46
/www/nrich/html/content/98/07/bbprob1/
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2011-02-01T00:00:01
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<p>Let's suppose that you are working in the school hall to help
with setting up the stage for a play or an assembly. Perhaps you've
got a stage that is raised up and there are some steps that lead up
to the stage. From the side the steps might look a bit like
this:-</p>
<p style="text-align: center;"><mdo:image height="124" width="174" src="Picture%201.jpg" alt="pic1"></mdo:image></p>
<p>These steps have been made up using blocks so that you go up a
total of 6 and along a total of 6. The steps could have been made
up another way:-</p>
<p style="text-align: center;"><mdo:image height="128" width="178" alt="pic2b" src="Pic2b.jpg"></mdo:image></p>
<p>In this case the steps are not equal and some of them need you
to make a big step up. Well, that does not matter.</p>
<p>Have a go at drawing some other steps that would take you up 6
and along 6.</p>
<h3>My challenge to you today is to design different step
arrangements, with the following rules:-</h3>
<p>You need to go along a distance of 6 on the steps, so this one
breaks the rule:-</p>
<p style="text-align: center;"><mdo:image height="123" width="133" src="Picture%203.jpg" alt="pic3"></mdo:image></p>
You need to end up at 6 high. So this next one breaks the rule and
is not allowed:-<br></br>
<div style="text-align: center;"><mdo:image height="99" width="172" src="Picture%204.jpg" alt="pic4"></mdo:image></div>
<br></br>
You cannot go DOWN once you have gone up. So this next one is not
allowed either:-<br></br>
<div style="text-align: center;"><mdo:image height="124" width="171" alt="pic5" src="Picture%205.jpg"></mdo:image></div>
<p>But, you can go along more than one block without going UP like
this one:-</p>
<p style="text-align: center;"><mdo:image height="122" width="173" src="Picture%206.jpg" alt="pic6"></mdo:image></p>
<p>As a little extra, you may have noticed that you can count how
many square blocks have been used for each design.</p>
<p>The very first example above used 21 and the last one used 33
square blocks.</p>
<p>Here are some questions you might like to think about to do with
the number of square blocks used:-</p>
<ol type="a">
<li>What is the largest number of square blocks that can be
used?</li>
<li>What is the smallest number of square blocks that can be
used?</li>
<li>Can you get to use all the numbers that go between your answer
to a. and your answer to b. ?</li>
</ol>
<span style="font-weight: bold;">Then</span> :-
<p style="font-weight: bold;">I wonder what would happen if it were
4 along and 4 up?</p>
<p style="font-weight: bold;">I wonder what would happen if it were
5 along and 5 up?</p>
<p style="font-weight: bold;">I wonder what would happen if it were
5 along and 6 up?</p>
<p style="font-weight: bold;">I wonder what would happen if it were
6 along and 5 up?</p>
<p style="font-weight: bold;">I wonder what would happen if
...?</p>
<p>Please send in your solutions and comments.</p>
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<p class="editorial">Here is a solution to STAIRS, which we very
happily received from Hannah of West Flegg Middle School, Norfolk.
She writes:</p>
<div>Numbers 1, 5, 8, 11, 12, 13, 14, 15, 16, 17 start with one up
then two across. Only number 10 starts with 2 up at the beginning.
