2282
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2011-02-01T00:00:01
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This problem follows on from <a href="http://nrich.maths.org/2281&part=">Number Pyramids</a>.<br></br>
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In the number pyramid below, the number in the bottom left hand corner determines all the other numbers.<br></br>
Try entering some different numbers.<br></br>
What patterns do you notice?<br></br>
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<a href="/content/id/2282/pyramids.swf">Full Screen Version</a><br></br>
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<div style="text-align: left;"><span style="font-weight: bold;">Here are some questions to consider:</span></div>
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<ul>
<li>Which numbers is it possible to get at the top, if you start with a whole number in the bottom left hand corner?</li>
<li>Can you <span style="font-weight: bold;">explain</span> why some numbers are impossible to get at the top, if you start with a whole number in the bottom left hand corner?</li>
<li>Given the number at the top, can you find a quick way of working out the number in the bottom left hand corner?</li>
</ul>
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Test out your observations and insights.<br></br>
How would your insights change if you used negative numbers? Decimals?<br></br>
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<span style="font-weight: bold;">Can you justify any generalisations that you have reached?</span><br></br>
Perhaps you could use algebra to explain your thinking.<br></br>
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<p><span style="font-weight: bold;">Can you adapt your insights so that they apply to pyramids with different sequences on the bottom layer?</span><br></br>
What if the numbers on the bottom layer go up in 2s? Or 3s? Or start at 17 and go up in 7s? Or...<br></br>
<a href="/content/id/2282/More%20number%20pyramids.xls">This spreadsheet</a> might be useful for exploring such pyramids with four or five layers.<br></br>
You could adapt it to work on even larger pyramids.<br></br></p>
<div> </div>
<div>For a challenging extension, why not explore <a href="http://nrich.maths.org/7531&part=">Function Pyramids</a>, in which the structure of the pyramid is based on a more complicated function than addition.</div>
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<p><span style="font-style: italic;">This problem features in Maths Trails - Generalising, one of the books in the Maths Trails series written by members of the NRICH Team and published by Cambridge University Press. For more details, please see our</span> <a style="font-style: italic;" href="http://nrich.maths.org/public/viewer.php?obj_id=4833&part=index">publications page</a> <span style="font-style: italic;">.</span></p>
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<p class="editorial"><span class="editorial">Well done to those of you who worked systematically with the interactive pyramid on the site. Sian from GSGW school and Scott noticed that the number at the top increases by 8 when you increase the number in the bottom left hand corner by 1. As Scott puts it:</span></p>
<p>I have found that each time you put a number in, eg. 1, your top number will be 20 but when you put in 2 your answer will be 28 so each time you put in a number one bigger the answer will be 8 bigger.</p>
<p> </p>
<p class="editorial"><span class="editorial">This table of results produced by Esther illustrates this well:</span></p>
<div class="editorial" style="text-align: center;"><mdo:image width="217" height="290" alt="table" src="http://nrich.maths.org/content/id/2282/number%20pyramids%201.JPG"></mdo:image></div>
<div class="editorial" style="text-align: center;"> </div>
<p class="editorial"><span class="editorial">Niamh from Brighouse High School and Saifuddin started to generalise the pyramid using pronumerals. This was continued by Thomas from New York who generalised the 4-level problem like this:</span></p>
<p class="editorial" style="text-align: center;"><mdo:image alt="" src="24.jpg" style="width: 430px; height: 249px;"></mdo:image></p>
<p class="editorial"><span class="editorial">Well done to Alicia, Laura, Sam, Amber, Chris, Jake, Sam and James from Colonel Frank Seely School, Matt from Newent, Jim and Sadia from Cooper School, Jen, Nikki, Cit, Molly, Ellie, Orla, Georgie, Hayley, Flora and Chloe from NGHS and Esther who also used algebra to describe the pyramid.</span></p>
<p class="editorial"> </p>
<p class="editorial"><span class="editorial">Tom from Carmel College started off by noticing that:</span></p>
<p>The numbers in the bottom row increase by 1, the numbers in the 2nd row increase by 2 and the numbers in the 3rd row increase by 4.</p>
<div> </div>
<div><span class="editorial">Then Paul commented that:</span></div>
<div>All the numbers at the top must be 20 or more and be a multiple of 4.</div>
<p class="editorial"> </p>
<p class="editorial"><span class="editorial">This was continued by Henry from the Latymer School, Karthik, Tom and Mitali who all algebraically showed that the top number is always a multiple of 4. Tom reasoned this by saying:</span></p>
<div>You only get multiples of <span class="MathJax" role="textbox" style=""><nobr><span class="math" id="MathJax-Span-25">4</span></nobr></span> at the top since the number at the top in relation to the bottom left number <em>x,</em> is $8x+12$ which can be factorised and written as $4(2x+3)$ proving it is always divisible by 4 (a multiple of 4).</div>
<div> </div>
<p class="editorial"><span class="editorial">Henry from Finton House School and Louisa from New Zealand gave good reasoning on how you can work backwards from the top number without using explicit algebra. Louisa reasoned that:</span></p>
<p>To quickly calculate the number in the bottom left hand corner when you know the top number you divide the top number by 4 and find two consecutive numbers that sum to this. Then subtract 1 from the smaller of the two numbers and this is the number in the bottom left corner.</p>
