2283
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2011-02-01T00:00:01
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<p>One block is needed to make an up-and-down staircase, with one
step up and one step down.</p>
<p style="text-align: center;"><mdo:image width="50" height="47" alt="single cube" src="onecube.gif"></mdo:image></p>
<p>$4$ blocks make an up-and-down staircase with $2$ steps up and
$2$ steps down.</p>
<p style="text-align: center;"><mdo:image width="130" height="84" alt="four cubes in 2 step up and down staircase" src="fourcubes.gif"></mdo:image></p>
<p>How many blocks would be needed to build an up-and-down
staircase with $5$ steps up and $5$ steps down?</p>
<p>Explain how you would work out the number of blocks needed to
build a staircase with any number of steps.</p>
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<p class="editorial">Many of you worked on this problem and sent us
solutions.</p>
<p class="editorial">Chloe from Watton Junior School explained how
she solved the first part of the question:</p>
<p>I worked this out practically, by building a model using lego
mega blocks. By doing this I was able to understand that however
many steps up/down you require you must therefore have that amount
of blocks in the middle and then you have one less block in the
columns on either side until the last step, which of course will be
one block.</p>
<p class="editorial">Kaan from Erenkoy Isik School in Turkey also
built staircases himself. He sent us this diagram:</p>
<p align="center"><mdo:image width="528" height="230" align="top" alt="images of staircases with 1 step, 2 steps, 3 steps etc" bgcolor="" src="solkaan.gif"></mdo:image></p>
<p class="editorial">So, Chloe and Kaan agree that for an
up-and-down staircase with 5 steps up and 5 steps down, you need 25
blocks.</p>
<p class="editorial">Kaan goes on to say:</p>
<p>I concluded that, if I multiply the number of steps with itself,
I can find the number of blocks. For example;</p>
<p>If I want to build</p>
<p>6 step up and down staircase, I will need 6 x 6 = 36 blocks</p>
<p>7 step up and down staircase, I will need 7 x 7 = 49 blocks</p>
<p>8 step up and down staircase, I will need 8 x 8 = 64 blocks and
...</p>
<p class="editorial">Rachel from Histon and Impington Infant School
adds a little more about why this works:</p>
<p class="standard">I noticed that if I take one half of the
staircase and put it on top of the other, I have a square shape.
This shape is 5 bricks across and 5 bricks down.</p>
<p>However many steps you have up and down is the same as the
number of bricks across and down in the square. So if the number of
steps is N, then the number of bricks is N x N.</p>
<p class="editorial">This is a fantastic explanation Rachel, thank
you. If you haven't seen it already, there is an animation of this
in the <a href="http://nrich.maths.org/public/viewer.php?obj_id=2283&part=clue">
Hint</a>. <a obj_id="2283" part="clue" href=""></a></p>
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<h2>Up and Down Staircases</h2>
<br></br>
<p>One block is needed to make an up-and-down staircase, with one step up and one step down.</p>
<p style="text-align: center;"><mdo:image alt="single cube" height="47" src="onecube.gif" width="50"></mdo:image></p>
<p>$4$ blocks make an up-and-down staircase with $2$ steps up and $2$ steps down.</p>
<p style="text-align: center;"><mdo:image alt="four cubes in 2 step up and down staircase" height="84" src="fourcubes.gif" width="130"></mdo:image></p>
<p>How many blocks would be needed to build an up-and-down staircase with $5$ steps up and $5$ steps down?</p>
<p>Explain how you would work out the number of blocks needed to build a staircase with any number of steps.</p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/public/viewer.php?obj_id=2283&part=index">This problem</a> could be a good way to introduce children to square numbers. It is an engaging, practical context in which to give them the opportunity to explore patterns and to generalise.</div>
<br></br>
<h3>Possible approach</h3>
<div>You could introduce this problem by building the staircases with one and two steps respectively. Explain how each is named and ask how many blocks are needed to build each. Ask the children to close their eyes and imagine the next staircase, which would have three steps up and three steps down. Can they visualise the number of blocks needed in total? Ask a few learners to explain how they
