7481
/www/nrich/html/content/id/7481/
1
2011-09-07T09:51:27
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<div 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.)</em></div>
<p><br></br>
Polygonal numbers are those that are arranged in shapes as they enlarge (starting with $1$). Here are examples of the $5$th polygonal numbers for the first $7$ shapes Triangle ($3$) through to Nonegon ($9$).</p>
<div style="text-align: center;"><mdo:image alt="polys" height="611" src="polys%20pics.jpg" width="530"></mdo:image></div>
<div style="text-align: left;">If we were to just look at the heagons we'd have a form of hexagonal numbers that grow $1,6, 15, 28, 45$.</div>
<div style="text-align: center;"> <mdo:image alt="ord hex grow 3" height="150" src="ord%20hex%20grow3.jpg" width="504"></mdo:image></div>
<div style="text-align: left;">There is another group that are called "Centred Polygonals" that look like these; <mdo:image alt="centre p 2" height="622" src="centre%20polys%202.jpg" width="620"></mdo:image></div>
<br></br>
<p>and for example, the centred hexagon numbers go $1, 7, 19, 37, 61$.</p>
<div style="text-align: center;"> <mdo:image alt="centred hex grow2" height="131" src="centred%20hex%20grow2.jpg" width="397"></mdo:image></div>
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<h4 style="text-align: center;"> Now it's over to you!</h4>
<p> <br></br>
This investigation invites you to explore these sets of numbers and explore relationships within ordinary polygonal numbers and/or centred polygonal numbers.<br></br>
You could also explore relationships between ordinary polygonal numbers and the centred polygonal numbers.<br></br>
<br></br>
For example, you could explore which different polygonals (both centred and ordinary) have the same number occuring in the series?</p>
<p><br></br>
KEY QUESTIONS:</p>
<p>What is the relationship between ordinary triangular polygonal numbers and others?</p>
<p>Can you re-arrange the dots from one polygonal to make another, and then generalise?<br></br>
Throughout your exploration the question "Why?" probably needs to be asked!<br></br>
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<h2>Polygonals</h2>
<br></br>
<div 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.)</em></div>
<p><br></br>
</p>
<p>Polygonal numbers are those that are arranged in shapes as they enlarge (starting with $1$). Here are examples of the $5$th polygonal numbers for the first $7$ shapes Triangle ($3$) through to Nonegon ($9$).</p>
<div style="text-align: center;"><mdo:image alt="polys" height="611" src="polys%20pics.jpg" width="530"></mdo:image></div>
<div style="text-align: left;">If we were to just look at the heagons we'd have a form of hexagonal numbers that grow $1,6, 15, 28, 45$.</div>
<div style="text-align: center;"> <mdo:image alt="ord hex grow 3" height="150" src="ord%20hex%20grow3.jpg" width="504"></mdo:image></div>
<div style="text-align: left;">There is another group that are called "Centred Polygonals" that look like these; <mdo:image alt="centre p 2" height="622" src="centre%20polys%202.jpg" width="620"></mdo:image></div>
<br></br>
<p>and for example, the centred hexagon numbers go $1, 7, 19, 37, 61$.</p>
<div style="text-align: center;"> <mdo:image alt="centred hex grow2" height="131" src="centred%20hex%20grow2.jpg" width="397"></mdo:image></div>
<br></br>
<h4 style="text-align: center;"> Now it's over to you!</h4>
<p> <br></br>
This investigation invites you to explore these sets of numbers and explore relationships within ordinary polygonal numbers and/or centred polygonal numbers.<br></br>
You could also explore relationships between ordinary polygonal numbers and the centred polygonal numbers.<br></br>
<br></br>
For example, you could explore which different polygonals (both centred and ordinary) have the same number occuring in the series?</p>
<p><br></br>
KEY QUESTIONS:</p>
<p>What is the relationship between ordinary triangular polygonal numbers and others?</p>
<p>Can you re-arrange the dots from one polygonal to make another, and then generalise.?<br></br>
Throughout your exploration the question "Why?" probably needs to be asked!<br></br>
</p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>This <a href="http://nrich.maths.org/7481&part=">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/7473&part=">3D Stacks</a> as polygonal numbers can occur in that investigation. 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 and patterns.</div>
<div> </div>
<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>
<br></br>
<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>
<h3><br></br>
Possible extension</h3>
<div>If your pupils have investigated this very thoroughly they might like to look at <a href="/8018">Steps to the Podium</a> and seek to find connections.</div>
<p><br></br>
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<mdoxml version="1.0">You may find that spreadsheet software would be useful.</mdoxml>
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<div style="text-align: center;">Here are somethings that may help
the teachers concerned</div>
<div style="text-align: center;"> <mdo:image width="544" height="605" src="QU%20Sp%20Cen%204%27s.jpg" alt="QU Sp Cen 4"></mdo:image></div>
<div style="text-align: center;">AND</div>
<div style="text-align: center;"> <mdo:image width="583" height="620" src="QU%20Ord%20%2B%20Cen%20to%20Tri%27s.jpg" alt="QU Ord & Cen"></mdo:image></div>
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polygonals
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Using, Applying and Reasoning about Mathematics
Investigations
Sequences, Functions and Graphs
Sequences
Using, Applying and Reasoning about Mathematics
Generalising