Number 18 and Number 2 have one up and then 3 across.</div>
<table cellspacing="20" border="0">
<tbody>
<tr>
<td>1. <mdo:image align="middle" src="1.aff.gif" alt=""></mdo:image></td>
<td>2. <mdo:image align="middle" src="2.aff.gif" alt=""></mdo:image></td>
<td>3. <mdo:image align="middle" src="3.aff.gif" alt=""></mdo:image></td>
<td>4. <mdo:image align="middle" src="4.aff.gif" alt=""></mdo:image></td>
</tr>
<tr>
<td>5. <mdo:image align="middle" src="5.aff.gif" alt=""></mdo:image></td>
<td>6. <mdo:image align="middle" src="6.aff.gif" alt=""></mdo:image></td>
<td>7. <mdo:image align="middle" src="7.aff.gif" alt=""></mdo:image></td>
<td>8. <mdo:image align="middle" src="8.aff.gif" alt=""></mdo:image></td>
</tr>
<tr>
<td>9. <mdo:image align="middle" src="9.aff.gif" alt=""></mdo:image></td>
<td>10. <mdo:image align="middle" src="10.aff.gif" alt=""></mdo:image></td>
<td>11. <mdo:image align="middle" src="11.aff.gif" alt=""></mdo:image></td>
<td>12. <mdo:image align="middle" src="12.aff.gif" alt=""></mdo:image></td>
</tr>
<tr>
<td>13. <mdo:image align="middle" src="13.aff.gif" alt=""></mdo:image></td>
<td>14. <mdo:image align="middle" src="14.aff.gif" alt=""></mdo:image></td>
<td>15. <mdo:image align="middle" src="15.aff.gif" alt=""></mdo:image></td>
<td>16. <mdo:image align="middle" src="16.aff.gif" alt=""></mdo:image></td>
</tr>
<tr>
<td>17. <mdo:image align="middle" src="17.aff.gif" alt=""></mdo:image></td>
<td>18. <mdo:image align="middle" src="18.aff.gif" alt=""></mdo:image></td>
<td>19. <mdo:image align="middle" src="19.aff.gif" alt=""></mdo:image></td>
<td>20. <mdo:image align="middle" src="20.aff.gif" alt=""></mdo:image></td>
</tr>
</tbody>
</table>
<p>Biggest and smallest:-</p>
<p>This is my biggest</p>
<p><mdo:image src="big.aff.gif" alt=""></mdo:image></p>
<p>This is my smallest</p>
<p><mdo:image src="small.aff.gif" alt=""></mdo:image></p>
<p>Then:-</p>
<p><mdo:image src="a12.aff.gif" alt=""></mdo:image></p>
<p><mdo:image src="a13.aff.gif" alt=""></mdo:image></p>
<p>14 on the first sheet - No 19.<br></br>
15 on the sheet - Nos. 14, 2, 3, 6.<br></br>
16 on the sheet - No. 9.<br></br>
17 on the sheet - Nos. 8, 13, 18.<br></br>
18 on the sheet - Nos. 4, 11.<br></br>
19 on the sheet - Nos. 1, 12, 13, 16.</p>
<p><mdo:image src="a20.aff.gif" alt=""></mdo:image></p>
<p><mdo:image src="a21.aff.gif" alt=""></mdo:image></p>
<p>22 on the sheet - No. 10.<br></br>
23 on the sheet - No. 17.</p>
<p><mdo:image src="a24.aff.gif" alt=""></mdo:image></p>
<p><mdo:image src="a25.aff.gif" alt=""></mdo:image></p>
<p>And so on... you can just add one to the last square, up the top
if you like. I added them up at the top of the last three.</p>
<p class="editorial">She went on to extend the challenge! WELL
DONE</p>
<p>I wonder what would happen if it were 5 along and four up?</p>
<p><mdo:image src="a.aff.gif" alt=""></mdo:image></p>
<p>The amount would be smaller all the time but the method would be
the same.</p>
<p class="editorial">She also looked at 5 by 6 and 6 by 5. She
finishes by saying:</p>
<p>I wonder what would happen if you could have two sets of
steps?</p>
<p><em>e.g.,</em></p>
<p><mdo:image src="eg.aff.gif" alt=""></mdo:image></p>
<p class="editorial">Bernard says "Well done, a good approach. I
liked the way you had a go, then sorted them into an order, put in
the missing ones, and saw how it was just going to go on. Then you
asked a further question, and explored it a little. And you
finished with a brand new question. Keep up this good work all of
you!''</p>
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<mdoxml version="1.0"><div class="embed">
<h2>Stairs</h2>
<br></br>
<p>Let's suppose that you are working in the school hall to help with setting up the stage for a play or an assembly. Perhaps you've got a stage that is raised up and there are some steps that lead up to the stage. From the side the steps might look a bit like this:-</p>
