48
/www/nrich/html/content/98/08/bbprob1/
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2011-02-01T00:00:01
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Imagine that you're walking along the beach, a rather nice sandy beach with just a few small pebbles in little groups here and there. You start off by collecting just four pebbles and you place them on the sand in the form of a square. The area inside is of course just $1$ square something, maybe $1$ square metre, $1$ square foot, $1$ square finger ... whatever.
<p style="text-align: center;"><mdo:image alt="1" height="97" src="1.jpg" width="97"></mdo:image><mdo:image alt="sq1" height="95" src="Sq1.jpg" width="98"></mdo:image></p>
<p>By adding another $2$ pebbles in line you double the area to $2$, like this:</p>
<p style="text-align: center;"><mdo:image alt="2" height="96" src="2.jpg" width="158"></mdo:image><mdo:image alt="sq2" height="96" src="Sq2.jpg" width="169"></mdo:image></p>
<p>The rule that's developing is that you keep the pebbles that are down already (not moving them to any new positions) and add as FEW pebbles as necessary to DOUBLE the PREVIOUS area, using RECTANGLES ONLY!</p>
<p>So, to continue, we add another three pebbles to get an area of $4$:</p>
<p style="text-align: center;"><mdo:image alt="3" height="156" src="3.jpg" width="157"></mdo:image><mdo:image alt="sq3" height="168" src="Sq3a.jpg" width="169"></mdo:image></p>
<p>You could have doubled the area by doing:</p>
<p style="text-align: center;"><mdo:image alt="4" height="95" src="4.jpg" width="274"></mdo:image><mdo:image alt="sq4" height="96" src="sq4a.jpg" width="311"></mdo:image></p>
<p>But this would not obey the rule that you must add as FEW pebbles as possible each time. So this one is not allowed.</p>
<p>Number 6 would look like this:</p>
<p style="text-align: center;"><mdo:image alt="5b" height="322" src="bigger%205%20b.jpg" width="602"></mdo:image></p>
<p style="text-align: center;"><mdo:image alt="sq5" height="311" src="Sq5a.jpg" width="599"></mdo:image></p>
<p>So remember:-</p>
<h4>The rule is that you keep the pebbles that are down already (not moving them to any new positions) and add as FEW pebbles as necessary to DOUBLE the PREVIOUS area.</h4>
<p>Well, now it's time for you to have a go.</p>
<p>"It's easy,'' I hear you say. Well, that's good. But what questions can we ask about the arrangements that we are getting?</p>
<p>We could make a start by saying "Stand back and look at the shapes you are getting. What do you see?'' I guess you may see quite a lot of different things.</p>
<p>It would be good for you to do some more of this pattern. See how far you can go. You may run out of pebbles, paper or whatever you may be using. (Multilink, pegboard, elastic bands with a nail board, <em><span style="font-style: normal;">etc.)</span></em></p>
<p>Well now, what about some questions to explore?<br></br>
Here are some I've thought of that look interesting:</p>
<ol type="a">
<li>How many extra pebbles are added each time? This starts off $2$, $3$, $6$ ...</li>
<li>How many are there around the edges? This starts off $4$, $6$, $8$ ...</li>
<li>How big is the area? This starts off $1$, $2$, $4$ ...</li>
<li>How many are there inside? This starts off $0$, $0$, $1$, $3$, $9$ ...</li>
</ol>
<p>Try to answer these, and any other questions you come up with, and perhaps put them in a kind of table/graph/spreadsheet <em style="font-style: normal;">etc.</em></p>
<p>Do let me see what you get - I'll be most interested.</p>
<p>Don't forget the all-important question to ask - "I wonder what would happen if I ...?''</p>
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<p class="editorial">A group of children from Manorfield Primary
School, Stoney Stanton sent in lots of ideas:</p>
<br></br>
<p><span class="editorial">T.K and A.H said</span>:</p>
The pattern: As you double it each time, the shape changes from a
square to a rectangle repetitively as each time you add a square to
make a rectangle (square $+$ square = rectangle) and add a
rectangle to make a square (rectangle $+$ rectangle =
square).<br></br>
<br></br>
