7366
/www/nrich/html/content/id/7366/
1
2011-02-01T00:00:01
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</ul>
<div> </div>
<br></br>
<p>The heptathlon is an athletics competition consisting of 7 events:</p>
<ul>
<li>200m sprint</li>
<li>800m run</li>
<li>100m hurdles</li>
<li>high jump</li>
<li>long jump</li>
<li>shot put</li>
<li>javelin</li>
</ul>
<div>The scoring system uses two types of equation:</div>
<div> </div>
<div style="text-align: center;">$y=a(b-x)^c$ (1)</div>
<div style="text-align: center;">$y=a(x-b)^c$ (2)</div>
<div> </div>
<div>$a$, $b$ and $c$ are constants, $x$ is the competitor's time or distance and $y$ is the number of points they are awarded.</div>
<div> </div>
<div>Which events do you think use equations of type (1)? Why?</div>
<div>Which events do you think use equations of type (2)? Why?</div>
<div> </div>
<div>For the running events, $x$ is the time in seconds. For the jumping events, $x$ is the distance/height in centimetres. For the throwing events, $x$ is the distance in metres.</div>
<br></br>
<p>The values for a, b, and c in each event are given below:<br></br>
<br></br>
</p>
<table border="1" cellpadding="0" cellspacing="0">
<colgroup>
<col width="132"></col>
<col span="3" width="64"></col></colgroup>
<tbody>
<tr>
<td width="132">Event </td>
<td style="text-align: center;" width="64">a </td>
<td style="text-align: center;" width="64">b </td>
<td style="text-align: center;" width="64">c</td>
</tr>
<tr>
<td>200 meters </td>
<td align="right">4.99087</td>
<td align="right">42.5</td>
<td align="right">1.81</td>
</tr>
<tr>
<td>800 meters </td>
<td align="right">0.11193</td>
<td align="right">254</td>
<td align="right">1.88</td>
</tr>
<tr>
<td>100 metres hurdles </td>
<td align="right">9.23076</td>
<td align="right">26.7</td>
<td align="right">1.835</td>
</tr>
<tr>
<td>High Jump </td>
<td align="right">1.84523</td>
<td align="right">75</td>
<td align="right">1.348</td>
</tr>
<tr>
<td>Long Jump </td>
<td align="right">0.188807</td>
<td align="right">210</td>
<td align="right">1.41</td>
</tr>
<tr>
<td>Shot Put </td>
<td align="right">56.0211</td>
<td align="right">1.5</td>
<td align="right">1.05</td>
</tr>
<tr>
<td>Javelin Throw </td>
<td align="right">15.9803</td>
<td align="right">3.8</td>
<td align="right">1.04</td>
</tr>
</tbody>
</table>
<br></br>
<p>In the table below are the best times and distances of an Olympic hopeful in training, as well as the World Records for each heptathlon event (as of April 2011).<br></br>
<br></br>
</p>
<table border="1">
<tbody>
<tr>
<td>Event </td>
<td>Olympic hopeful</td>
<td>World records</td>
</tr>
<tr>
<td>200m</td>
<td>25.34s </td>
<td>21.34s</td>
</tr>
<tr>
<td>800m</td>
<td>2min 13.00s</td>
<td>1min 53.28s</td>
</tr>
<tr>
<td>100m hurdles</td>
<td>13.65s</td>
<td>12.21s</td>
</tr>
<tr>
<td>High jump</td>
<td>1.43m</td>
<td>2.09m</td>
</tr>
<tr>
<td>Long jump</td>
<td>5.67m</td>
<td>7.52m</td>
</tr>
<tr>
<td>Shot put</td>
<td>12.45m</td>
<td>22.63m</td>
</tr>
<tr>
<td>Javelin</td>
<td>45.05m </td>
<td>72.28m</td>
</tr>
</tbody>
</table>
<div> </div>
<div>In order to work out a suitable training schedule for her, work out her score in each event.</div>
<div> </div>
<div>The world record for each event can be taken as a theoretical maximum/minimum.</div>
<div>Suppose she could close the gap between her current performance in each event and the world record by 10%. How would that affect her progress towards her target heptathlon score of 6000 points?</div>
<div> </div>
<div>Instead, she could put together an alternative training schedule aiming to close the gap by 20% in some events. However, this extra training would have to be at the expense of her training for other events (so for every event she chooses to improve by 20%, she must choose another where she forfeits the 10% gain and instead maintains her current level).</div>
<div> </div>
<div>Could this training strategy lead to a better score?</div>
<div>Can she reach the target of 6000 points?</div>
<br></br>
<p> <br></br>
</p>
<div class="framework"><br></br>
NOTES AND BACKGROUND<br></br>
<br></br>
Data source: <a href="http://en.wikipedia.org/wiki/Heptathlon">http://en.wikipedia.org/wiki/Heptathlon</a></div>
<br></br></mdoxml>
<mdoxml version="1.0"><br></br>
<span class="editorial">Joseph from Park View Community School
sent us the following solution:</span><br></br>
<br></br>
Find the percentage away from the world record for each event.