<p style="text-align: center;"><mdo:image alt="pic1" height="124" src="Picture%201.jpg" width="174"></mdo:image></p>
<p>These steps have been made up using blocks so that you go up a total of 6 and along a total of 6. The steps could have been made up another way:-</p>
<p style="text-align: center;"><mdo:image alt="pic2b" height="128" src="Pic2b.jpg" width="178"></mdo:image></p>
<p>In this case the steps are not equal and some of them need you to make a big step up. Well, that does not matter.</p>
<p>Have a go at drawing some other steps that would take you up 6 and along 6.</p>
<h3>My challenge to you today is to design different step arrangements, with the following rules:-</h3>
<p>You need to go along a distance of 6 on the steps, so this one breaks the rule:-</p>
<p style="text-align: center;"><mdo:image alt="pic3" height="123" src="Picture%203.jpg" width="133"></mdo:image></p>
You need to end up at 6 high. So this next one breaks the rule and is not allowed:-<br></br>
<div style="text-align: center;"><mdo:image alt="pic4" height="99" src="Picture%204.jpg" width="172"></mdo:image></div>
<br></br>
You cannot go DOWN once you have gone up. So this next one is not allowed either:-<br></br>
<div style="text-align: center;"><mdo:image alt="pic5" height="124" src="Picture%205.jpg" width="171"></mdo:image></div>
<p>But, you can go along more than one block without going UP like this one:-</p>
<p style="text-align: center;"><mdo:image alt="pic6" height="122" src="Picture%206.jpg" width="173"></mdo:image></p>
<p>As a little extra, you may have noticed that you can count how many square blocks have been used for each design.</p>
<p>The very first example above used 21 and the last one used 33 square blocks.</p>
<p>Here are some questions you might like to think about to do with the number of square blocks used:-</p>
<ol type="a">
<li>What is the largest number of square blocks that can be used?</li>
<li>What is the smallest number of square blocks that can be used?</li>
<li>Can you get to use all the numbers that go between your answer to a. and your answer to b. ?</li>
</ol>
<span style="font-weight: bold;">Then</span> :-
<p style="font-weight: bold;">I wonder what would happen if it were 4 along and 4 up?</p>
<p style="font-weight: bold;">I wonder what would happen if it were 5 along and 5 up?</p>
<p style="font-weight: bold;">I wonder what would happen if it were 5 along and 6 up?</p>
<p style="font-weight: bold;">I wonder what would happen if it were 6 along and 5 up?</p>
<p style="font-weight: bold;">I wonder what would happen if ...?</p>
<p>Please send in your solutions and comments.</p>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div>Use this <a href="http://nrich.maths.org/public/viewer.php?obj_id=46&part=">activity</a> to introduce the youngsters to a simple problem solving activity and to allow them to develop very individual approaches to gaining solutions.</div>
<br></br>
<h3>Possible approach</h3>
<div>Try some practical approches with blocks and get a discussion going to reinforce what is and what is not allowed.</div>
<br></br>
<h3>Key questions</h3>
<div>How many across have you gone?</div>
<div>Can you find a way of doing this so that if you had the time you'd know you had found all the ways?</div>
<br></br>
<h3>Possible extension</h3>
<div>Explore variations like 3 across, 4 up etc.</div>
<br></br>
<h3>Possible support</h3>
<div>Using 3D blocks all through the the work will help all children access this challenge.</div>
<br></br>
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Trying out the stairs using blocks, or drawing them on squared
paper, will help.<br></br>
What will the last block have to look like?<br></br>
What might the first block look like?<br></br></mdoxml>
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Stairs
This challenge is to design different step arrangements, which must
go along a distance of 6 on the steps and must end up at 6 high.
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