<p><span class="editorial">A.H and E.R said:</span></p>
We have noticed that each shape changes from a square to a
rectangle because if you add a square to a square you get a
rectangle e.g.:<br></br>
$4\times4 + 4\times4 = 8\times4$ - a rectangle<br></br>
<br></br>
<p><span class="editorial">S.B. and N.L. produced the following
table of results:</span></p>
<br></br>
<table border="1">
<tbody>
<tr>
<td style="text-align: center;">Number</td>
<td style="text-align: center;">Shape</td>
<td style="text-align: center;">Number of pebbles on side</td>
<td style="text-align: center;">Area of shape</td>
<td style="text-align: center;">Perimeter of shape</td>
</tr>
<tr>
<td style="text-align: center;">1</td>
<td style="text-align: center;">Square</td>
<td style="text-align: center;">2x2</td>
<td style="text-align: center;">1cm$^2$</td>
<td style="text-align: center;">4cm</td>
</tr>
<tr>
<td style="text-align: center;">2</td>
<td style="text-align: center;">Rectangle</td>
<td style="text-align: center;">2x3</td>
<td style="text-align: center;">2cm$^2$</td>
<td style="text-align: center;">6cm</td>
</tr>
<tr>
<td style="text-align: center;">3</td>
<td style="text-align: center;">Square</td>
<td style="text-align: center;">3x3</td>
<td style="text-align: center;">4cm$^2$</td>
<td style="text-align: center;">8cm</td>
</tr>
<tr>
<td style="text-align: center;">4</td>
<td style="text-align: center;">Rectangle</td>
<td style="text-align: center;">3x5</td>
<td style="text-align: center;">8cm$^2$</td>
<td style="text-align: center;">12cm</td>
</tr>
<tr>
<td style="text-align: center;">5</td>
<td style="text-align: center;">Square</td>
<td style="text-align: center;">5x5</td>
<td style="text-align: center;">16cm$^2$</td>
<td style="text-align: center;">16cm</td>
</tr>
<tr>
<td style="text-align: center;">6</td>
<td style="text-align: center;">Rectangle</td>
<td style="text-align: center;">5x9</td>
<td style="text-align: center;">32cm$^2$</td>
<td style="text-align: center;">24cm</td>
</tr>
<tr>
<td style="text-align: center;">7</td>
<td style="text-align: center;">Square</td>
<td style="text-align: center;">9x9</td>
<td style="text-align: center;">64cm$^2$</td>
<td style="text-align: center;">32cm</td>
</tr>
<tr>
<td style="text-align: center;">8</td>
<td style="text-align: center;">Rectangle</td>
<td style="text-align: center;">9x17</td>
<td style="text-align: center;">128cm$^2$</td>
<td style="text-align: center;">48cm</td>
</tr>
</tbody>
</table>
<br></br>
<div>PATTERNS AND FORMULAE</div>
<div>It was said in whole class discussion that the pattern for the
size of the shapes was:</div>
<div>Squares: each side is the same length as the longest side of
previous rectangle</div>
<div>Rectangle: one side is the same length as the side of the
previous square, the other side is a "new length"</div>
<br></br>
<p class="editorial">A.H and E.R also said:</p>
<div>The formula for the area = $2$ to the power of $(n-1)$.</div>
<div>The pattern for the new side of the rectangles is $+1$, $+2$,
$+4$, $+8$, $+16$ ... (it doubles)</div>
<div>The pattern for the perimeter is $+2+2$, $+4+4$, $+8+8$,
$+16+16$ ... (it doubles)</div>
<br></br>
<div>PREDICTIONS</div>
<div>S.W. and L.B filled in their table for the first six shapes,
and then predicted:</div>
<div>For shape number seven, the area will be double the area of
shape six and the perimeter will be $8$ more than the perimeter of
shape six. And for shape eight, the area will be double the area of
shape seven and the perimeter will be $16$ more than shape
seven.</div>
<br></br>
<p class="editorial">Well done all of you - you obviously worked
hard on this problem. </p>
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<mdoxml version="1.0"><div class="embed">
<h2>Pebbles</h2>
<br></br>
Imagine that you're walking along the beach, a rather nice sandy beach with just a few small pebbles in little groups here and there. You start off by collecting just four pebbles and you place them on the sand in the form of a square. The area inside is of course just $1$ square something, maybe $1$ square metre, $1$ square foot, $1$ square finger ... whatever.