Increase the 3 lowest events by 20% and accordingly maintain
current levels for the 3 greatest events. increase performance by
10% on the remaining event. I calculated all points including all
percentage increases as well using excel.<br></br>
<br></br>
<span class="editorial">You can download Joseph's spreadsheet to
see how he calculated this solution</span> <a class="editorial" href="/content/id/7366/Joseph%20solution.xls">here</a><span class="editorial">.</span> <br></br></mdoxml>
<mdoxml version="1.0"><br></br>
<h3>Why do this problem?</h3>
<div>When we watch sports coverage of the multi-discipline events
such as the heptathlon, the scoring mechanism is not usually made
explicit. This problem explores some of the maths behind the
scoring system and invites students to optimise an athlete's
performance by choosing a suitable training schedule. Along the
way, students can practise substituting into formulas, make sense
of functions, and use spreadsheets to repeat routine calculations
quickly.</div>
<br></br>
<h3>Possible approach</h3>
<div>Set the scene by introducing the seven heptathlon events
(perhaps asking students if they can name the events). Then display
the two equations:</div>
<div> </div>
<div style="text-align: center;">
$y=a(b-x)^c$ (1)</div>
<div style="text-align: center;">$y=a(x-b)^c$
(2) </div>
<div> </div>
<div>"These are the equations used to calculate the points scored
in the different heptathlon events. Why do you think there are two
equations? Which equation do you think is used for each
event?"</div>
<div> </div>
<div>Give students time to discuss their answers in pairs, and then
share as a class.</div>
<div> </div>
<div>Now display the rest of the problem and/or hand out <a href="/content/id/7366/Training%20Schedule.pdf">this worksheet</a>.
Before they get started, take some time to talk through the
problem, and check students understand the task.</div>
<div> </div>
<div>As there is quite a bit of repeated computation involved, it
may be useful to use spreadsheets if computers are available.
Alternatively, students could work in small groups and share out
the calculations among themselves.</div>
<div> </div>
<div>Finally, gather the class together to share the different
training schedules they have devised. </div>
<h3>Key questions</h3>
<div>What happens to the number of points given by each equation as
$x$ increases/decreases?</div>
<h3>Possible extension</h3>
<div>Imagine the heptathlon was to become an octathlon. Choose an
eighth event and design and justify a scoring formula which would
allocate points consistent with the other events.</div>
<br></br>
<h3>Possible support</h3>
Students may opt to use a calculator to solve this problem, but it
is much more efficient to use a spreadsheet. In order to make the
most of the task, it may be worthwhile spending some time with the
whole class talking about how to set up formulas in a spreadsheet
and how to make changes to investigate different training
schedules.<br></br></mdoxml>
<mdoxml version="1.0"><br></br>
Why not use a spreadsheet to calculate quickly the score for
each event given the athlete's times or distances?<br></br>
Then try tweaking her times and distances to see how her scores can
be affected by different training schedules.<br></br></mdoxml>
<mdoxml version="1.0"><br></br>
<div class="provisional">
<div>The scoring system uses two types of equation:</div>
<div> </div>
<div style="text-align: center;">$y=a(b-x)^c$
(1)</div>
<div style="text-align: center;"> </div>
<div style="text-align: center;">200m sprint </div>
<div style="text-align: center;">800m run </div>
<div style="text-align: center;">100m hurdles</div>
<div style="text-align: center;"> </div>
<div style="text-align: center;"> </div>
<div style="text-align: center;"> </div>
<div style="text-align: center;"> </div>
<div style="text-align: center;">$y=a(x-b)^c$
(2)</div>
<div style="text-align: center;"> </div>
<div style="text-align: center;">high jump</div>
<div style="text-align: center;">long jump</div>
<div style="text-align: center;">shot put</div>
<div style="text-align: center;">javelin </div>
<div style="text-align: center;"> </div>
An Olympic hopeful has personal best times and distances as
follows:<br></br>
<br></br>
<table border="1">
<tbody>
<tr>
<td>Event </td>
<td>Time/Score</td>
<td>Score (to hide)</td>
</tr>
<tr>
<td>200m</td>
<td>25.34m </td>
<td>(856)</td>
</tr>
<tr>
<td>800m</td>
<td>2m13s</td>
<td>(922)</td>
</tr>
<tr>
<td>100m hurdles</td>
<td>13.65s</td>
<td>(1029)</td>
</tr>
<tr>
<td>High jump</td>
<td>1.43m</td>
<td>(545)</td>
</tr>
<tr>
<td>Long jump</td>
<td>5.67m</td>
<td>(750)</td>
</tr>
<tr>
<td>Shot put</td>
<td>12.45m</td>
<td>(691)</td>
</tr>
<tr>
<td>Javelin</td>
<td>45.05m </td>
<td>(765)</td>
</tr>
</tbody>
</table>
<div> </div>
<div>What would be her score? (5558)</div>
<div> </div>
<div>Increasing everything by 10% increased the total score to
5892.</div>
</div>
<div> </div>
<div> </div>
<div>Increasing High jump, Long jump and Shot by 20%, Javelin by
10% and maintaing the track events at their current level, achieved
a score of 6005.</div>
<div> </div>
<div> </div>
<br></br></mdoxml>
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Training schedule
The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?
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