p>
<p style="text-align: center;"><mdo:image alt="" src="F4a.jpg"></mdo:image></p>
<p>I was not altogether happy with this, but maybe you are, and that's O.K.</p>
<p>I decided to remove that centre line and come up with:-</p>
<p style="text-align: center;"><mdo:image alt="" src="F4.jpg"></mdo:image></p>
<p>Mind you, you could just continue with things in a straight line. What do you think?</p>
<p style="text-align: center;"><mdo:image alt="" src="F4b.jpg"></mdo:image></p>
<p>Try this out for yourself and maybe extend the pattern for even greater numbers of squares.</p>
<p>Someone might be asking "What about smaller sized squares?" You then might have something like :-</p>
<p style="text-align: center;"><mdo:image alt="" src="F1Sm.jpg"></mdo:image></p>
<p>You could try some more like these.<br></br>
What rules do you decide on, so that the pattern works out in the same way?</p>
<p>Perhaps something rather different appears with different sized squares and you could follow that line!!</p>
<p>And for those who really are fed up with squares why not look at the same idea with triangles!!</p>
<p>Yes, here we are :-</p>
<p style="text-align: center;"><mdo:image alt="" src="tri1.gif"></mdo:image></p>
<p>for the small one and my next size up is like this :-</p>
<p style="text-align: center;"><mdo:image alt="" src="tri4.gif"></mdo:image></p>
<p>But what happens when we want more of the central triangles?</p>
<p>I wonder??</p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>Do this <a href="http://nrich.maths.org/public/viewer.php?obj_id=70&part=">activity</a> to encourage pupils to look at patterns around them and find the many patterns that are usually there to discover. The Great Tiling Count can be used as a gentle introduction to investigations that bridge the spatial and the numerical aspects of mathematics. Younger children may be encouraged to
explore the situations very practically with real "tiles" of some kind, even if they are coloured cubes. Computer graphics packages will also be of use for those inclined towards using that technology.</div>
<br></br>
<h3>Possible approach</h3>
<div>If possible use this activity to actually get out and observe the patterns that are in your locality.</div>
<div>Using the problems as set out, there will be a need for most children to have access to squared paper and possibly triangular paper.</div>
<br></br>
<div>Some of the arithmetic that may come out of the exploration will be concerned with counting and addition as well as multiplication and the use of halves [as in the edge tiles]. Particular additions, as in the case of the first picture counting the triangular pieces, could result in discussions about adding up:-</div>
<div>$1 + 3 + 5 + 7 + 9$</div>
<div>and then when the smaller versions are considered, adding up:-</div>
<div>$1 + 3 + 5$</div>
<div>The significance of there always being four quarter pieces may be thought about.</div>
<br></br>
<div>There should be many opportunities for looking at the occurrence of even numbers ... and why?!</div>
<br></br>
<h3>Key questions</h3>
<div>What shapes can you see?</div>
<div>What would you like to count?</div>
<div>What would a larger one look like?</div>
<br></br>
<h3>Possible extension</h3>
<div>Explore mosaic and tiling patterns that can be found online. Learners can create their own mosaic/tiling patterns and be able to describe the tiles that will be necessary.</div>
<br></br>
<h3>Possible support</h3>
<div>Some pupils will need a nudge to get started on the mathematics that is associated with a pattern, also help in counting accurately where appropriate.</div>
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The Great Tiling Count
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
Cambridge.
Using, Applying and Reasoning about Mathematics
Investigations
Using, Applying and Reasoning about Mathematics
Generalising
Sequences, Functions and Graphs
Arithmetic sequence
Sequences, Functions and Graphs
Geometric sequence