<p style="text-align: center;"><mdo:image alt="1" height="97" src="1.jpg" width="97"></mdo:image><mdo:image alt="sq1" height="95" src="Sq1.jpg" width="98"></mdo:image></p>
<p>By adding another $2$ pebbles in line you double the area to $2$, like this:</p>
<p style="text-align: center;"><mdo:image alt="2" height="96" src="2.jpg" width="158"></mdo:image><mdo:image alt="sq2" height="96" src="Sq2.jpg" width="169"></mdo:image></p>
<p>The rule that's developing is that you keep the pebbles that are down already (not moving them to any new positions) and add as FEW pebbles as necessary to DOUBLE the PREVIOUS area, using RECTANGLES ONLY!</p>
<p>So, to continue, we add another three pebbles to get an area of $4$:</p>
<p style="text-align: center;"><mdo:image alt="3" height="156" src="3.jpg" width="157"></mdo:image><mdo:image alt="sq3" height="168" src="Sq3a.jpg" width="169"></mdo:image></p>
<p>You could have doubled the area by doing:</p>
<p style="text-align: center;"><mdo:image alt="4" height="95" src="4.jpg" width="274"></mdo:image><mdo:image alt="sq4" height="96" src="sq4a.jpg" width="311"></mdo:image></p>
<p>But this would not obey the rule that you must add as FEW pebbles as possible each time. So this one is not allowed.</p>
<p>Number 6 would look like this:</p>
<p style="text-align: center;"><mdo:image alt="5b" height="322" src="bigger%205%20b.jpg" width="602"></mdo:image></p>
<p style="text-align: center;"><mdo:image alt="sq5" height="311" src="Sq5a.jpg" width="599"></mdo:image></p>
<p>So remember:-</p>
<h4>The rule is that you keep the pebbles that are down already (not moving them to any new positions) and add as FEW pebbles as necessary to DOUBLE the PREVIOUS area.</h4>
<p>Well, now it's time for you to have a go.</p>
<p>"It's easy,'' I hear you say. Well, that's good. But what questions can we ask about the arrangements that we are getting?</p>
<p>We could make a start by saying "Stand back and look at the shapes you are getting. What do you see?'' I guess you may see quite a lot of different things.</p>
<p>It would be good for you to do some more of this pattern. See how far you can go. You may run out of pebbles, paper or whatever you may be using. (Multilink, pegboard, elastic bands with a nail board, <em><span style="font-style: normal;">etc.)</span></em></p>
<p>Well now, what about some questions to explore?<br></br>
Here are some I've thought of that look interesting:</p>
<ol type="a">
<li>How many extra pebbles are added each time? This starts off $2$, $3$, $6$ ...</li>
<li>How many are there around the edges? This starts off $4$, $6$, $8$ ...</li>
<li>How big is the area? This starts off $1$, $2$, $4$ ...</li>
<li>How many are there inside? This starts off $0$, $0$, $1$, $3$, $9$ ...</li>
</ol>
<p>Try to answer these, and any other questions you come up with, and perhaps put them in a kind of table/graph/spreadsheet <em style="font-style: normal;">etc.</em></p>
<p>Do let me see what you get - I'll be most interested.</p>
<p>Don't forget the all-important question to ask - "I wonder what would happen if I ...?''</p>
</div>
<br></br>
<h3>Why do this problem?</h3>
<div>Use <a href="http://nrich.maths.org/public/viewer.php?obj_id=48&part=">this activity</a> to introduce the youngsters to an investigation that mixes both shape and space work with number work. You could also introduce learners to this extended piece of work to help you look at perseverence and persistence.</div>
<br></br>
<h3>Possible approach</h3>
<div>A good introduction can be had with the whole class by making the first two or three arrangements altogether. It is useful to have squared and dotted (squares) paper available whilst some pupils may benefit from using blocks (such as multilink) to represent the pebbles. You may also find it helpful to use the <a href="http://nrich.maths.org/public/viewer.php?obj_id=2883&part=index">Virtual Geoboard</a> for sharing ideas amongst the whole group.</div>
