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/www/nrich/html/content/00/01/penta3/
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2011-02-01T00:00:01
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<p>On a digital $24$ hour clock, at certain times, all the digits
are consecutive (in counting order). You can count forwards or
backwards.</p>
<p>For example, <strong style="font-weight: 400;">1:23</strong> or
<strong style="font-weight: 400;">5:43</strong>.</p>
<blockquote>How many times like this are there between midnight and
7:00?<br></br>
How many are there between 7:00 and midday?<br></br>
How many are there between midday and midnight?</blockquote>
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<strong class="editorial" style="font-weight: bold;">Jay</strong>
<span class="editorial">and</span> <strong class="editorial" style="font-weight: bold;">Ben</strong> <span class="editorial">(Mile
Cross Middle School) sent in a correct solution:</span>
<p>Between midnight and 7:00 we found ten</p>
<blockquote>
<pre>
<strong>0:12
1:23
5:43
6:54
2:34
3:45
3:21
4:32
4:56
2:10
</strong>
</pre></blockquote>
<p>There are <strong style="font-weight: 400;">no times between
7:00 and midday</strong> .<br></br>
We found two times between midday and midnight. These are
<strong style="font-weight: 400;">12:34</strong> and <strong style="font-weight: 400;">23:45.</strong></p>
<p><strong class="editorial" style="font-weight: bold;">Syed</strong> <span class="editorial">(Foxford
School and Community College) agrees with this answer and makes a
statement about why you don't get times containing a 7, 8, or 9 in
the solution:</span></p>
<p>The largest the tens digit of the minute number can be is 5, so
the largest unit of the hour number is 6 in order for the time to
have consecutive digits.</p>
<p><strong><span class="editorial" style="font-weight: bold;">George</span> <span class="editorial">(</span></strong><span class="editorial">Rosebank
Primary School, Leeds) also solved this one and explained his
thinking well.</span></p>
<p><strong class="editorial" style="font-weight: bold;">Jason</strong> <span class="editorial">(Priory
Middle School, Dunstable) took a different view of this problem.
Instead of only looking at single digit numbers, he also looked for
consecutive two-digit numbers. This is what he found:</span></p>
<table border="1">
<tbody>
<tr>
<td>For midnight to 7am</td>
<td>For 7am to midday</td>
<td colspan="2">For midday to midnight</td>
</tr>
<tr>
<td>1:23</td>
<td>10:11</td>
<td>12:13</td>
<td>23:22</td>
</tr>
<tr>
<td>2:34</td>
<td>11:12</td>
<td>13:14</td>
<td>22:21</td>
</tr>
<tr>
<td>3:45</td>
<td>11:10</td>
<td>14:15</td>
<td>21:20</td>
</tr>
<tr>
<td>4:56</td>
<td><br></br></td>
<td>15:16</td>
<td>20:19</td>
</tr>
<tr>
<td>2:10</td>
<td><br></br></td>
<td>16:17</td>
<td>19:18</td>
</tr>
<tr>
<td>3:21</td>
<td><br></br></td>
<td>17:18</td>
<td>18:17</td>
</tr>
<tr>
<td>4:32</td>
<td><br></br></td>
<td>18:19</td>
<td>17:16</td>
</tr>
<tr>
<td>5:43</td>
<td><br></br></td>
<td>19:20</td>
<td>16:15</td>
</tr>
<tr>
<td>6:54</td>
<td><br></br></td>
<td>20:21</td>
<td>15:14</td>
</tr>
<tr>
<td><br></br></td>
<td><br></br></td>
<td>21:22</td>
<td>14:13</td>
</tr>
<tr>
<td><br></br></td>
<td><br></br></td>
<td>22:23</td>
<td>13:12</td>
</tr>
<tr>
<td><br></br></td>
<td><br></br></td>
<td>23:24</td>
<td>12:11</td>
</tr>
<tr>
<td><br></br></td>
<td><br></br></td>
<td>0:23</td>
<td><br></br></td>
</tr>
</tbody>
</table>
<br></br>
<p class="editorial">Some interesting patterns here Jason!</p>
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<h2>How Many Times?</h2>
<br></br>
<p>On a digital $24$ hour clock, at certain times, all the digits are consecutive (in counting order). You can count forwards or backwards.</p>
<p>For example, <strong style="font-weight: 400;">1:23</strong> or <strong style="font-weight: 400;">5:43</strong>.</p>
<blockquote>How many times like this are there between midnight and 7:00?<br></br>
How many are there between 7:00 and midday?<br></br>
How many are there between midday and midnight?</blockquote>
</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=981&part=index">This problem</a> will help consolidate children's understanding of the $24$ hour clock notation. It could also be used to focus on ways of working systematically.</div>
<br></br>
<h3>Possible approach</h3>
<div>It would be good to have an interactive digital clock on the whiteboard for the duration of this lesson so that you and the class can refer to it whenever necessary. (<a href="http://www.oswego.org/ocsd-web/games/ClassClock/clockres.html">This free version</a> would be suitable, for example. Click on the arrows button to switch to a digital display.) You may want to begin by asking a few
oral questions based on the clock before moving on to the problem as it stands.</div>
<br></br>
<div>Explain the challenge to the class and ask children to suggest a few examples so that it is clear what is meant by consecutive. You may need to clarify that <span style="font-style: italic;">all</span> the digits in the time need to be consecutive so, for example, 13:45 wouldn't count, as it only has three consecutive digits. Invite pairs of children to begin working on the first part of the
problem. They could use mini-whiteboards to keep a record of the times they find.</div>
<br></br>
<div>After a short time, draw the group together to share ways of working. Some children may be recording answers as they occur to them, others may have some sort of system - for example starting with the earliest time and working 'upwards'. Discuss the benefit of a systematic approach - it means that we know when we have found all the solutions. Having talked about this, children will be able to
apply a system to the other parts of the question.</div>
<br></br>
<div>In the plenary, as well as sharing solutions, encourage children to articulate reasons for their findings.</div>
<br></br>
<h3>Key questions</h3>
<div>Which digits will be possible? Why?</div>
<div>How will you know you've got all the different times?</div>
<br></br>
<h3>Possible extension</h3>
<div>Children could also investigate the times which have just three consecutive digits. <a href="http://nrich.maths.org/public/viewer.php?obj_id=1981&part=index">5 on the Clock</a> is a problem that requires a similar systematic approach and also involves digital time.</div>
<br></br>
<h3>Possible support</h3>
<div>It might be useful for some children to have access to an interactive version of a digital clock themselves, perhaps at a shared computer.</div>
<br></br>
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<mdoxml version="1.0">Which digits will be possible? Why? <br></br>
How will you know you've got all the different times?<br></br></mdoxml>
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1
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How many Times?
On a digital 24 hour clock, at certain times, all the digits are
consecutive. How many times like this are there between midnight
and 7 a.m.?
Using, Applying and Reasoning about Mathematics
Working systematically
Measures and Mensuration
Time
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