were picturing the staircase in their head and how they knew the total number of blocks.</div>
<br></br>
<div>Set up the challenge of wanting to know the number of blocks in a five-step staircase and encourage pupils to work in pairs. Some may like to use blocks to physically make the staircase, others may want to draw it, or part of it, others may be happy visualising and calculating. Draw the group back together again and share their different methods.</div>
<br></br>
<div>Suggest that you'd like to know how many blocks would be needed to build a much bigger staircase, for example twenty steps up and twenty down. Invite pupils to suggest how they might answer this question if they didn't have enough cubes. Some may say to draw it, but you could protest that this would take too long! Encourage them to look carefully at the numbers they have found so far,
perhaps by drawing a table on the board:</div>
<br></br>
<table style="" border="1">
<tbody>
<tr>
<td>Number of steps up</td>
<td>Total number of blocks</td>
</tr>
<tr>
<td>1</td>
<td>1</td>
</tr>
<tr>
<td>2</td>
<td>4</td>
</tr>
<tr>
<td>3</td>
<td>9</td>
</tr>
<tr>
<td>5</td>
<td>25</td>
</tr>
</tbody>
</table>
<br></br>
<br></br>
<div>Children do not necessarily need to know about squaring numbers in order to express the relationship, it can be explained in terms of "multiplying a number by itself". You may find they need to create more staircases before being able to generalise fully. Once the relationship is articulated, they will enjoy working out the number of cubes needed for huge staircases! As a final challenge,
ask them if they can see why square numbers are produced. You may like to show the interactivity in the <a href="http://nrich.maths.org/public/viewer.php?obj_id=2283&part=clue">hints</a> , or use cubes to show the same thing.</div>
<br></br>
<h3>Key questions</h3>
<div>Can you predict the number of cubes in the next staircase? How did you know?</div>
<div>Do you notice any patterns in the number of steps compared with the total number of cubes?</div>
<div>How could you record your results for each staircase?</div>
<br></br>
<h3>Possible extension</h3>
<div><a href="/content/id/2283/StaircasesExtension.doc">This document</a> gives details of two possible extension ideas. Children could explore the numbers of cubes in each 'column' of the staircases, or investigate other kinds of staircases.</div>
<br></br>
<h3>Possible support</h3>
<div>You could supply some children with a table ready to be filled in.</div>
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<p>How many blocks are needed for a $3$ step up-and-down
staircase?</p>
<p>Try making a table of the number of blocks needed for each
numbers of steps.</p>
<p>Can you rearrange the blocks for each staircase to make a square
pattern? How might this help?</p>
<p><mdo:flash height="400" width="550"><param value="top" name="align" ></param><param value="" name="bgcolor" ></param><param value="newseriessum1.swf" name="movie" ></param><param value="animation of squares" name="alt" ></param></mdo:flash></p>
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<table border="1" style="border-style: solid; width: 80%;">
<tbody>
<tr>
<td valign="top">
<p>Number of stairs</p>
</td>
<td valign="top">
<p>Number of cubes</p>
</td>
</tr>
<tr>
<td valign="top">
<p>1</p>
</td>
<td valign="top">
<p>1</p>
</td>
</tr>
<tr>
<td valign="top">
<p>2</p>
</td>
<td valign="top">
<p>4</p>
</td>
</tr>
<tr>
<td valign="top">
<p>3</p>
</td>
<td valign="top">
<p>9</p>
</td>
</tr>
</tbody>
</table>
<p>etc</p>
<p>number of cubes = number of stairs squared</p>
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Up and Down Staircases
One block is needed to make an up-and-down staircase, with one step
up and one step down. How many blocks would be needed to build an
up-and-down staircase with 5 steps up and 5 steps down?
Algebra
Introducing algebra
Numbers and the Number System
Square numbers
Using, Applying and Reasoning about Mathematics
Generalising
Mathematics Tools
Wooden/plastic cubes
Admin
Upper primary mapping document