7733
/www/nrich/html/content/id/7733/
1
2011-12-13T11:33:11
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<p style="text-align: center;"><em>This activity has been particularly created for the most able. (The pupils that you come across in many classrooms just once every few years.) It is seen as a possible follow on from</em></p>
<p style="text-align: center;"><em>"<a href="http://nrich.maths.org/7544">Tiles in the Garden</a>".</em></p>
<p>This activity takes "Tiles in the Garden", much further. We can keep the main ideas the same - </p>
<ul>
<li>Square tiles</li>
<li>A corner of a tile at each corner of the rectangle</li>
<li>The ability to slice a tile into parts so as to use each part</li>
</ul>
<p style="text-align: center;"><mdo:image src="Tiles%20Slants.jpg"></mdo:image></p>
<p>So this one used $26$ and the slope was generated by going along $1$ and up $5$.</p>
<p> </p>
<p>This time let's put on a limit of using less than $100$ tiles.</p>
<p>What sizes of rectangles could be filled obeying the three rules?</p>
<p>How many tiles for each rectangle you find?</p>
<p>Are there any numbers of tiles between $10$ and $100$ for which there cannot be a rectangle?</p>
<p> </p>
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<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>This <a href="http://nrich.maths.org/7733">activity</a> is specially designed for the highest-attaining pupils that you ever come across. It may act as a further extension to <a href="http://nrich.maths.org/7544">Tiling the Garden</a>. It's an activity that is intended to give opportunities for those pupils to explore more deeply using their intuition and flair in the areas of both spatial
awareness and number relationships .</div>
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<h3>Possible approach</h3>
<div>As this is designed for the highest attaining, it might be presented as on the website or in a one-to-one situation, encouraging discussion between adult and pupil. The pupils may need access to a spreadsheet once many number results are being acquired.</div>
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<h3>Key questions</h3>
<div>Tell me about what you have found?</div>
<div>Can you describe the ways that you arrived at these numbers?</div>
<div>How did you construct this on the spreadsheet you used?</div>
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<p>Here is a table showing the number of tiles used possibilities down the left column and then the types of slopes of the sides that produce those number of tiles inside the rectangle.</p>
<p><mdo:image src="tile%20rects%20possibilities.jpg"></mdo:image></p>
<p> </p>
<p>The meaning of the three numbers arre as follows,</p>
<p><mdo:image src="Slope%204%20tiles.jpg"></mdo:image></p>
<p>So if the rectangle were as follows;</p>
<p><mdo:image src="canon%20help.jpg"></mdo:image></p>
<p>The shortest side (red) is $3$ across and $2$ up and just one length of that slope so $3,2,1$</p>
<p>but the other side (green) is two of that length so its $3,2,2$) . On the table above it show that $26$ tiles are needed for $3,2,2$.</p>
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Tiling into slanted rectangles
Using, Applying and Reasoning about Mathematics
Investigations
Measures and Mensuration
Area
Measures and Mensuration
Surface and surface area