<p> </p>
<p><span class="editorial">We had many explanations on how to find the bottom left number from the top number using algebra. James and David from Colonel Frank Seely School gave a good explanation:</span></p>
<p>Let <em>t</em> = top number and <em>n</em> = bottom left number. Then $ t=8n+12 \Rightarrow n=\frac{t-12}{8} $</p>
<p><span class="editorial">Thanks also to Alicia from Colonel Frank Seely School, Ng Xing Yu from Singapore, Karthik from India and Umarah, Meera, Tehillah, Sara, Olga, Daisy, June, Mo and Kassie from NGHS.</span></p>
<p> </p>
<p><span class="editorial">Mitali continued this by showing that if the number at the top is 48 then the starting number cannot be an integer:</span> $$\begin{align*} 8x+12&=48 \\ 8x&=36 \\ x&=4.5 \end{align*}$$</p>
<p><span class="editorial">Nathan from Rushmore Primary explained this by noting:</span></p>
<p>If we only use whole numbers, the numbers on the top can only be a number in the 4 times tables, but not the 8 times tables. And 48 is a multiple of 8.</p>
<p> </p>
<p><span class="editorial">Sophie from Putney High School began to extend to 6-layer Pyramids. Thomas then said:</span></p>
<p>The topmost number for five tiers is given by $16x+32$ The topmost number for six tiers is given by $32x+80$.</p>
<p><span class="editorial">This was extended by Brittany Howe, Amy Johnson, Livvy Bolton and Abi B from NGHS who gave an example below:</span></p>
<p class="editorial" style="text-align: center;"><mdo:image alt="" src="Pyramid%20larger.jpg" style="width: 432px; height: 345px;"></mdo:image></p>
<p class="editorial">Matt from Newent began to consider an <em>n</em>-level pyramid, and correctly realised that the coefficient of <em>x</em> in the top term, will be $2^{n-1}$. Using Pascal's triangle below, can you work out the rest of the general formula for the top term in an <em>n</em>-level pyramid?</p>
<p class="editorial" style="text-align: center;"><mdo:image alt="" src="pascals.jpg" style="width: 460px; height: 316px;"></mdo:image></p>
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<h3>Why do this problem?</h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=2282">This
problem</a> offers students the opportunity to notice
patterns, make conjectures, explain what they notice and prove
their conjectures. Generalisation provokes the need to use
algebraic techniques such as collecting like terms and representing
number sequences algebraically.</div>
<br></br>
<h3>Possible approach</h3>
<div>This problem follows on nicely from <a href="http://nrich.maths.org/public/viewer.php?obj_id=2281&part=">
Number Pyramids</a></div>
<div>What follows could be done in a classroom with students
working on paper, or in a computer room so that students can make
use of the interactivity and spreadsheet.</div>
<div> </div>
<div>Start by showing the interactivity:</div>
<div>"I'm going to type in a number (2), and I'd like you to watch
what happens. Can you work out what is going on? Do you notice
anything interesting?"</div>
<div>Allow students a short time to discuss in pairs what they
saw.</div>
<div>"In a moment, I'm going to type in the number 7. Can you
predict what will happen?"</div>
<div>Give pairs a little time to discuss and decide, then show what
happens.</div>
<div> </div>
<div>"In a while, I'm going to ask you to share anything
interesting you have noticed, and any questions that have arisen.
You might want to try some more examples to test out your ideas or
to give you more data before looking for patterns. Or you might
like to think about different ways of representing what's going on
in the pyramid."</div>
<div> </div>
<div>After students have had plenty of time to explore, bring the
class together and share noticings and conjectures. If no-one has
considered using algebra, this would be a good time to suggest
representing the bottom left corner with $n$ for example, and
working out the other entries in terms of $n$.</div>
<div> </div>
<div>Once the class have an algebraic expression for the top
number, this can be used in two ways:</div>
<ul>
<li>Can they explain why it's impossible for some numbers to appear
at the top (when an integer is entered at the bottom)?</li>
<li>Given a top number, can they use their expression to find what
number should be entered at the bottom to generate it?</li>
</ul>
<div>"In these number pyramids, the bottom layer is always a set of
consecutive numbers, but there's no reason why the bottom layer
couldn't be any other number sequence - starting at 13 and going up
in 4s for example. Is there a quick way to work out what the top
number will be? Explore some different sequences and use algebra to
help you predict and explain what happens."</div>
<div> </div>
If students are in a computer room, <a href="/content/id/2282/More%20number%20pyramids.xls">this
spreadsheet</a> can be used to explore different number
sequences.<br></br>
<h3>Key questions</h3>
<div>Can you work out what is going on in this pyramid of
numbers?</div>
<div>What do you notice about the numbers on each row of the
pyramid?</div>
<div>How do we know that $8x+12$ is always a multiple of $4$ but
never a multiple of $8$?</div>
(for integer values of $x$)<br></br>
<h3>Possible extension</h3>
Given the top number and <span style="font-weight: bold;">either</span> the starting number <span style="font-weight: bold;">or</span> the difference between the
numbers on the bottom layer, can students work out the missing
piece of information?<br></br>
<h3>Possible support</h3>
<div>Students could work on <a href="http://nrich.maths.org/2281&part=">Number Pyramids</a>
first in order to gain some familiarity with the structure
underlying the problem.</div>
<div> </div>
<div>The group could be split so that some investigate sequences
that go up in 2s, some 3s, some 4s and so on. Then the class could
come together to share what they have found out before generalising
to any sequence.</div>
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Why not take a look at the problem <a href="http://nrich.maths.org/2281&part=">Number Pyramids</a>
first?<br></br>
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