7531
/www/nrich/html/content/id/7531/
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2011-06-10T12:29:06
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<span style="font-style: italic;">This problem explores structures
like those found in</span> <a style="font-style: italic;" href="http://nrich.maths.org/2281&part=">Number Pyramids</a>
<span style="font-style: italic;">and</span> <a style="font-style: italic;" href="http://nrich.maths.org/2282&part=">More Number
Pyramids</a><span style="font-style: italic;">.</span><br></br>
<br></br>
A function pyramid is a structure where each entry in the pyramid
is determined by the two entries below it, together with some
function:<br></br>
<div style="text-align: center;"> <mdo:image height="237" width="281" src="fnpyramid.jpg" alt="function pyramid"></mdo:image></div>
The function used in the problems <a href="http://nrich.maths.org/2281&part=">Number Pyramids</a>
and <a href="http://nrich.maths.org/2282&part=">More Number
Pyramids</a> could be expressed as:<br></br>
$$f(a,b) = a+b$$<br></br>
<br></br>
Here is a function pyramid for you to explore. Type some numbers
into the three spaces on the bottom layer. <br></br><br></br><span style="font-weight: bold;">Can you figure out how
the rest of the pyramid is generated?</span> <br></br> <br></br>
<div style="position:relative;">
<img id="pyramid" src="pyramid.jpg"/>
<span id="top" > </span>
<span id="middle1" > </span>
<span id="middle2" > </span>
<input id="bottom1" type="text" onkeyup="populate();" />
<input id="bottom2" type="text" onkeyup="populate();" />
<input id="bottom3" type="text" onkeyup="populate();" />
</div>
<br></br><span style="font-weight: bold;">Here are some questions you might like to consider:</span><br></br>
<br></br>
Can you choose numbers for the bottom layer so that the number 1
appears in the top cell?<br></br>
Can you choose numbers for the bottom layer so that the number 5
appears in the top cell?<br></br>
Can you identify the function used to determine the next layer,
given the bottom layer?<br></br>
Can you choose numbers for the bottom layer so that the number in
the top cell is negative? <br></br>
<br></br>
<br></br>
Why not make up some function pyramids of your own, and ask
yourself some similar questions?<br></br>
<br></br>
<script src="7531-script.js" type="text/javascript"></script>
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<p><span class="editorial">Arti from the USA found a way to get 1 and 5 in the top cell:</span></p>
<p>To get 1 at the top cell use 2, 2, 1 for the three bottom cells values. To get 5 at the top cell use 16, 16, 1 for the three bottom cells values.</p>
<p> </p>
<p><span class="editorial">Arti, along with James from Newmarket College and Felipe from St Pauls School in Brazil all worked out the function. Here is Felipe's explanation:</span></p>
<p>The function for calculating the number above is given by $log_2$ of the product of the two numbers below it. This can be found by constructing a table where you put all the whole numbers you can make. A pattern will soon appear and you will notice that the values you input for a and b to form these whole numbers are in fact a logarithmic scale of base 2 and the obtained value is the sum of
the $log_2$ of these numbers. Therefore the expression for calculating the number above is: $log_2(a) + log_2(b)$ , which simplified is: $log_2(ab)$</p>
<p> </p>
<p><span class="editorial">Another solver from St Pauls explained how he found the answer:</span></p>
<p>$16= 2^4$ and in the next cell $4= 2^2$ will give the answer $6$ in the space above both these numbers.</p>
<p>To get negative numbers you need to have decimals such as $0.5$ which is $2^-1$</p>
<p>To get a negative number on the top bracket you need to have numbers such as 1.1 in the spaces given as these will give decimals which will then give negative numbers in the top space.</p>
<p> </p>
<p><span class="editorial">Christopher from Sale Grammar School sent this picture to illustrate his answer:</span></p>
<p><mdo:image src="Christopher.png"></mdo:image></p>
<p><span class="editorial">Finally, Daniel from Savile Park school send us</span> <a href="/content/id/7531/Daniel.pdf">this clear and well-explained answer</a> <span class="editorial">.</span></p>
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<h3>Why do this problem?</h3>
This problem offers an opportunity to explore functions of
functions. Although the context is powers and logs, the same
structure can be used to explore other functions. By considering
the questions in the task, students will gain a clearer
understanding of the functions in question.<br></br>
<h3>Possible approach</h3>
<div>If students all have access to computers, they could explore
the interactivity on their own or in pairs to make sense of how the
upper layers of the pyramid are generated. Alternatively,
display the interactivity to the whole class, and ask for
suggestions of numbers to enter in the bottom layer, and give them
time to discuss in pairs what they think is going on.</div>
<div>Once they have some ideas, discuss as a class what they have
noticed.</div>
<div> </div>
<div>If they have not yet met logarithms, they might express the
relationship as something along the lines of:</div>
<div>"It's the power of two that the product of the numbers on the
layer below is", and this is a good opportunity to introduce the
notation of logarithms, and perhaps to start deducing some of the
laws of logarithms.</div>
<div> </div>
<div>Once the class have established how the pyramid works, set
them the four challenges from the problem, to work on away from the
computers, so they have to work out the answers rather than relying
on trial and error. </div>
<div> </div>
<div>Finally, bring the group together and share the methods they
used for answering each challenge, before checking their answers
using the interactivity.</div>
<h3>Key questions</h3>
<div>Can you find numbers for the bottom layer that give you
whole numbers on the second layer?</div>
<div>What is special about these numbers?</div>
<div>What happens if you change <span style="font-weight: bold;">just one</span> of the numbers on the
bottom layer?</div>
<br></br>
<h3>Possible extension</h3>
This <a href="/content/id/7531/Function%20Pyramid.xls">spreadsheet</a> offers
the same activity as the interativity in the problem, but could be
adapted to create other function pyramids to investigate.<br></br>
<h3>Possible support</h3>
Suggest students start by putting 4s in every cell on the bottom
layer.<br></br>
<br></br></mdoxml>
<mdoxml version="1.0">Can you find numbers for the bottom layer that give you whole
numbers on the second layer? What is special about these numbers?
<br></br>
What happens if you change just one of the numbers on the bottom
layer? <br></br></mdoxml>
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Function Pyramids
A function pyramid is a structure where each entry in the pyramid
is determined by the two entries below it. Can you figure out how
the pyramid is generated?
Sequences, Functions and Graphs
Functions and their inverses
Sequences, Functions and Graphs
Laws of logarithms
Numbers and the Number System
Powers & roots
Using, Applying and Reasoning about Mathematics
Working systematically