<br></br>
<div>After children have worked in pairs for a time, investigating subsequent arrangements, you can pose some of the suggested questions (for example looking at the number of pebbles added each time) and invite them to ask and explore their own questions. Encourage record keeping in whatever form the pupils feel is appropriate.</div>
<br></br>
<h3>Key questions</h3>
<div>What are you counting? <span style="font-style: italic;">(Sometimes there is confusion about the counting of the pebbles and the counting of the spaces in between them - particularly along the lengths of sides.)</span></div>
<div>Is this rectangle double the size of the last one?</div>
<div>How are you recording what you have done?</div>
<br></br>
<h3>Possible extension</h3>
<div>Some pupils may produce a table or a spreadsheet of their results which would enable them to explore further.</div>
<br></br>
<div>Here is an example of many results that lead to the consideration of the digital roots (d.r.):</div>
<br></br>
<div style="text-align: center;"><mdo:image alt="latesTable" height="323" src="latest%20pebbleTable.jpg" width="604"></mdo:image></div>
<br></br>
<h3 style="text-align: left;">For the exceptionally mathematically able</h3>
<div style="text-align: left;">Go to More Pebbles <a href="http://nrich.maths.org/7083&part=">here</a>.</div>
<br></br>
<br></br>
<br></br>
<h3>Possible support</h3>
<div>Children may benefit from adult support in keeping track of where they are in their exploration. They could be helped to proceed as if it were a game.</div>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0">You could use cubes or counters for pebbles, with
squared paper.<br></br>
Can you make a rectangle with just one more pebble? Two
more? Three more ...?<br></br>
Have you checked that each rectangle is double the size of the last
one?<br></br>
How are you recording what you have done?<br></br></mdoxml>
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<mdoxml version="1.0"><br></br>
<p class="editorial">There are lots of solutions to this problem,
depending on the questions you choose to ask.</p>
<br></br>
<p><span class="editorial">Have a go yourself, and if you discover
anything interesting then</span> <a href="mailto:nrich@damtp.cam.ac.uk" class="editorial">let us know</a>
<span class="editorial">!Please don't worry that your solution is
not "complete" - we'd like to hear about anything you have
tried.</span></p>
<br></br>
<p><span class="editorial">I present this table of results and the
visual one just as an example of the kinds of solutions you may
get. Obviously the numbers can then be explored much
further.</span></p>
<div style="text-align: center;"><mdo:image width="482" height="438" alt="PebSpread2" src="pebSpread2.jpg"></mdo:image></div>
<p class="editorial">or, if you prefer;</p>
<p style="text-align: center;"><mdo:image width="596" height="313" src="pebPic.jpg" alt="pebPic"></mdo:image></p>
<p style="text-align: center;"> </p>
<br></br>
<br></br>
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Pebbles
Place four pebbles on the sand in the form of a square. Keep adding
as few pebbles as necessary to double the area. How many extra
pebbles are added each time?
Sequences, Functions and Graphs
Sequences
2D Geometry, Shape and Space
Rectangles
Numbers and the Number System
Factors and multiples
Measures and Mensuration
Area
Using, Applying and Reasoning about Mathematics
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Secondary Mapping Document
Patterns and sequences LS
Secondary Mapping Document
Area and volume LS
Secondary Mapping Document
Factors, multiples and primes